idx int64 1 56k | question stringlengths 15 155 | answer stringlengths 2 29.2k ⌀ | question_cut stringlengths 15 100 | answer_cut stringlengths 2 200 ⌀ | conversation stringlengths 47 29.3k | conversation_cut stringlengths 47 301 |
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12,401 | The proof of equivalent formulas of ridge regression | It's perhaps worth reading about Lagrangian duality and a broader relation (at times equivalence) between:
optimization subject to hard (i.e. inviolable) constraints
optimization with penalties for violating constraints.
Quick intro to weak duality and strong duality
Assume we have some function $f(x,y)$ of two varia... | The proof of equivalent formulas of ridge regression | It's perhaps worth reading about Lagrangian duality and a broader relation (at times equivalence) between:
optimization subject to hard (i.e. inviolable) constraints
optimization with penalties for v | The proof of equivalent formulas of ridge regression
It's perhaps worth reading about Lagrangian duality and a broader relation (at times equivalence) between:
optimization subject to hard (i.e. inviolable) constraints
optimization with penalties for violating constraints.
Quick intro to weak duality and strong duali... | The proof of equivalent formulas of ridge regression
It's perhaps worth reading about Lagrangian duality and a broader relation (at times equivalence) between:
optimization subject to hard (i.e. inviolable) constraints
optimization with penalties for v |
12,402 | The proof of equivalent formulas of ridge regression | They are not equivalent.
For a constrained minimization problem
$$\min_{\mathbf b} \sum_{i=1}^n (y - \mathbf{x}'_i \cdot \mathbf{b})^2\\
s.t. \sum_{j=1}^p b_j^2 \leq t,\;\;\; \mathbf b = (b_1,...,b_p) \tag{1}$$
we solve by minimize over $\mathbf b$ the corresponding Lagrangean
$$\Lambda = \sum_{i=1}^n (y - \mathbf{... | The proof of equivalent formulas of ridge regression | They are not equivalent.
For a constrained minimization problem
$$\min_{\mathbf b} \sum_{i=1}^n (y - \mathbf{x}'_i \cdot \mathbf{b})^2\\
s.t. \sum_{j=1}^p b_j^2 \leq t,\;\;\; \mathbf b = (b_1,...,b | The proof of equivalent formulas of ridge regression
They are not equivalent.
For a constrained minimization problem
$$\min_{\mathbf b} \sum_{i=1}^n (y - \mathbf{x}'_i \cdot \mathbf{b})^2\\
s.t. \sum_{j=1}^p b_j^2 \leq t,\;\;\; \mathbf b = (b_1,...,b_p) \tag{1}$$
we solve by minimize over $\mathbf b$ the correspondi... | The proof of equivalent formulas of ridge regression
They are not equivalent.
For a constrained minimization problem
$$\min_{\mathbf b} \sum_{i=1}^n (y - \mathbf{x}'_i \cdot \mathbf{b})^2\\
s.t. \sum_{j=1}^p b_j^2 \leq t,\;\;\; \mathbf b = (b_1,...,b |
12,403 | How do I figure out what kind of distribution represents this data on ping response times? | Weibull is sometimes used for modelling ping time. try a weibull distribution. To fit one in R:
x<-rweibull(n=1000,shape=2,scale=100)
#generate a weibull (this should be your data).
hist(x)
#this is an histogram of your data.
library(survival)
a1<-survreg(Surv(x,rep(1,1000))~1,dist='weibull')
exp(a1$coef) #this is the ... | How do I figure out what kind of distribution represents this data on ping response times? | Weibull is sometimes used for modelling ping time. try a weibull distribution. To fit one in R:
x<-rweibull(n=1000,shape=2,scale=100)
#generate a weibull (this should be your data).
hist(x)
#this is a | How do I figure out what kind of distribution represents this data on ping response times?
Weibull is sometimes used for modelling ping time. try a weibull distribution. To fit one in R:
x<-rweibull(n=1000,shape=2,scale=100)
#generate a weibull (this should be your data).
hist(x)
#this is an histogram of your data.
lib... | How do I figure out what kind of distribution represents this data on ping response times?
Weibull is sometimes used for modelling ping time. try a weibull distribution. To fit one in R:
x<-rweibull(n=1000,shape=2,scale=100)
#generate a weibull (this should be your data).
hist(x)
#this is a |
12,404 | How do I figure out what kind of distribution represents this data on ping response times? | Let me ask a more basic question: what do you want to do with this distributional information?
The reason I ask is because it may well make more sense to approximate the distribution with some sort of kernel density estimator, rather than insist that it fit into one of the (possibly shifted) exponential family distrib... | How do I figure out what kind of distribution represents this data on ping response times? | Let me ask a more basic question: what do you want to do with this distributional information?
The reason I ask is because it may well make more sense to approximate the distribution with some sort o | How do I figure out what kind of distribution represents this data on ping response times?
Let me ask a more basic question: what do you want to do with this distributional information?
The reason I ask is because it may well make more sense to approximate the distribution with some sort of kernel density estimator, r... | How do I figure out what kind of distribution represents this data on ping response times?
Let me ask a more basic question: what do you want to do with this distributional information?
The reason I ask is because it may well make more sense to approximate the distribution with some sort o |
12,405 | How do I figure out what kind of distribution represents this data on ping response times? | There is no reason to expect that any real world data set will fit a known distributional form...especially from such a known messy data source.
What you want to do with the answers will largely indicate an approach. For example, if you want to know when the ping times have changed significantly, then trending the empi... | How do I figure out what kind of distribution represents this data on ping response times? | There is no reason to expect that any real world data set will fit a known distributional form...especially from such a known messy data source.
What you want to do with the answers will largely indic | How do I figure out what kind of distribution represents this data on ping response times?
There is no reason to expect that any real world data set will fit a known distributional form...especially from such a known messy data source.
What you want to do with the answers will largely indicate an approach. For example,... | How do I figure out what kind of distribution represents this data on ping response times?
There is no reason to expect that any real world data set will fit a known distributional form...especially from such a known messy data source.
What you want to do with the answers will largely indic |
12,406 | How do I figure out what kind of distribution represents this data on ping response times? | A simpler approach might be to transform the data. After transforming, it might be close to Gaussian.
One common way to do so is by taking the logarithm of all values.
My guess is that in this case the distribution of the reciprocal of the round-trip times will be more symmetrical and perhaps close to Gaussian. By ta... | How do I figure out what kind of distribution represents this data on ping response times? | A simpler approach might be to transform the data. After transforming, it might be close to Gaussian.
One common way to do so is by taking the logarithm of all values.
My guess is that in this case | How do I figure out what kind of distribution represents this data on ping response times?
A simpler approach might be to transform the data. After transforming, it might be close to Gaussian.
One common way to do so is by taking the logarithm of all values.
My guess is that in this case the distribution of the recip... | How do I figure out what kind of distribution represents this data on ping response times?
A simpler approach might be to transform the data. After transforming, it might be close to Gaussian.
One common way to do so is by taking the logarithm of all values.
My guess is that in this case |
12,407 | How do I figure out what kind of distribution represents this data on ping response times? | Try the gamma distribution which is parametrized as $x \sim Gamma(k,\theta)$. If you see these pdf plots of the gamma from the wiki you will see that there are some plots that look similar to what you have.
Update- Estimation Process
The estimation via maximum likelihood is tricky but possible. I imagine you can start ... | How do I figure out what kind of distribution represents this data on ping response times? | Try the gamma distribution which is parametrized as $x \sim Gamma(k,\theta)$. If you see these pdf plots of the gamma from the wiki you will see that there are some plots that look similar to what you | How do I figure out what kind of distribution represents this data on ping response times?
Try the gamma distribution which is parametrized as $x \sim Gamma(k,\theta)$. If you see these pdf plots of the gamma from the wiki you will see that there are some plots that look similar to what you have.
Update- Estimation Pro... | How do I figure out what kind of distribution represents this data on ping response times?
Try the gamma distribution which is parametrized as $x \sim Gamma(k,\theta)$. If you see these pdf plots of the gamma from the wiki you will see that there are some plots that look similar to what you |
12,408 | How do I figure out what kind of distribution represents this data on ping response times? | Another approach, that is more justified by network considerations, is to try to fit a sum of independent exponentials with different parameters. A reasonable assumption would be that each node in the path of the ping the delay would be an independent exponential, with different parameters. A reference to the distribut... | How do I figure out what kind of distribution represents this data on ping response times? | Another approach, that is more justified by network considerations, is to try to fit a sum of independent exponentials with different parameters. A reasonable assumption would be that each node in the | How do I figure out what kind of distribution represents this data on ping response times?
Another approach, that is more justified by network considerations, is to try to fit a sum of independent exponentials with different parameters. A reasonable assumption would be that each node in the path of the ping the delay w... | How do I figure out what kind of distribution represents this data on ping response times?
Another approach, that is more justified by network considerations, is to try to fit a sum of independent exponentials with different parameters. A reasonable assumption would be that each node in the |
12,409 | How do I figure out what kind of distribution represents this data on ping response times? | Looking at it I would say a skew-normal or possibly a binormal distribution may fit it well.
In R you could use the sn library to deal with skew-normal distribution and use nls or mle to do a non-linear least square or a maximum likelihood extimation fit of your data.
===
EDIT: rereading your question/comments I would ... | How do I figure out what kind of distribution represents this data on ping response times? | Looking at it I would say a skew-normal or possibly a binormal distribution may fit it well.
In R you could use the sn library to deal with skew-normal distribution and use nls or mle to do a non-line | How do I figure out what kind of distribution represents this data on ping response times?
Looking at it I would say a skew-normal or possibly a binormal distribution may fit it well.
In R you could use the sn library to deal with skew-normal distribution and use nls or mle to do a non-linear least square or a maximum ... | How do I figure out what kind of distribution represents this data on ping response times?
Looking at it I would say a skew-normal or possibly a binormal distribution may fit it well.
In R you could use the sn library to deal with skew-normal distribution and use nls or mle to do a non-line |
12,410 | How do I figure out what kind of distribution represents this data on ping response times? | Based on your comment "Really i want to draw the mathematical curve that follows the distribution. Granted it might not be a known distribution; but i can't imagine that this hasn't been investigated before." I am providing a function that sort of fits.
Take a look at ExtremeValueDistribution
I added an amplitude and m... | How do I figure out what kind of distribution represents this data on ping response times? | Based on your comment "Really i want to draw the mathematical curve that follows the distribution. Granted it might not be a known distribution; but i can't imagine that this hasn't been investigated | How do I figure out what kind of distribution represents this data on ping response times?
Based on your comment "Really i want to draw the mathematical curve that follows the distribution. Granted it might not be a known distribution; but i can't imagine that this hasn't been investigated before." I am providing a fun... | How do I figure out what kind of distribution represents this data on ping response times?
Based on your comment "Really i want to draw the mathematical curve that follows the distribution. Granted it might not be a known distribution; but i can't imagine that this hasn't been investigated |
12,411 | How do I figure out what kind of distribution represents this data on ping response times? | The distribution looks log-normal to me.
You can fit your data using two parameters: scale and location. These can be fitted in much the same way as a normal distribution using expectation maximisation.
http://en.wikipedia.org/wiki/Log-normal_distribution | How do I figure out what kind of distribution represents this data on ping response times? | The distribution looks log-normal to me.
You can fit your data using two parameters: scale and location. These can be fitted in much the same way as a normal distribution using expectation maximisatio | How do I figure out what kind of distribution represents this data on ping response times?
The distribution looks log-normal to me.
You can fit your data using two parameters: scale and location. These can be fitted in much the same way as a normal distribution using expectation maximisation.
http://en.wikipedia.org/wi... | How do I figure out what kind of distribution represents this data on ping response times?
The distribution looks log-normal to me.
You can fit your data using two parameters: scale and location. These can be fitted in much the same way as a normal distribution using expectation maximisatio |
12,412 | How do I figure out what kind of distribution represents this data on ping response times? | Perhaps the distribution is multimodal, i.e. the distributions could be a sum of different conditions and hence multiple distributions, e.g. such as a system that is idle vs. a system that is doing heavy single file downloading vs. a system that is streaming requiring regular intervals of high data throughput. The tim... | How do I figure out what kind of distribution represents this data on ping response times? | Perhaps the distribution is multimodal, i.e. the distributions could be a sum of different conditions and hence multiple distributions, e.g. such as a system that is idle vs. a system that is doing he | How do I figure out what kind of distribution represents this data on ping response times?
Perhaps the distribution is multimodal, i.e. the distributions could be a sum of different conditions and hence multiple distributions, e.g. such as a system that is idle vs. a system that is doing heavy single file downloading v... | How do I figure out what kind of distribution represents this data on ping response times?
Perhaps the distribution is multimodal, i.e. the distributions could be a sum of different conditions and hence multiple distributions, e.g. such as a system that is idle vs. a system that is doing he |
12,413 | Why doesn't the exponential family include all distributions? | First, note there is a terminology problem in your title: the exponential family seems to imply one exponential family. You should say a exponential family, there are many exponential families.
Well, one consequence of your definition: $$p(\mathbf x|\boldsymbol \eta) = h(\mathbf x) g(\boldsymbol \eta) \exp \{\boldsymbo... | Why doesn't the exponential family include all distributions? | First, note there is a terminology problem in your title: the exponential family seems to imply one exponential family. You should say a exponential family, there are many exponential families.
Well, | Why doesn't the exponential family include all distributions?
First, note there is a terminology problem in your title: the exponential family seems to imply one exponential family. You should say a exponential family, there are many exponential families.
Well, one consequence of your definition: $$p(\mathbf x|\boldsym... | Why doesn't the exponential family include all distributions?
First, note there is a terminology problem in your title: the exponential family seems to imply one exponential family. You should say a exponential family, there are many exponential families.
Well, |
12,414 | Why doesn't the exponential family include all distributions? | Consider the non-central Laplace distribution
$$
f(x; \mu, \sigma) \propto \exp \left(-| x - \mu | / \sigma \right).
$$
Unless $\mu = 0$ you won't be able to write $|x - \mu|$ as an inner product between $\mu$ and some function of $x$.
The exponential family does include the vast majority of the nice named distributio... | Why doesn't the exponential family include all distributions? | Consider the non-central Laplace distribution
$$
f(x; \mu, \sigma) \propto \exp \left(-| x - \mu | / \sigma \right).
$$
Unless $\mu = 0$ you won't be able to write $|x - \mu|$ as an inner product bet | Why doesn't the exponential family include all distributions?
Consider the non-central Laplace distribution
$$
f(x; \mu, \sigma) \propto \exp \left(-| x - \mu | / \sigma \right).
$$
Unless $\mu = 0$ you won't be able to write $|x - \mu|$ as an inner product between $\mu$ and some function of $x$.
The exponential famil... | Why doesn't the exponential family include all distributions?
Consider the non-central Laplace distribution
$$
f(x; \mu, \sigma) \propto \exp \left(-| x - \mu | / \sigma \right).
$$
Unless $\mu = 0$ you won't be able to write $|x - \mu|$ as an inner product bet |
12,415 | Why doesn't the exponential family include all distributions? | Both the existing answers are good, but just to try add a bit of intuition about what is going on here.
The equation you have written defines how to make an exponential family of distributions. Fixing $h$, $g$ and $u$ will give you a set of distributions that have parameter $\eta$. The correct choice of $h$, $g$ and ... | Why doesn't the exponential family include all distributions? | Both the existing answers are good, but just to try add a bit of intuition about what is going on here.
The equation you have written defines how to make an exponential family of distributions. Fixin | Why doesn't the exponential family include all distributions?
Both the existing answers are good, but just to try add a bit of intuition about what is going on here.
The equation you have written defines how to make an exponential family of distributions. Fixing $h$, $g$ and $u$ will give you a set of distributions th... | Why doesn't the exponential family include all distributions?
Both the existing answers are good, but just to try add a bit of intuition about what is going on here.
The equation you have written defines how to make an exponential family of distributions. Fixin |
12,416 | What is the difference between decision_function, predict_proba, and predict function for logistic regression problem? | Recall that the functional form of logistic regression is
$$ f(x) = \frac{1}{1 + e^{-(\beta_0 + \beta_1 x_1 + \cdots + \beta_k x_k)}} $$
This is what is returned by predict_proba.
The term inside the exponential
$$ d(x) = \beta_0 + \beta_1 x_1 + \cdots + \beta_k x_k $$
is what is returned by decision_function. The "hy... | What is the difference between decision_function, predict_proba, and predict function for logistic r | Recall that the functional form of logistic regression is
$$ f(x) = \frac{1}{1 + e^{-(\beta_0 + \beta_1 x_1 + \cdots + \beta_k x_k)}} $$
This is what is returned by predict_proba.
The term inside the | What is the difference between decision_function, predict_proba, and predict function for logistic regression problem?
Recall that the functional form of logistic regression is
$$ f(x) = \frac{1}{1 + e^{-(\beta_0 + \beta_1 x_1 + \cdots + \beta_k x_k)}} $$
This is what is returned by predict_proba.
The term inside the e... | What is the difference between decision_function, predict_proba, and predict function for logistic r
Recall that the functional form of logistic regression is
$$ f(x) = \frac{1}{1 + e^{-(\beta_0 + \beta_1 x_1 + \cdots + \beta_k x_k)}} $$
This is what is returned by predict_proba.
The term inside the |
12,417 | Does simple linear regression imply causation? | The quick answer is, no. You can easily come up with non-related data that when regressed, will pass all sorts of statistical tests. Below is an old picture from Wikipedia (which, for some reason has recently been removed) that has been used to illustrate data-driven "causality".
We need more pirates to cool the... | Does simple linear regression imply causation? | The quick answer is, no. You can easily come up with non-related data that when regressed, will pass all sorts of statistical tests. Below is an old picture from Wikipedia (which, for some reason | Does simple linear regression imply causation?
The quick answer is, no. You can easily come up with non-related data that when regressed, will pass all sorts of statistical tests. Below is an old picture from Wikipedia (which, for some reason has recently been removed) that has been used to illustrate data-driven "... | Does simple linear regression imply causation?
The quick answer is, no. You can easily come up with non-related data that when regressed, will pass all sorts of statistical tests. Below is an old picture from Wikipedia (which, for some reason |
12,418 | Does simple linear regression imply causation? | There is nothing explicit in the mathematics of regression that state causal relationships, and hence one need not explicitly interpret the slope (strength and direction) nor the p-values (i.e. the probability a relation as strong as or stronger would have been observed if the relationship were zero in the population) ... | Does simple linear regression imply causation? | There is nothing explicit in the mathematics of regression that state causal relationships, and hence one need not explicitly interpret the slope (strength and direction) nor the p-values (i.e. the pr | Does simple linear regression imply causation?
There is nothing explicit in the mathematics of regression that state causal relationships, and hence one need not explicitly interpret the slope (strength and direction) nor the p-values (i.e. the probability a relation as strong as or stronger would have been observed if... | Does simple linear regression imply causation?
There is nothing explicit in the mathematics of regression that state causal relationships, and hence one need not explicitly interpret the slope (strength and direction) nor the p-values (i.e. the pr |
12,419 | Does simple linear regression imply causation? | Neither correlation nor regression can indicate causation (as is illustrated by @bill_080's answer) but as @Andy W indicates regression is often based on an explicitly fixed (i.e., independent) variable and an explicit (i.e., random) dependent variable. These designations are not appropriate in correlation analysis.
T... | Does simple linear regression imply causation? | Neither correlation nor regression can indicate causation (as is illustrated by @bill_080's answer) but as @Andy W indicates regression is often based on an explicitly fixed (i.e., independent) variab | Does simple linear regression imply causation?
Neither correlation nor regression can indicate causation (as is illustrated by @bill_080's answer) but as @Andy W indicates regression is often based on an explicitly fixed (i.e., independent) variable and an explicit (i.e., random) dependent variable. These designations... | Does simple linear regression imply causation?
Neither correlation nor regression can indicate causation (as is illustrated by @bill_080's answer) but as @Andy W indicates regression is often based on an explicitly fixed (i.e., independent) variab |
12,420 | Does simple linear regression imply causation? | From a semantic perspective, an alternative goal is to build evidence for a good predictive model instead of proving causation. A simple procedure for building evidence for the predictive value of a regression model is to divide your data in 2 parts and fit your regression with one part of the data and with the other p... | Does simple linear regression imply causation? | From a semantic perspective, an alternative goal is to build evidence for a good predictive model instead of proving causation. A simple procedure for building evidence for the predictive value of a r | Does simple linear regression imply causation?
From a semantic perspective, an alternative goal is to build evidence for a good predictive model instead of proving causation. A simple procedure for building evidence for the predictive value of a regression model is to divide your data in 2 parts and fit your regression... | Does simple linear regression imply causation?
From a semantic perspective, an alternative goal is to build evidence for a good predictive model instead of proving causation. A simple procedure for building evidence for the predictive value of a r |
12,421 | Does simple linear regression imply causation? | If you think of regression coefficients:
$$ \hat{\beta} = \frac{Cov(x,y)}{Var(x)} $$
$$\hat{\alpha}= \bar{y}-\hat{\beta}\bar{x} ,$$
where Var(.) and Cov(.) are estimates from sample (data).
Consequently, these parameters themselves are nothing else than some functions of correlation between x and y. Especially, beta i... | Does simple linear regression imply causation? | If you think of regression coefficients:
$$ \hat{\beta} = \frac{Cov(x,y)}{Var(x)} $$
$$\hat{\alpha}= \bar{y}-\hat{\beta}\bar{x} ,$$
where Var(.) and Cov(.) are estimates from sample (data).
Consequen | Does simple linear regression imply causation?
If you think of regression coefficients:
$$ \hat{\beta} = \frac{Cov(x,y)}{Var(x)} $$
$$\hat{\alpha}= \bar{y}-\hat{\beta}\bar{x} ,$$
where Var(.) and Cov(.) are estimates from sample (data).
Consequently, these parameters themselves are nothing else than some functions of ... | Does simple linear regression imply causation?
If you think of regression coefficients:
$$ \hat{\beta} = \frac{Cov(x,y)}{Var(x)} $$
$$\hat{\alpha}= \bar{y}-\hat{\beta}\bar{x} ,$$
where Var(.) and Cov(.) are estimates from sample (data).
Consequen |
12,422 | Does simple linear regression imply causation? | My understanding (I'm a causality beginner) is the following:
Linear regression implies causality if your covariates are from a controlled experiment, and your experiment isolates the hypothesized causal factor well (see Linear regression and causality in a randomized controlled experiment).
Alternatively, (updated th... | Does simple linear regression imply causation? | My understanding (I'm a causality beginner) is the following:
Linear regression implies causality if your covariates are from a controlled experiment, and your experiment isolates the hypothesized ca | Does simple linear regression imply causation?
My understanding (I'm a causality beginner) is the following:
Linear regression implies causality if your covariates are from a controlled experiment, and your experiment isolates the hypothesized causal factor well (see Linear regression and causality in a randomized con... | Does simple linear regression imply causation?
My understanding (I'm a causality beginner) is the following:
Linear regression implies causality if your covariates are from a controlled experiment, and your experiment isolates the hypothesized ca |
12,423 | Graph theory -- analysis and visualization | iGraph is a very interesting cross-language (R, Python, Ruby, C) library.
It allows you to work with unidirected and directed graphs and has quite a few analysis algorithms already implemented. | Graph theory -- analysis and visualization | iGraph is a very interesting cross-language (R, Python, Ruby, C) library.
It allows you to work with unidirected and directed graphs and has quite a few analysis algorithms already implemented. | Graph theory -- analysis and visualization
iGraph is a very interesting cross-language (R, Python, Ruby, C) library.
It allows you to work with unidirected and directed graphs and has quite a few analysis algorithms already implemented. | Graph theory -- analysis and visualization
iGraph is a very interesting cross-language (R, Python, Ruby, C) library.
It allows you to work with unidirected and directed graphs and has quite a few analysis algorithms already implemented. |
12,424 | Graph theory -- analysis and visualization | There are various packages for representing directed and undirected graphs, incidence/adjacency matrix, etc. in addition to graph$^\dagger$; look for example at the gR Task view.
For visualization and basic computation, I think the igraph package is the reliable one, in addition to Rgraphviz (on BioC as pointed out by ... | Graph theory -- analysis and visualization | There are various packages for representing directed and undirected graphs, incidence/adjacency matrix, etc. in addition to graph$^\dagger$; look for example at the gR Task view.
For visualization and | Graph theory -- analysis and visualization
There are various packages for representing directed and undirected graphs, incidence/adjacency matrix, etc. in addition to graph$^\dagger$; look for example at the gR Task view.
For visualization and basic computation, I think the igraph package is the reliable one, in additi... | Graph theory -- analysis and visualization
There are various packages for representing directed and undirected graphs, incidence/adjacency matrix, etc. in addition to graph$^\dagger$; look for example at the gR Task view.
For visualization and |
12,425 | Graph theory -- analysis and visualization | Aside from what has been said, for the vusualization task alone (and outside from R), you might be interested in checking Gephi. | Graph theory -- analysis and visualization | Aside from what has been said, for the vusualization task alone (and outside from R), you might be interested in checking Gephi. | Graph theory -- analysis and visualization
Aside from what has been said, for the vusualization task alone (and outside from R), you might be interested in checking Gephi. | Graph theory -- analysis and visualization
Aside from what has been said, for the vusualization task alone (and outside from R), you might be interested in checking Gephi. |
12,426 | Graph theory -- analysis and visualization | Another option is the statnet package. Statnet has functions for all the commonly used measures in SNA, and can also estimate ERG models. If you have your data in an edge list, read in the data as follows (assuming your data frame is labelled "edgelist"):
net <- as.network(edgelist, matrix.type = "edgelist", directed =... | Graph theory -- analysis and visualization | Another option is the statnet package. Statnet has functions for all the commonly used measures in SNA, and can also estimate ERG models. If you have your data in an edge list, read in the data as fol | Graph theory -- analysis and visualization
Another option is the statnet package. Statnet has functions for all the commonly used measures in SNA, and can also estimate ERG models. If you have your data in an edge list, read in the data as follows (assuming your data frame is labelled "edgelist"):
net <- as.network(edg... | Graph theory -- analysis and visualization
Another option is the statnet package. Statnet has functions for all the commonly used measures in SNA, and can also estimate ERG models. If you have your data in an edge list, read in the data as fol |
12,427 | Graph theory -- analysis and visualization | A similar question was asked on cstheory, also if you are specifically interested in planar graphs, or bibliographic visualization.
Gephi was already mentioned here, but it was also recommended by a few on cstheory, so I think that is a good choice.
Other cool options include:
Flare provides some really cool visualiza... | Graph theory -- analysis and visualization | A similar question was asked on cstheory, also if you are specifically interested in planar graphs, or bibliographic visualization.
Gephi was already mentioned here, but it was also recommended by a f | Graph theory -- analysis and visualization
A similar question was asked on cstheory, also if you are specifically interested in planar graphs, or bibliographic visualization.
Gephi was already mentioned here, but it was also recommended by a few on cstheory, so I think that is a good choice.
Other cool options include:... | Graph theory -- analysis and visualization
A similar question was asked on cstheory, also if you are specifically interested in planar graphs, or bibliographic visualization.
Gephi was already mentioned here, but it was also recommended by a f |
12,428 | Graph theory -- analysis and visualization | I found NodeXL very helpful and easy to use. It is an MS Excel template that provides easy import / export of a graph, formatting of edges / vertices, calculates some metrics, has some clustering algorithms. You can easily use custom images as vertices.
Another helpful tool for me was Microsoft Automatic Graph layout w... | Graph theory -- analysis and visualization | I found NodeXL very helpful and easy to use. It is an MS Excel template that provides easy import / export of a graph, formatting of edges / vertices, calculates some metrics, has some clustering algo | Graph theory -- analysis and visualization
I found NodeXL very helpful and easy to use. It is an MS Excel template that provides easy import / export of a graph, formatting of edges / vertices, calculates some metrics, has some clustering algorithms. You can easily use custom images as vertices.
Another helpful tool fo... | Graph theory -- analysis and visualization
I found NodeXL very helpful and easy to use. It is an MS Excel template that provides easy import / export of a graph, formatting of edges / vertices, calculates some metrics, has some clustering algo |
12,429 | Why does the US death rate not "match" life expectancy | In short
The discrepancy between death rate and the reciprocal of the life expectancy generally occurs when the age distribution of the population is not the same as the survival curve, which relates to a hypothetical population on which the life expectancy is based (and more specifically the population is younger tha... | Why does the US death rate not "match" life expectancy | In short
The discrepancy between death rate and the reciprocal of the life expectancy generally occurs when the age distribution of the population is not the same as the survival curve, which relates | Why does the US death rate not "match" life expectancy
In short
The discrepancy between death rate and the reciprocal of the life expectancy generally occurs when the age distribution of the population is not the same as the survival curve, which relates to a hypothetical population on which the life expectancy is base... | Why does the US death rate not "match" life expectancy
In short
The discrepancy between death rate and the reciprocal of the life expectancy generally occurs when the age distribution of the population is not the same as the survival curve, which relates |
12,430 | Why does the US death rate not "match" life expectancy | There is a simpler way to understand this than the (detailed and correct) explanation in another answer.
The life expectancy now depends on the death rate in the past.
As an extreme example, suppose that some disaster infected every person in a country with a medical condition which was incurable and had a 50% fatality... | Why does the US death rate not "match" life expectancy | There is a simpler way to understand this than the (detailed and correct) explanation in another answer.
The life expectancy now depends on the death rate in the past.
As an extreme example, suppose t | Why does the US death rate not "match" life expectancy
There is a simpler way to understand this than the (detailed and correct) explanation in another answer.
The life expectancy now depends on the death rate in the past.
As an extreme example, suppose that some disaster infected every person in a country with a medic... | Why does the US death rate not "match" life expectancy
There is a simpler way to understand this than the (detailed and correct) explanation in another answer.
The life expectancy now depends on the death rate in the past.
As an extreme example, suppose t |
12,431 | Overfitting and Underfitting | I'll try to answer in the simplest way. Each of those problems has its own main origin:
Overfitting: Data is noisy, meaning that there are some deviations from reality (because of measurement errors, influentially random factors, unobserved variables and rubbish correlations) that makes it harder for us to see their tr... | Overfitting and Underfitting | I'll try to answer in the simplest way. Each of those problems has its own main origin:
Overfitting: Data is noisy, meaning that there are some deviations from reality (because of measurement errors, | Overfitting and Underfitting
I'll try to answer in the simplest way. Each of those problems has its own main origin:
Overfitting: Data is noisy, meaning that there are some deviations from reality (because of measurement errors, influentially random factors, unobserved variables and rubbish correlations) that makes it ... | Overfitting and Underfitting
I'll try to answer in the simplest way. Each of those problems has its own main origin:
Overfitting: Data is noisy, meaning that there are some deviations from reality (because of measurement errors, |
12,432 | Overfitting and Underfitting | Overfitting is when a model estimates the variable you are modeling really well on the original data, but it does not estimate well on new data set (hold out, cross validation, forecasting, etc.). You have too many variables or estimators in your model (dummy variables, etc.) and these cause your model to become too s... | Overfitting and Underfitting | Overfitting is when a model estimates the variable you are modeling really well on the original data, but it does not estimate well on new data set (hold out, cross validation, forecasting, etc.). Yo | Overfitting and Underfitting
Overfitting is when a model estimates the variable you are modeling really well on the original data, but it does not estimate well on new data set (hold out, cross validation, forecasting, etc.). You have too many variables or estimators in your model (dummy variables, etc.) and these cau... | Overfitting and Underfitting
Overfitting is when a model estimates the variable you are modeling really well on the original data, but it does not estimate well on new data set (hold out, cross validation, forecasting, etc.). Yo |
12,433 | Overfitting and Underfitting | Perhaps during your research you came across the following equation:
Error = IrreducibleError + Bias² + Variance.
Why do we face these two problems in training a model ?
The learning problem itself is basically a trade-off between bias and variance.
What are the main reasons for overfitting and underfitting ?
Sho... | Overfitting and Underfitting | Perhaps during your research you came across the following equation:
Error = IrreducibleError + Bias² + Variance.
Why do we face these two problems in training a model ?
The learning problem itself | Overfitting and Underfitting
Perhaps during your research you came across the following equation:
Error = IrreducibleError + Bias² + Variance.
Why do we face these two problems in training a model ?
The learning problem itself is basically a trade-off between bias and variance.
What are the main reasons for overfi... | Overfitting and Underfitting
Perhaps during your research you came across the following equation:
Error = IrreducibleError + Bias² + Variance.
Why do we face these two problems in training a model ?
The learning problem itself |
12,434 | Overfitting and Underfitting | Almost all statistical problems can be stated in the following form:
Given the data $(y, x)$ find $\hat{f}$ which produces $\hat{y}=\hat{f}(x)$.
Make this $\hat{f}$ as close as possible to "true" $f$, where $f$ is defined as
$$y = f(x) + \varepsilon$$
The temptation is always to make $\hat{f}$ produce $\hat{y}$ whic... | Overfitting and Underfitting | Almost all statistical problems can be stated in the following form:
Given the data $(y, x)$ find $\hat{f}$ which produces $\hat{y}=\hat{f}(x)$.
Make this $\hat{f}$ as close as possible to "true" $f | Overfitting and Underfitting
Almost all statistical problems can be stated in the following form:
Given the data $(y, x)$ find $\hat{f}$ which produces $\hat{y}=\hat{f}(x)$.
Make this $\hat{f}$ as close as possible to "true" $f$, where $f$ is defined as
$$y = f(x) + \varepsilon$$
The temptation is always to make $\h... | Overfitting and Underfitting
Almost all statistical problems can be stated in the following form:
Given the data $(y, x)$ find $\hat{f}$ which produces $\hat{y}=\hat{f}(x)$.
Make this $\hat{f}$ as close as possible to "true" $f |
12,435 | Overfitting and Underfitting | What are the main reasons for overfitting and underfitting ?
For overfitting, the model is too complex to fit the training data well. For underfitting, the model is too simple.
Why do we face these two problems in training a model ?
It is hard to pick the "just right" model and parameters for the data. | Overfitting and Underfitting | What are the main reasons for overfitting and underfitting ?
For overfitting, the model is too complex to fit the training data well. For underfitting, the model is too simple.
Why do we face these | Overfitting and Underfitting
What are the main reasons for overfitting and underfitting ?
For overfitting, the model is too complex to fit the training data well. For underfitting, the model is too simple.
Why do we face these two problems in training a model ?
It is hard to pick the "just right" model and parameter... | Overfitting and Underfitting
What are the main reasons for overfitting and underfitting ?
For overfitting, the model is too complex to fit the training data well. For underfitting, the model is too simple.
Why do we face these |
12,436 | Overfitting and Underfitting | Overfitting and underfitting are basically inadequate explanations of the data by an hypothesized model and can be seen as the model overexplaining or underexplaining the data. This is created by the relationship between the model used to explain the data and the model generating the data. In our attempt to explain we ... | Overfitting and Underfitting | Overfitting and underfitting are basically inadequate explanations of the data by an hypothesized model and can be seen as the model overexplaining or underexplaining the data. This is created by the | Overfitting and Underfitting
Overfitting and underfitting are basically inadequate explanations of the data by an hypothesized model and can be seen as the model overexplaining or underexplaining the data. This is created by the relationship between the model used to explain the data and the model generating the data. ... | Overfitting and Underfitting
Overfitting and underfitting are basically inadequate explanations of the data by an hypothesized model and can be seen as the model overexplaining or underexplaining the data. This is created by the |
12,437 | Overfitting and Underfitting | In a nutshell, overfiitting appears as a consequence of patterns that appear in your training dataset but are not present on the entire population (they appeared out of luck) If your use a simple model (think for linear regression for instance), risk of overfitting is low, as the number of possible patterns it can dete... | Overfitting and Underfitting | In a nutshell, overfiitting appears as a consequence of patterns that appear in your training dataset but are not present on the entire population (they appeared out of luck) If your use a simple mode | Overfitting and Underfitting
In a nutshell, overfiitting appears as a consequence of patterns that appear in your training dataset but are not present on the entire population (they appeared out of luck) If your use a simple model (think for linear regression for instance), risk of overfitting is low, as the number of ... | Overfitting and Underfitting
In a nutshell, overfiitting appears as a consequence of patterns that appear in your training dataset but are not present on the entire population (they appeared out of luck) If your use a simple mode |
12,438 | Overfitting and Underfitting | Short answer:
The main reason for overfitting is using a complex model when you have a small training set.
The main reason for underfitting is using a model that is too simple and can not perform well on the training set.
Main reason for overfitting?
Models with high capacity can overfit by memorizing properties of ... | Overfitting and Underfitting | Short answer:
The main reason for overfitting is using a complex model when you have a small training set.
The main reason for underfitting is using a model that is too simple and can not perform well | Overfitting and Underfitting
Short answer:
The main reason for overfitting is using a complex model when you have a small training set.
The main reason for underfitting is using a model that is too simple and can not perform well on the training set.
Main reason for overfitting?
Models with high capacity can overfit... | Overfitting and Underfitting
Short answer:
The main reason for overfitting is using a complex model when you have a small training set.
The main reason for underfitting is using a model that is too simple and can not perform well |
12,439 | Overfitting and Underfitting | Consider an example where we have a hypothesis/model equation,
y=q*X+c,
where X=feature list, y=label and q and c are the coefficients that we have to trained.
If we come up with the coefficient values such that its large enough and start suppressing the features value(i.e. X) in such a case we always get the consta... | Overfitting and Underfitting | Consider an example where we have a hypothesis/model equation,
y=q*X+c,
where X=feature list, y=label and q and c are the coefficients that we have to trained.
If we come up with the coefficient va | Overfitting and Underfitting
Consider an example where we have a hypothesis/model equation,
y=q*X+c,
where X=feature list, y=label and q and c are the coefficients that we have to trained.
If we come up with the coefficient values such that its large enough and start suppressing the features value(i.e. X) in such a ... | Overfitting and Underfitting
Consider an example where we have a hypothesis/model equation,
y=q*X+c,
where X=feature list, y=label and q and c are the coefficients that we have to trained.
If we come up with the coefficient va |
12,440 | Lasso vs. adaptive Lasso | Brief answers to your questions:
Lasso and adaptive lasso are different. (Check Zou (2006) to see how adaptive lasso differs from standard lasso.)
Lasso is a special case of elastic net. (See Zou & Hastie (2005).)
Adaptive lasso is not a special case of elastic net.
Elastic net is not a special case of lasso or adapti... | Lasso vs. adaptive Lasso | Brief answers to your questions:
Lasso and adaptive lasso are different. (Check Zou (2006) to see how adaptive lasso differs from standard lasso.)
Lasso is a special case of elastic net. (See Zou & H | Lasso vs. adaptive Lasso
Brief answers to your questions:
Lasso and adaptive lasso are different. (Check Zou (2006) to see how adaptive lasso differs from standard lasso.)
Lasso is a special case of elastic net. (See Zou & Hastie (2005).)
Adaptive lasso is not a special case of elastic net.
Elastic net is not a specia... | Lasso vs. adaptive Lasso
Brief answers to your questions:
Lasso and adaptive lasso are different. (Check Zou (2006) to see how adaptive lasso differs from standard lasso.)
Lasso is a special case of elastic net. (See Zou & H |
12,441 | Lasso vs. adaptive Lasso | LASSO solutions are solutions that minimize
$$Q(\beta|X,y) = \dfrac{1}{2n}||y-X\beta||^2 + \lambda\sum_{j}|\beta_j|$$
the adaptive lasso simply adds weights to this to try to counteract the known issue of LASSO estimates being biased.
$$Q_a(\beta|X,y,w) = \dfrac{1}{2n}||y-X\beta||^2 + \lambda\sum_{j}w_j|\beta_j|$$
Oft... | Lasso vs. adaptive Lasso | LASSO solutions are solutions that minimize
$$Q(\beta|X,y) = \dfrac{1}{2n}||y-X\beta||^2 + \lambda\sum_{j}|\beta_j|$$
the adaptive lasso simply adds weights to this to try to counteract the known issu | Lasso vs. adaptive Lasso
LASSO solutions are solutions that minimize
$$Q(\beta|X,y) = \dfrac{1}{2n}||y-X\beta||^2 + \lambda\sum_{j}|\beta_j|$$
the adaptive lasso simply adds weights to this to try to counteract the known issue of LASSO estimates being biased.
$$Q_a(\beta|X,y,w) = \dfrac{1}{2n}||y-X\beta||^2 + \lambda\... | Lasso vs. adaptive Lasso
LASSO solutions are solutions that minimize
$$Q(\beta|X,y) = \dfrac{1}{2n}||y-X\beta||^2 + \lambda\sum_{j}|\beta_j|$$
the adaptive lasso simply adds weights to this to try to counteract the known issu |
12,442 | Lasso vs. adaptive Lasso | Adaptive LASSO is used for consistent variable selection. The problems we encounter when using the LASSO for variable selection are:
The shrinkage parameter must be larger for selection than prediction
Large nonzero parameters will be too small so that the bias is too large
Small nonzero parameters cannot be detected ... | Lasso vs. adaptive Lasso | Adaptive LASSO is used for consistent variable selection. The problems we encounter when using the LASSO for variable selection are:
The shrinkage parameter must be larger for selection than predicti | Lasso vs. adaptive Lasso
Adaptive LASSO is used for consistent variable selection. The problems we encounter when using the LASSO for variable selection are:
The shrinkage parameter must be larger for selection than prediction
Large nonzero parameters will be too small so that the bias is too large
Small nonzero param... | Lasso vs. adaptive Lasso
Adaptive LASSO is used for consistent variable selection. The problems we encounter when using the LASSO for variable selection are:
The shrinkage parameter must be larger for selection than predicti |
12,443 | Should the mean be used when data are skewed? | I disagree with the advice as a flat out rule. (It's not common to all books.)
The issues are more subtle.
If you're actually interested in making inference about the population mean, the sample mean is at least an unbiased estimator of it, and has a number of other advantages. In fact, see the Gauss-Markov theorem - i... | Should the mean be used when data are skewed? | I disagree with the advice as a flat out rule. (It's not common to all books.)
The issues are more subtle.
If you're actually interested in making inference about the population mean, the sample mean | Should the mean be used when data are skewed?
I disagree with the advice as a flat out rule. (It's not common to all books.)
The issues are more subtle.
If you're actually interested in making inference about the population mean, the sample mean is at least an unbiased estimator of it, and has a number of other advanta... | Should the mean be used when data are skewed?
I disagree with the advice as a flat out rule. (It's not common to all books.)
The issues are more subtle.
If you're actually interested in making inference about the population mean, the sample mean |
12,444 | Should the mean be used when data are skewed? | In real life, we should choose a measure of central tendency based on what we are trying to find out; and yes, sometimes the mode is the right thing to use. Sometimes it's the Winsorized or trimmed mean. Sometimes the geometric or harmonic mean. Sometimes there is no good measure of central tendency.
Intro books are wr... | Should the mean be used when data are skewed? | In real life, we should choose a measure of central tendency based on what we are trying to find out; and yes, sometimes the mode is the right thing to use. Sometimes it's the Winsorized or trimmed me | Should the mean be used when data are skewed?
In real life, we should choose a measure of central tendency based on what we are trying to find out; and yes, sometimes the mode is the right thing to use. Sometimes it's the Winsorized or trimmed mean. Sometimes the geometric or harmonic mean. Sometimes there is no good m... | Should the mean be used when data are skewed?
In real life, we should choose a measure of central tendency based on what we are trying to find out; and yes, sometimes the mode is the right thing to use. Sometimes it's the Winsorized or trimmed me |
12,445 | Should the mean be used when data are skewed? | Even when data are skewed (e.g., health care costs calculated alongside a clinical trial, where few patients totalled zero cost because they die just after the enrollment, and few patients accrued tons of cost due to side effects of a given health care programme under investigation), mean may be preferred to median for... | Should the mean be used when data are skewed? | Even when data are skewed (e.g., health care costs calculated alongside a clinical trial, where few patients totalled zero cost because they die just after the enrollment, and few patients accrued ton | Should the mean be used when data are skewed?
Even when data are skewed (e.g., health care costs calculated alongside a clinical trial, where few patients totalled zero cost because they die just after the enrollment, and few patients accrued tons of cost due to side effects of a given health care programme under inves... | Should the mean be used when data are skewed?
Even when data are skewed (e.g., health care costs calculated alongside a clinical trial, where few patients totalled zero cost because they die just after the enrollment, and few patients accrued ton |
12,446 | Should the mean be used when data are skewed? | I think that what's missing from the question as well as both the answers so far is that the discussion of mean vs median in introductory statistics books generally occurs early on in a chapter about how to numerically summarize a distribution. As opposed to inferential statistics, this is generally about producing de... | Should the mean be used when data are skewed? | I think that what's missing from the question as well as both the answers so far is that the discussion of mean vs median in introductory statistics books generally occurs early on in a chapter about | Should the mean be used when data are skewed?
I think that what's missing from the question as well as both the answers so far is that the discussion of mean vs median in introductory statistics books generally occurs early on in a chapter about how to numerically summarize a distribution. As opposed to inferential st... | Should the mean be used when data are skewed?
I think that what's missing from the question as well as both the answers so far is that the discussion of mean vs median in introductory statistics books generally occurs early on in a chapter about |
12,447 | Gentler approach to Bayesian statistics | The use of probability density functions in calculations. In other words how to evaluate such equations.
I think you're still thinking of this from a frequentist perspective: if you're looking for a point estimate, the posterior won't give it to you. You put PDFs in, you get PDFs out. You can derive point estimates by... | Gentler approach to Bayesian statistics | The use of probability density functions in calculations. In other words how to evaluate such equations.
I think you're still thinking of this from a frequentist perspective: if you're looking for a | Gentler approach to Bayesian statistics
The use of probability density functions in calculations. In other words how to evaluate such equations.
I think you're still thinking of this from a frequentist perspective: if you're looking for a point estimate, the posterior won't give it to you. You put PDFs in, you get PDF... | Gentler approach to Bayesian statistics
The use of probability density functions in calculations. In other words how to evaluate such equations.
I think you're still thinking of this from a frequentist perspective: if you're looking for a |
12,448 | Gentler approach to Bayesian statistics | A beta distribution has the form $p(\theta)=\frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)}\theta^{\alpha-1}(1-\theta)^{\beta-1}$. A beta(1,1) distribution has parameters $(\alpha, \beta)=(1,1)$. (Unfortunately, this kind of statistical short-hand places a burden on the reader to know how the particular model ... | Gentler approach to Bayesian statistics | A beta distribution has the form $p(\theta)=\frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)}\theta^{\alpha-1}(1-\theta)^{\beta-1}$. A beta(1,1) distribution has parameters $(\alpha, \beta)=(1, | Gentler approach to Bayesian statistics
A beta distribution has the form $p(\theta)=\frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)}\theta^{\alpha-1}(1-\theta)^{\beta-1}$. A beta(1,1) distribution has parameters $(\alpha, \beta)=(1,1)$. (Unfortunately, this kind of statistical short-hand places a burden on the ... | Gentler approach to Bayesian statistics
A beta distribution has the form $p(\theta)=\frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)}\theta^{\alpha-1}(1-\theta)^{\beta-1}$. A beta(1,1) distribution has parameters $(\alpha, \beta)=(1, |
12,449 | Gentler approach to Bayesian statistics | If you are looking for a gentler approach I can highly recommend the book by Kruschke which uses R to explain the core concepts. It is a very practical and hands-on approach into learning Bayesian statistics and on his website you can find all of the codes used.
Someone also recommended the text by Cam.Davidson.Pilon ... | Gentler approach to Bayesian statistics | If you are looking for a gentler approach I can highly recommend the book by Kruschke which uses R to explain the core concepts. It is a very practical and hands-on approach into learning Bayesian sta | Gentler approach to Bayesian statistics
If you are looking for a gentler approach I can highly recommend the book by Kruschke which uses R to explain the core concepts. It is a very practical and hands-on approach into learning Bayesian statistics and on his website you can find all of the codes used.
Someone also rec... | Gentler approach to Bayesian statistics
If you are looking for a gentler approach I can highly recommend the book by Kruschke which uses R to explain the core concepts. It is a very practical and hands-on approach into learning Bayesian sta |
12,450 | Why are there extra terms $-p_i+q_i$ in SciPy's implementation of Kullback-Leibler divergence? | The other answer tells us why we don't usually see the $-p_i+q_i$ term: $p$ and $q$ are usually residents of the simplex and so sum to one, so this leads to $\sum - [p_i - q_i] = \sum - p_i + \sum q_i = -1 + 1 = 0$.
In this answer, I want to show why those terms are there in the first place, by viewing KL divergence as... | Why are there extra terms $-p_i+q_i$ in SciPy's implementation of Kullback-Leibler divergence? | The other answer tells us why we don't usually see the $-p_i+q_i$ term: $p$ and $q$ are usually residents of the simplex and so sum to one, so this leads to $\sum - [p_i - q_i] = \sum - p_i + \sum q_i | Why are there extra terms $-p_i+q_i$ in SciPy's implementation of Kullback-Leibler divergence?
The other answer tells us why we don't usually see the $-p_i+q_i$ term: $p$ and $q$ are usually residents of the simplex and so sum to one, so this leads to $\sum - [p_i - q_i] = \sum - p_i + \sum q_i = -1 + 1 = 0$.
In this a... | Why are there extra terms $-p_i+q_i$ in SciPy's implementation of Kullback-Leibler divergence?
The other answer tells us why we don't usually see the $-p_i+q_i$ term: $p$ and $q$ are usually residents of the simplex and so sum to one, so this leads to $\sum - [p_i - q_i] = \sum - p_i + \sum q_i |
12,451 | Why are there extra terms $-p_i+q_i$ in SciPy's implementation of Kullback-Leibler divergence? | The referenced book has a free pdf on Boyd's site: https://web.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf
On page 90, formula 3.17 gives this definition. I suspect the reason for the added terms is that in convex optimization, the two vectors needn't be probability distributions; the authors say
Note that the relative ... | Why are there extra terms $-p_i+q_i$ in SciPy's implementation of Kullback-Leibler divergence? | The referenced book has a free pdf on Boyd's site: https://web.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf
On page 90, formula 3.17 gives this definition. I suspect the reason for the added terms is tha | Why are there extra terms $-p_i+q_i$ in SciPy's implementation of Kullback-Leibler divergence?
The referenced book has a free pdf on Boyd's site: https://web.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf
On page 90, formula 3.17 gives this definition. I suspect the reason for the added terms is that in convex optimization,... | Why are there extra terms $-p_i+q_i$ in SciPy's implementation of Kullback-Leibler divergence?
The referenced book has a free pdf on Boyd's site: https://web.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf
On page 90, formula 3.17 gives this definition. I suspect the reason for the added terms is tha |
12,452 | How do I analytically calculate variance of a recursive random variable? | Call the next chests as $X_1,X_2$. With $0.4$ probability, our new variable is $X_1+X_2$ and with $0.6$ probability, it is $1$. So,
$$\begin{align}E[X^2]&=0.4\times E[(X_1+X_2)^2]+0.6\times1^2\\&=0.4\times E[X_1^2+X_2^2+2X_1X_2]+0.6\\&=0.4\times(2E[X^2]+2E[X]^2)+0.6\\&=0.8\times E[X^2]+7.8\rightarrow E[X^2]=39\rightarr... | How do I analytically calculate variance of a recursive random variable? | Call the next chests as $X_1,X_2$. With $0.4$ probability, our new variable is $X_1+X_2$ and with $0.6$ probability, it is $1$. So,
$$\begin{align}E[X^2]&=0.4\times E[(X_1+X_2)^2]+0.6\times1^2\\&=0.4\ | How do I analytically calculate variance of a recursive random variable?
Call the next chests as $X_1,X_2$. With $0.4$ probability, our new variable is $X_1+X_2$ and with $0.6$ probability, it is $1$. So,
$$\begin{align}E[X^2]&=0.4\times E[(X_1+X_2)^2]+0.6\times1^2\\&=0.4\times E[X_1^2+X_2^2+2X_1X_2]+0.6\\&=0.4\times(2... | How do I analytically calculate variance of a recursive random variable?
Call the next chests as $X_1,X_2$. With $0.4$ probability, our new variable is $X_1+X_2$ and with $0.6$ probability, it is $1$. So,
$$\begin{align}E[X^2]&=0.4\times E[(X_1+X_2)^2]+0.6\times1^2\\&=0.4\ |
12,453 | How do I analytically calculate variance of a recursive random variable? | Actually, it's relatively simple to obtain formulas for the entire distribution as well as an easy procedure to compute any moment of it.
For $n=1,2,3,\ldots,$ let $f_n(p) = \Pr(X=n)$ with $p=0.6.$ Define
$$F_p(t) = f_1(p)t + f_2(p)t^2 + \cdots + f_n(p)t^n + \cdots$$
(the probability generating function). The problem... | How do I analytically calculate variance of a recursive random variable? | Actually, it's relatively simple to obtain formulas for the entire distribution as well as an easy procedure to compute any moment of it.
For $n=1,2,3,\ldots,$ let $f_n(p) = \Pr(X=n)$ with $p=0.6.$ D | How do I analytically calculate variance of a recursive random variable?
Actually, it's relatively simple to obtain formulas for the entire distribution as well as an easy procedure to compute any moment of it.
For $n=1,2,3,\ldots,$ let $f_n(p) = \Pr(X=n)$ with $p=0.6.$ Define
$$F_p(t) = f_1(p)t + f_2(p)t^2 + \cdots +... | How do I analytically calculate variance of a recursive random variable?
Actually, it's relatively simple to obtain formulas for the entire distribution as well as an easy procedure to compute any moment of it.
For $n=1,2,3,\ldots,$ let $f_n(p) = \Pr(X=n)$ with $p=0.6.$ D |
12,454 | When we calculate mean and variance, do we assume data are normally distributed? | No, those equations come directly from the mean and variance formulae in terms of expected value, considering the collected data as a population.
$$\mu = \mathbb{E}\big[X\big]$$
$$\sigma^2 = \mathbb{E}\big[\big(X-\mu\big)^2\big]$$
Since you have a finite number of observations, the distribution is discrete,$^{\dagger}$... | When we calculate mean and variance, do we assume data are normally distributed? | No, those equations come directly from the mean and variance formulae in terms of expected value, considering the collected data as a population.
$$\mu = \mathbb{E}\big[X\big]$$
$$\sigma^2 = \mathbb{E | When we calculate mean and variance, do we assume data are normally distributed?
No, those equations come directly from the mean and variance formulae in terms of expected value, considering the collected data as a population.
$$\mu = \mathbb{E}\big[X\big]$$
$$\sigma^2 = \mathbb{E}\big[\big(X-\mu\big)^2\big]$$
Since yo... | When we calculate mean and variance, do we assume data are normally distributed?
No, those equations come directly from the mean and variance formulae in terms of expected value, considering the collected data as a population.
$$\mu = \mathbb{E}\big[X\big]$$
$$\sigma^2 = \mathbb{E |
12,455 | When we calculate mean and variance, do we assume data are normally distributed? | You are mixing descriptive statistics of a sample (such as $\bar X, S)$ with parameters of a population (such as $\mu, \sigma),$
and description of a sample with estimation of parameters.
Describing sample center or location.
The correct version of the sample mean of a sample $X_i, X_2, \dots X_n$ of size $n$ is $\bar ... | When we calculate mean and variance, do we assume data are normally distributed? | You are mixing descriptive statistics of a sample (such as $\bar X, S)$ with parameters of a population (such as $\mu, \sigma),$
and description of a sample with estimation of parameters.
Describing s | When we calculate mean and variance, do we assume data are normally distributed?
You are mixing descriptive statistics of a sample (such as $\bar X, S)$ with parameters of a population (such as $\mu, \sigma),$
and description of a sample with estimation of parameters.
Describing sample center or location.
The correct v... | When we calculate mean and variance, do we assume data are normally distributed?
You are mixing descriptive statistics of a sample (such as $\bar X, S)$ with parameters of a population (such as $\mu, \sigma),$
and description of a sample with estimation of parameters.
Describing s |
12,456 | When we calculate mean and variance, do we assume data are normally distributed? | Normality is an ideal case for the ordinary sample mean and variance (as well as other common statistics like least squares regression estimates), but it is certainly not a requirement. After all, normality is never true in practice for any any real data-generating process, nor is it ever true for actual data. So if no... | When we calculate mean and variance, do we assume data are normally distributed? | Normality is an ideal case for the ordinary sample mean and variance (as well as other common statistics like least squares regression estimates), but it is certainly not a requirement. After all, nor | When we calculate mean and variance, do we assume data are normally distributed?
Normality is an ideal case for the ordinary sample mean and variance (as well as other common statistics like least squares regression estimates), but it is certainly not a requirement. After all, normality is never true in practice for an... | When we calculate mean and variance, do we assume data are normally distributed?
Normality is an ideal case for the ordinary sample mean and variance (as well as other common statistics like least squares regression estimates), but it is certainly not a requirement. After all, nor |
12,457 | When we calculate mean and variance, do we assume data are normally distributed? | $\mu$ is, indeed, the value that maximizes the Gaussian likelihood, but you have to assume Gaussianity first. You could assume other distributions instead.
Assume you are flipping coins, and you want to estimate the probability $p$ of heads.
You toss it $n$ times, and measure $\hat p \times n$ tails.
This process assum... | When we calculate mean and variance, do we assume data are normally distributed? | $\mu$ is, indeed, the value that maximizes the Gaussian likelihood, but you have to assume Gaussianity first. You could assume other distributions instead.
Assume you are flipping coins, and you want | When we calculate mean and variance, do we assume data are normally distributed?
$\mu$ is, indeed, the value that maximizes the Gaussian likelihood, but you have to assume Gaussianity first. You could assume other distributions instead.
Assume you are flipping coins, and you want to estimate the probability $p$ of head... | When we calculate mean and variance, do we assume data are normally distributed?
$\mu$ is, indeed, the value that maximizes the Gaussian likelihood, but you have to assume Gaussianity first. You could assume other distributions instead.
Assume you are flipping coins, and you want |
12,458 | When we calculate mean and variance, do we assume data are normally distributed? | One thing that's only been touched on in the answers so far is the (widespread) use of sample mean & variance to estimate their population analogues without making the assumption that the data are from a distribution in any particular parametric family.
For independently, identically distributed observations from distr... | When we calculate mean and variance, do we assume data are normally distributed? | One thing that's only been touched on in the answers so far is the (widespread) use of sample mean & variance to estimate their population analogues without making the assumption that the data are fro | When we calculate mean and variance, do we assume data are normally distributed?
One thing that's only been touched on in the answers so far is the (widespread) use of sample mean & variance to estimate their population analogues without making the assumption that the data are from a distribution in any particular para... | When we calculate mean and variance, do we assume data are normally distributed?
One thing that's only been touched on in the answers so far is the (widespread) use of sample mean & variance to estimate their population analogues without making the assumption that the data are fro |
12,459 | When we calculate mean and variance, do we assume data are normally distributed? | Method of moments
The expressions on the right are sample moments and on the left are parameters of a distribution (in terms of moments of the distribution).
$$\begin{array}{ccl}
\overbrace{\mu}^{\substack{\text{parameters of}\\\text{population distribution}\\\text{in terms of moments}}} &=& \overbrace{\frac{1}{N}\sum_... | When we calculate mean and variance, do we assume data are normally distributed? | Method of moments
The expressions on the right are sample moments and on the left are parameters of a distribution (in terms of moments of the distribution).
$$\begin{array}{ccl}
\overbrace{\mu}^{\sub | When we calculate mean and variance, do we assume data are normally distributed?
Method of moments
The expressions on the right are sample moments and on the left are parameters of a distribution (in terms of moments of the distribution).
$$\begin{array}{ccl}
\overbrace{\mu}^{\substack{\text{parameters of}\\\text{popul... | When we calculate mean and variance, do we assume data are normally distributed?
Method of moments
The expressions on the right are sample moments and on the left are parameters of a distribution (in terms of moments of the distribution).
$$\begin{array}{ccl}
\overbrace{\mu}^{\sub |
12,460 | Obtaining a formula for prediction limits in a linear model (i.e.: prediction intervals) | You will need matrix arithmetic. I'm not sure how Excel will go with that. Anyway, here are the details.
Suppose your regression is written as $\mathbf{y} = \mathbf{X}\mathbf{\beta} + \mathbf{e}$.
Let $\mathbf{X}^*$ be a row vector containing the values of the predictors for the forecasts (in the same format as $\mathb... | Obtaining a formula for prediction limits in a linear model (i.e.: prediction intervals) | You will need matrix arithmetic. I'm not sure how Excel will go with that. Anyway, here are the details.
Suppose your regression is written as $\mathbf{y} = \mathbf{X}\mathbf{\beta} + \mathbf{e}$.
Let | Obtaining a formula for prediction limits in a linear model (i.e.: prediction intervals)
You will need matrix arithmetic. I'm not sure how Excel will go with that. Anyway, here are the details.
Suppose your regression is written as $\mathbf{y} = \mathbf{X}\mathbf{\beta} + \mathbf{e}$.
Let $\mathbf{X}^*$ be a row vector... | Obtaining a formula for prediction limits in a linear model (i.e.: prediction intervals)
You will need matrix arithmetic. I'm not sure how Excel will go with that. Anyway, here are the details.
Suppose your regression is written as $\mathbf{y} = \mathbf{X}\mathbf{\beta} + \mathbf{e}$.
Let |
12,461 | Obtaining a formula for prediction limits in a linear model (i.e.: prediction intervals) | Are you by chance after the different types of prediction intervals? The predict.lm manual page has
## S3 method for class 'lm'
predict(object, newdata, se.fit = FALSE, scale = NULL, df = Inf,
interval = c("none", "confidence", "prediction"),
level = 0.95, type = c("response", "terms"),
te... | Obtaining a formula for prediction limits in a linear model (i.e.: prediction intervals) | Are you by chance after the different types of prediction intervals? The predict.lm manual page has
## S3 method for class 'lm'
predict(object, newdata, se.fit = FALSE, scale = NULL, df = Inf,
| Obtaining a formula for prediction limits in a linear model (i.e.: prediction intervals)
Are you by chance after the different types of prediction intervals? The predict.lm manual page has
## S3 method for class 'lm'
predict(object, newdata, se.fit = FALSE, scale = NULL, df = Inf,
interval = c("none", "conf... | Obtaining a formula for prediction limits in a linear model (i.e.: prediction intervals)
Are you by chance after the different types of prediction intervals? The predict.lm manual page has
## S3 method for class 'lm'
predict(object, newdata, se.fit = FALSE, scale = NULL, df = Inf,
|
12,462 | Obtaining a formula for prediction limits in a linear model (i.e.: prediction intervals) | @Tal: Might I suggest Kutner et al as a fabulous source for linear models.
There is the distinction between
a prediction of $Y$ from an individual new observation $X_{vec}$,
the expected value of a $Y$ conditioned on $X_{vec}$, $E(Y|X_{vec})$ and
$Y$ from several instances of $x_{vec}$
They are all covered in detail ... | Obtaining a formula for prediction limits in a linear model (i.e.: prediction intervals) | @Tal: Might I suggest Kutner et al as a fabulous source for linear models.
There is the distinction between
a prediction of $Y$ from an individual new observation $X_{vec}$,
the expected value of a $ | Obtaining a formula for prediction limits in a linear model (i.e.: prediction intervals)
@Tal: Might I suggest Kutner et al as a fabulous source for linear models.
There is the distinction between
a prediction of $Y$ from an individual new observation $X_{vec}$,
the expected value of a $Y$ conditioned on $X_{vec}$, $E... | Obtaining a formula for prediction limits in a linear model (i.e.: prediction intervals)
@Tal: Might I suggest Kutner et al as a fabulous source for linear models.
There is the distinction between
a prediction of $Y$ from an individual new observation $X_{vec}$,
the expected value of a $ |
12,463 | Alternatives to Bayesian statistics when distributions are unknown | From what you’re saying is that you want something Bayesian, but you can’t define the likelihood. For such cases there is approximate Bayesian computation (see abc), where in place of likelihood you use some summary statistics.
As a side note, for using proper Bayesian analysis you don’t need to know the “exact” distri... | Alternatives to Bayesian statistics when distributions are unknown | From what you’re saying is that you want something Bayesian, but you can’t define the likelihood. For such cases there is approximate Bayesian computation (see abc), where in place of likelihood you u | Alternatives to Bayesian statistics when distributions are unknown
From what you’re saying is that you want something Bayesian, but you can’t define the likelihood. For such cases there is approximate Bayesian computation (see abc), where in place of likelihood you use some summary statistics.
As a side note, for using... | Alternatives to Bayesian statistics when distributions are unknown
From what you’re saying is that you want something Bayesian, but you can’t define the likelihood. For such cases there is approximate Bayesian computation (see abc), where in place of likelihood you u |
12,464 | Alternatives to Bayesian statistics when distributions are unknown | If the distribution of the data is unknown, the Bayesian way of handling this uncertainty is to put a prior on it. There exists a huge literature on Bayesian non-parametrics, including the Fundamentals of Nonparametric Bayesian inference by Ghosal and van der Vaart. The default priors in such settings are distributions... | Alternatives to Bayesian statistics when distributions are unknown | If the distribution of the data is unknown, the Bayesian way of handling this uncertainty is to put a prior on it. There exists a huge literature on Bayesian non-parametrics, including the Fundamental | Alternatives to Bayesian statistics when distributions are unknown
If the distribution of the data is unknown, the Bayesian way of handling this uncertainty is to put a prior on it. There exists a huge literature on Bayesian non-parametrics, including the Fundamentals of Nonparametric Bayesian inference by Ghosal and v... | Alternatives to Bayesian statistics when distributions are unknown
If the distribution of the data is unknown, the Bayesian way of handling this uncertainty is to put a prior on it. There exists a huge literature on Bayesian non-parametrics, including the Fundamental |
12,465 | Alternatives to Bayesian statistics when distributions are unknown | Although the other answers are ok, I think they might be overkill for yours or for many other problems.
Without changing the paradigm, if you know the distribution has heavier tails than Gaussian, you can fit t-distribution with either fixed heaviness of tails or with the heaviness of tails that are estimated from the ... | Alternatives to Bayesian statistics when distributions are unknown | Although the other answers are ok, I think they might be overkill for yours or for many other problems.
Without changing the paradigm, if you know the distribution has heavier tails than Gaussian, you | Alternatives to Bayesian statistics when distributions are unknown
Although the other answers are ok, I think they might be overkill for yours or for many other problems.
Without changing the paradigm, if you know the distribution has heavier tails than Gaussian, you can fit t-distribution with either fixed heaviness o... | Alternatives to Bayesian statistics when distributions are unknown
Although the other answers are ok, I think they might be overkill for yours or for many other problems.
Without changing the paradigm, if you know the distribution has heavier tails than Gaussian, you |
12,466 | Alternatives to Bayesian statistics when distributions are unknown | I suggest as alternative bootstrap based methods, which can be parametric and not, and works with very few assumptions and very few data (only 5-7 samples suffice); it works on non gaussian distributions and for many measures (eg., mean, etc). One can "rebuild" a distribution by bootstraping the mean, if it makes sense... | Alternatives to Bayesian statistics when distributions are unknown | I suggest as alternative bootstrap based methods, which can be parametric and not, and works with very few assumptions and very few data (only 5-7 samples suffice); it works on non gaussian distributi | Alternatives to Bayesian statistics when distributions are unknown
I suggest as alternative bootstrap based methods, which can be parametric and not, and works with very few assumptions and very few data (only 5-7 samples suffice); it works on non gaussian distributions and for many measures (eg., mean, etc). One can "... | Alternatives to Bayesian statistics when distributions are unknown
I suggest as alternative bootstrap based methods, which can be parametric and not, and works with very few assumptions and very few data (only 5-7 samples suffice); it works on non gaussian distributi |
12,467 | Why does feature engineering work ? | You question title and the content seems mismatched to me. If you are using linear model, add a total feature in addition to attack and defense will make things worse.
First I would answer why feature engineering work in general.
A picture is worth a thousand words. This figure may tell you some insights on feature eng... | Why does feature engineering work ? | You question title and the content seems mismatched to me. If you are using linear model, add a total feature in addition to attack and defense will make things worse.
First I would answer why feature | Why does feature engineering work ?
You question title and the content seems mismatched to me. If you are using linear model, add a total feature in addition to attack and defense will make things worse.
First I would answer why feature engineering work in general.
A picture is worth a thousand words. This figure may t... | Why does feature engineering work ?
You question title and the content seems mismatched to me. If you are using linear model, add a total feature in addition to attack and defense will make things worse.
First I would answer why feature |
12,468 | Why does feature engineering work ? | The type of model we are using might not be very efficient at learning certain combinations of existing features.
For example, consider your example where features are a and d, and we are using a decision tree to predict a binary outcome that happens to be $0$ if $a+d<0$ and $1$ if $a+d\geq0$.
Since decision trees can ... | Why does feature engineering work ? | The type of model we are using might not be very efficient at learning certain combinations of existing features.
For example, consider your example where features are a and d, and we are using a deci | Why does feature engineering work ?
The type of model we are using might not be very efficient at learning certain combinations of existing features.
For example, consider your example where features are a and d, and we are using a decision tree to predict a binary outcome that happens to be $0$ if $a+d<0$ and $1$ if $... | Why does feature engineering work ?
The type of model we are using might not be very efficient at learning certain combinations of existing features.
For example, consider your example where features are a and d, and we are using a deci |
12,469 | Why does feature engineering work ? | A constructed feature like total can still be predictively useful if it isn't strongly correlated with other features in the same model. total in particular need not be strongly correlated with attack or defense. For example, if attack is (8, 0, 4) and defense is (1, 9, 6), then the correlation of total with attack is ... | Why does feature engineering work ? | A constructed feature like total can still be predictively useful if it isn't strongly correlated with other features in the same model. total in particular need not be strongly correlated with attack | Why does feature engineering work ?
A constructed feature like total can still be predictively useful if it isn't strongly correlated with other features in the same model. total in particular need not be strongly correlated with attack or defense. For example, if attack is (8, 0, 4) and defense is (1, 9, 6), then the ... | Why does feature engineering work ?
A constructed feature like total can still be predictively useful if it isn't strongly correlated with other features in the same model. total in particular need not be strongly correlated with attack |
12,470 | Why does feature engineering work ? | To give a general answer, feature engineering in most cases is about extracting meaningful features from your data, so if you give more information to your model, it obviously should behave better. Say that your data consists of e-mail addresses in form ‘name.surname@domain.country-code’. If you used them as-is in your... | Why does feature engineering work ? | To give a general answer, feature engineering in most cases is about extracting meaningful features from your data, so if you give more information to your model, it obviously should behave better. Sa | Why does feature engineering work ?
To give a general answer, feature engineering in most cases is about extracting meaningful features from your data, so if you give more information to your model, it obviously should behave better. Say that your data consists of e-mail addresses in form ‘name.surname@domain.country-c... | Why does feature engineering work ?
To give a general answer, feature engineering in most cases is about extracting meaningful features from your data, so if you give more information to your model, it obviously should behave better. Sa |
12,471 | Why does feature engineering work ? | What are you trying to accomplish with your "feature" total? If you're merely comparing heroes, attack and defense might be more useful. If you would find the type of build (how offensively-oriented versus how defensively-oriented) to be useful, perhaps attack / defense would be more useful. Or maybe MyAttack - YourDef... | Why does feature engineering work ? | What are you trying to accomplish with your "feature" total? If you're merely comparing heroes, attack and defense might be more useful. If you would find the type of build (how offensively-oriented v | Why does feature engineering work ?
What are you trying to accomplish with your "feature" total? If you're merely comparing heroes, attack and defense might be more useful. If you would find the type of build (how offensively-oriented versus how defensively-oriented) to be useful, perhaps attack / defense would be more... | Why does feature engineering work ?
What are you trying to accomplish with your "feature" total? If you're merely comparing heroes, attack and defense might be more useful. If you would find the type of build (how offensively-oriented v |
12,472 | Paired versus unpaired t-test | I agree with the points that both Frank and Peter make but I think there is a simple formula that gets to the heart of the issue and may be worthwhile for the OP to consider.
Let $X$ and $Y$ be two random variables whose correlation is unknown.
Let $Z=X-Y$
What is the variance of $Z$?
Here is the simple formula:
$$
\t... | Paired versus unpaired t-test | I agree with the points that both Frank and Peter make but I think there is a simple formula that gets to the heart of the issue and may be worthwhile for the OP to consider.
Let $X$ and $Y$ be two ra | Paired versus unpaired t-test
I agree with the points that both Frank and Peter make but I think there is a simple formula that gets to the heart of the issue and may be worthwhile for the OP to consider.
Let $X$ and $Y$ be two random variables whose correlation is unknown.
Let $Z=X-Y$
What is the variance of $Z$?
Her... | Paired versus unpaired t-test
I agree with the points that both Frank and Peter make but I think there is a simple formula that gets to the heart of the issue and may be worthwhile for the OP to consider.
Let $X$ and $Y$ be two ra |
12,473 | Paired versus unpaired t-test | Rather than pairing it is probably better to understand the underlying data model. If pairing is done to deal with uncontrolled heterogeneity, it is usually the case (except in twin studies) that the pairing only partially controls this source of variability and multiple regression would do better. This is because ma... | Paired versus unpaired t-test | Rather than pairing it is probably better to understand the underlying data model. If pairing is done to deal with uncontrolled heterogeneity, it is usually the case (except in twin studies) that the | Paired versus unpaired t-test
Rather than pairing it is probably better to understand the underlying data model. If pairing is done to deal with uncontrolled heterogeneity, it is usually the case (except in twin studies) that the pairing only partially controls this source of variability and multiple regression would ... | Paired versus unpaired t-test
Rather than pairing it is probably better to understand the underlying data model. If pairing is done to deal with uncontrolled heterogeneity, it is usually the case (except in twin studies) that the |
12,474 | Paired versus unpaired t-test | The two tests (paired and unpaired) ask different questions so they can get different answers. Correct pairing nearly always is more powerful than unpaired - that's really the point of pairing. So, since you say the pairing is correct, it is likely that the p-value for your paired test is lower than for the same data u... | Paired versus unpaired t-test | The two tests (paired and unpaired) ask different questions so they can get different answers. Correct pairing nearly always is more powerful than unpaired - that's really the point of pairing. So, si | Paired versus unpaired t-test
The two tests (paired and unpaired) ask different questions so they can get different answers. Correct pairing nearly always is more powerful than unpaired - that's really the point of pairing. So, since you say the pairing is correct, it is likely that the p-value for your paired test is ... | Paired versus unpaired t-test
The two tests (paired and unpaired) ask different questions so they can get different answers. Correct pairing nearly always is more powerful than unpaired - that's really the point of pairing. So, si |
12,475 | Paired versus unpaired t-test | I now understand much better what was worrying me about paired versus unpaired t-tests, and associated p-values. Finding out has been an interesting journey, and there have been many surprises along the way. One surprise has resulted from an investigation of Michael's contribution. This is irreproachable in terms of pr... | Paired versus unpaired t-test | I now understand much better what was worrying me about paired versus unpaired t-tests, and associated p-values. Finding out has been an interesting journey, and there have been many surprises along t | Paired versus unpaired t-test
I now understand much better what was worrying me about paired versus unpaired t-tests, and associated p-values. Finding out has been an interesting journey, and there have been many surprises along the way. One surprise has resulted from an investigation of Michael's contribution. This is... | Paired versus unpaired t-test
I now understand much better what was worrying me about paired versus unpaired t-tests, and associated p-values. Finding out has been an interesting journey, and there have been many surprises along t |
12,476 | Paired versus unpaired t-test | I would offer another perspective. Often, pairing is done do reduce bias. Suppose that you are interested in whether exposure E is a risk factor for a continuous outcome Y. For each E+ subject, you get an age and sex matched subject who is E-. Now, we could do either a paired t-test or an unpaired t-test. I think w... | Paired versus unpaired t-test | I would offer another perspective. Often, pairing is done do reduce bias. Suppose that you are interested in whether exposure E is a risk factor for a continuous outcome Y. For each E+ subject, you | Paired versus unpaired t-test
I would offer another perspective. Often, pairing is done do reduce bias. Suppose that you are interested in whether exposure E is a risk factor for a continuous outcome Y. For each E+ subject, you get an age and sex matched subject who is E-. Now, we could do either a paired t-test or ... | Paired versus unpaired t-test
I would offer another perspective. Often, pairing is done do reduce bias. Suppose that you are interested in whether exposure E is a risk factor for a continuous outcome Y. For each E+ subject, you |
12,477 | Is there a formula for an s-shaped curve with domain and range [0,1]? | Here's one:
$y=\frac{1}{1+\left ( \frac{x}{1-x} \right )^{-\beta}}$
where $\beta$ is $>0$
]2 | Is there a formula for an s-shaped curve with domain and range [0,1]? | Here's one:
$y=\frac{1}{1+\left ( \frac{x}{1-x} \right )^{-\beta}}$
where $\beta$ is $>0$
]2 | Is there a formula for an s-shaped curve with domain and range [0,1]?
Here's one:
$y=\frac{1}{1+\left ( \frac{x}{1-x} \right )^{-\beta}}$
where $\beta$ is $>0$
]2 | Is there a formula for an s-shaped curve with domain and range [0,1]?
Here's one:
$y=\frac{1}{1+\left ( \frac{x}{1-x} \right )^{-\beta}}$
where $\beta$ is $>0$
]2 |
12,478 | Is there a formula for an s-shaped curve with domain and range [0,1]? | What you're trying to use this for is not particularly clear to me so I can't say whether it makes sense to do but fulfilling all your criteria seems to be fairly trivial.
s-shaped curve
parametric function
0 maps to 0, 1 maps to 1, strictly increasing
simple derivative
So why not just take any convenient speci... | Is there a formula for an s-shaped curve with domain and range [0,1]? | What you're trying to use this for is not particularly clear to me so I can't say whether it makes sense to do but fulfilling all your criteria seems to be fairly trivial.
s-shaped curve
parametric | Is there a formula for an s-shaped curve with domain and range [0,1]?
What you're trying to use this for is not particularly clear to me so I can't say whether it makes sense to do but fulfilling all your criteria seems to be fairly trivial.
s-shaped curve
parametric function
0 maps to 0, 1 maps to 1, strictly incr... | Is there a formula for an s-shaped curve with domain and range [0,1]?
What you're trying to use this for is not particularly clear to me so I can't say whether it makes sense to do but fulfilling all your criteria seems to be fairly trivial.
s-shaped curve
parametric |
12,479 | Is there a formula for an s-shaped curve with domain and range [0,1]? | As already commented by @whuber the function $ f(x)=x $ satisfies the three requirements you mentioned (i.e. 0 maps to 0, 1 maps to 1 and the function is strictly increasing). In the title of your question, you seem to indicate that you are also interested in the function being S-shaped, as in Sigmoid/Logistic curve. ... | Is there a formula for an s-shaped curve with domain and range [0,1]? | As already commented by @whuber the function $ f(x)=x $ satisfies the three requirements you mentioned (i.e. 0 maps to 0, 1 maps to 1 and the function is strictly increasing). In the title of your qu | Is there a formula for an s-shaped curve with domain and range [0,1]?
As already commented by @whuber the function $ f(x)=x $ satisfies the three requirements you mentioned (i.e. 0 maps to 0, 1 maps to 1 and the function is strictly increasing). In the title of your question, you seem to indicate that you are also int... | Is there a formula for an s-shaped curve with domain and range [0,1]?
As already commented by @whuber the function $ f(x)=x $ satisfies the three requirements you mentioned (i.e. 0 maps to 0, 1 maps to 1 and the function is strictly increasing). In the title of your qu |
12,480 | Is there a formula for an s-shaped curve with domain and range [0,1]? | Let me offer the most general solution consistent with the requirements: that will give you the most flexibility to choose and optimize.
We may interpret "S-shaped" as a monotonically increasing curve (because the transformation ought to be one-to-one) consisting of one part that is concave upwards and another part tha... | Is there a formula for an s-shaped curve with domain and range [0,1]? | Let me offer the most general solution consistent with the requirements: that will give you the most flexibility to choose and optimize.
We may interpret "S-shaped" as a monotonically increasing curve | Is there a formula for an s-shaped curve with domain and range [0,1]?
Let me offer the most general solution consistent with the requirements: that will give you the most flexibility to choose and optimize.
We may interpret "S-shaped" as a monotonically increasing curve (because the transformation ought to be one-to-on... | Is there a formula for an s-shaped curve with domain and range [0,1]?
Let me offer the most general solution consistent with the requirements: that will give you the most flexibility to choose and optimize.
We may interpret "S-shaped" as a monotonically increasing curve |
12,481 | What does " Y axis" mean in continuous probability distribution? | There are two common ways to represent a probability distribution, the probability density function (PDF) and cumulative distribution function (CDF). I suspect you're wondering most about the former. For the latter, the distribution is plotted as cumulative from zero to one, so the y-axis is the sum of the distribution... | What does " Y axis" mean in continuous probability distribution? | There are two common ways to represent a probability distribution, the probability density function (PDF) and cumulative distribution function (CDF). I suspect you're wondering most about the former. | What does " Y axis" mean in continuous probability distribution?
There are two common ways to represent a probability distribution, the probability density function (PDF) and cumulative distribution function (CDF). I suspect you're wondering most about the former. For the latter, the distribution is plotted as cumulati... | What does " Y axis" mean in continuous probability distribution?
There are two common ways to represent a probability distribution, the probability density function (PDF) and cumulative distribution function (CDF). I suspect you're wondering most about the former. |
12,482 | What does " Y axis" mean in continuous probability distribution? | I suppose you have a moderately large or large random sample from a continuous distribution, and that you want to make a plot of the data
that suggests the shape of the population distribution.
Then a starting point would be to make a 'density' histogram in which the total area of all bars adds to unity $(1).$ [Similar... | What does " Y axis" mean in continuous probability distribution? | I suppose you have a moderately large or large random sample from a continuous distribution, and that you want to make a plot of the data
that suggests the shape of the population distribution.
Then a | What does " Y axis" mean in continuous probability distribution?
I suppose you have a moderately large or large random sample from a continuous distribution, and that you want to make a plot of the data
that suggests the shape of the population distribution.
Then a starting point would be to make a 'density' histogram ... | What does " Y axis" mean in continuous probability distribution?
I suppose you have a moderately large or large random sample from a continuous distribution, and that you want to make a plot of the data
that suggests the shape of the population distribution.
Then a |
12,483 | What does " Y axis" mean in continuous probability distribution? | Probability density is a density, and may be understood as such.
Although this way of thinking is touched on in other answers, and at greater length in other threads, I find it helpful when trying to teach the topic, and to build on what people should already know about density generally and indeed long since.
Thus in ... | What does " Y axis" mean in continuous probability distribution? | Probability density is a density, and may be understood as such.
Although this way of thinking is touched on in other answers, and at greater length in other threads, I find it helpful when trying to | What does " Y axis" mean in continuous probability distribution?
Probability density is a density, and may be understood as such.
Although this way of thinking is touched on in other answers, and at greater length in other threads, I find it helpful when trying to teach the topic, and to build on what people should alr... | What does " Y axis" mean in continuous probability distribution?
Probability density is a density, and may be understood as such.
Although this way of thinking is touched on in other answers, and at greater length in other threads, I find it helpful when trying to |
12,484 | What does " Y axis" mean in continuous probability distribution? | Just to make it clear with an equation, the probability density function (PDF) $f_X(x)$ of a random variable $X$ is defined as:
$$ dP_X(x) \equiv f_X(x) dx,$$
where $dP_X(x)$ is the infinitesimal probability that the random variable $X$ takes the value $x$ and $dx$ is a differential of the random variable $X$. In other... | What does " Y axis" mean in continuous probability distribution? | Just to make it clear with an equation, the probability density function (PDF) $f_X(x)$ of a random variable $X$ is defined as:
$$ dP_X(x) \equiv f_X(x) dx,$$
where $dP_X(x)$ is the infinitesimal prob | What does " Y axis" mean in continuous probability distribution?
Just to make it clear with an equation, the probability density function (PDF) $f_X(x)$ of a random variable $X$ is defined as:
$$ dP_X(x) \equiv f_X(x) dx,$$
where $dP_X(x)$ is the infinitesimal probability that the random variable $X$ takes the value $x... | What does " Y axis" mean in continuous probability distribution?
Just to make it clear with an equation, the probability density function (PDF) $f_X(x)$ of a random variable $X$ is defined as:
$$ dP_X(x) \equiv f_X(x) dx,$$
where $dP_X(x)$ is the infinitesimal prob |
12,485 | What does " Y axis" mean in continuous probability distribution? | PDF is the derivative of CDF, i.e., the rate of CDF's change, just like speed is the derivative of moving distance. When you are driving a car, at any particular time moment the passing distance is zero, but the speed (i.e. the rate of the distance's change) is not zero. Do you think that speed is a meaningful measure ... | What does " Y axis" mean in continuous probability distribution? | PDF is the derivative of CDF, i.e., the rate of CDF's change, just like speed is the derivative of moving distance. When you are driving a car, at any particular time moment the passing distance is ze | What does " Y axis" mean in continuous probability distribution?
PDF is the derivative of CDF, i.e., the rate of CDF's change, just like speed is the derivative of moving distance. When you are driving a car, at any particular time moment the passing distance is zero, but the speed (i.e. the rate of the distance's chan... | What does " Y axis" mean in continuous probability distribution?
PDF is the derivative of CDF, i.e., the rate of CDF's change, just like speed is the derivative of moving distance. When you are driving a car, at any particular time moment the passing distance is ze |
12,486 | PCA and k-fold cross-validation in caret package in R | I didn't see the lecture, so I can't comment on what was said.
My $0.02: If you want to get good estimates of performance using resampling, you should really do all of the operations during resampling instead of prior. This is really true of feature selection [1] as well as non-trivial operations like PCA. If it adds ... | PCA and k-fold cross-validation in caret package in R | I didn't see the lecture, so I can't comment on what was said.
My $0.02: If you want to get good estimates of performance using resampling, you should really do all of the operations during resamplin | PCA and k-fold cross-validation in caret package in R
I didn't see the lecture, so I can't comment on what was said.
My $0.02: If you want to get good estimates of performance using resampling, you should really do all of the operations during resampling instead of prior. This is really true of feature selection [1] a... | PCA and k-fold cross-validation in caret package in R
I didn't see the lecture, so I can't comment on what was said.
My $0.02: If you want to get good estimates of performance using resampling, you should really do all of the operations during resamplin |
12,487 | PCA and k-fold cross-validation in caret package in R | Don't preProcess the data prior to running the train function! Use the preProcess argument for the train function, and the pre-processing will be applied to each re-sampling iteration.
e.g. don't do this:
library(caret)
dat <- iris
pp <- preProcess(dat[,-5], method="pca")
dat[,-5] <- predict(pp, dat[,-5])
knnFit1 <- ... | PCA and k-fold cross-validation in caret package in R | Don't preProcess the data prior to running the train function! Use the preProcess argument for the train function, and the pre-processing will be applied to each re-sampling iteration.
e.g. don't do | PCA and k-fold cross-validation in caret package in R
Don't preProcess the data prior to running the train function! Use the preProcess argument for the train function, and the pre-processing will be applied to each re-sampling iteration.
e.g. don't do this:
library(caret)
dat <- iris
pp <- preProcess(dat[,-5], metho... | PCA and k-fold cross-validation in caret package in R
Don't preProcess the data prior to running the train function! Use the preProcess argument for the train function, and the pre-processing will be applied to each re-sampling iteration.
e.g. don't do |
12,488 | Does random forest need input variables to be scaled or centered? | No.
Random Forests are based on tree partitioning algorithms.
As such, there's no analogue to a coefficient one obtain in general regression strategies, which would depend on the units of the independent variables. Instead, one obtain a collection of partition rules, basically a decision given a threshold, and this sho... | Does random forest need input variables to be scaled or centered? | No.
Random Forests are based on tree partitioning algorithms.
As such, there's no analogue to a coefficient one obtain in general regression strategies, which would depend on the units of the independ | Does random forest need input variables to be scaled or centered?
No.
Random Forests are based on tree partitioning algorithms.
As such, there's no analogue to a coefficient one obtain in general regression strategies, which would depend on the units of the independent variables. Instead, one obtain a collection of par... | Does random forest need input variables to be scaled or centered?
No.
Random Forests are based on tree partitioning algorithms.
As such, there's no analogue to a coefficient one obtain in general regression strategies, which would depend on the units of the independ |
12,489 | Does random forest need input variables to be scaled or centered? | Overall I agree with Firebug, but there could be some value in standardizing your variables if you're interested in predictor importance scores. RF will tend to favour highly variable continuous predictors because there are more opportunities to partition the data. A better way to deal with this issue, however, is to u... | Does random forest need input variables to be scaled or centered? | Overall I agree with Firebug, but there could be some value in standardizing your variables if you're interested in predictor importance scores. RF will tend to favour highly variable continuous predi | Does random forest need input variables to be scaled or centered?
Overall I agree with Firebug, but there could be some value in standardizing your variables if you're interested in predictor importance scores. RF will tend to favour highly variable continuous predictors because there are more opportunities to partitio... | Does random forest need input variables to be scaled or centered?
Overall I agree with Firebug, but there could be some value in standardizing your variables if you're interested in predictor importance scores. RF will tend to favour highly variable continuous predi |
12,490 | Regression in $p>n$ setting: how to choose regularization method (Lasso, PLS, PCR, ridge)? | I think there is no single answer to your question - it depends upon many situation, data and what you are trying to do. Some of the modification can be or should be modified to achieve the goal. However the following general discussion can help.
Before jumping to into the more advanced methods let's discussion of ba... | Regression in $p>n$ setting: how to choose regularization method (Lasso, PLS, PCR, ridge)? | I think there is no single answer to your question - it depends upon many situation, data and what you are trying to do. Some of the modification can be or should be modified to achieve the goal. How | Regression in $p>n$ setting: how to choose regularization method (Lasso, PLS, PCR, ridge)?
I think there is no single answer to your question - it depends upon many situation, data and what you are trying to do. Some of the modification can be or should be modified to achieve the goal. However the following general di... | Regression in $p>n$ setting: how to choose regularization method (Lasso, PLS, PCR, ridge)?
I think there is no single answer to your question - it depends upon many situation, data and what you are trying to do. Some of the modification can be or should be modified to achieve the goal. How |
12,491 | Two negative main effects yet positive interaction effect? | This situation is certainly possible. As a simple example, consider an experiment where you are adding certain volumes of hot (V1) and cold (V2) water to a fish tank that begins at the correct temperature. The response variable (V3) is the number of fish that survive after a day. Intuitively, if you add only hot wate... | Two negative main effects yet positive interaction effect? | This situation is certainly possible. As a simple example, consider an experiment where you are adding certain volumes of hot (V1) and cold (V2) water to a fish tank that begins at the correct temper | Two negative main effects yet positive interaction effect?
This situation is certainly possible. As a simple example, consider an experiment where you are adding certain volumes of hot (V1) and cold (V2) water to a fish tank that begins at the correct temperature. The response variable (V3) is the number of fish that ... | Two negative main effects yet positive interaction effect?
This situation is certainly possible. As a simple example, consider an experiment where you are adding certain volumes of hot (V1) and cold (V2) water to a fish tank that begins at the correct temper |
12,492 | Two negative main effects yet positive interaction effect? | An alternative way of looking at the situation to @underminer's brilliant example is to note that under least squares regression, your fitted values satisfy "correlation constraints"
$$\sum_{ i=1}^nx_{ik}\hat{y}_i=\sum_{ i=1}^nx_{ik}y_i$$
Where $ x_{ik} $ is the value of the kth (independent/explanatory/predictor/etc) ... | Two negative main effects yet positive interaction effect? | An alternative way of looking at the situation to @underminer's brilliant example is to note that under least squares regression, your fitted values satisfy "correlation constraints"
$$\sum_{ i=1}^nx_ | Two negative main effects yet positive interaction effect?
An alternative way of looking at the situation to @underminer's brilliant example is to note that under least squares regression, your fitted values satisfy "correlation constraints"
$$\sum_{ i=1}^nx_{ik}\hat{y}_i=\sum_{ i=1}^nx_{ik}y_i$$
Where $ x_{ik} $ is th... | Two negative main effects yet positive interaction effect?
An alternative way of looking at the situation to @underminer's brilliant example is to note that under least squares regression, your fitted values satisfy "correlation constraints"
$$\sum_{ i=1}^nx_ |
12,493 | How does the standard error work? | Yes, the standard error of the mean (SEM) is the standard deviation (SD) of the means. (Standard error is another way to say SD of a sampling distribution. In this case, the sampling distribution is means for samples of a fixed size, say N.) There is a mathematical relationship between the SEM and the population SD: ... | How does the standard error work? | Yes, the standard error of the mean (SEM) is the standard deviation (SD) of the means. (Standard error is another way to say SD of a sampling distribution. In this case, the sampling distribution is | How does the standard error work?
Yes, the standard error of the mean (SEM) is the standard deviation (SD) of the means. (Standard error is another way to say SD of a sampling distribution. In this case, the sampling distribution is means for samples of a fixed size, say N.) There is a mathematical relationship betwe... | How does the standard error work?
Yes, the standard error of the mean (SEM) is the standard deviation (SD) of the means. (Standard error is another way to say SD of a sampling distribution. In this case, the sampling distribution is |
12,494 | How does the standard error work? | Suppose $X_1, X_2, \ldots, X_n$ are independent and identically distributed. This is the situation I am pretty sure you are referring to. Let their common mean be $\mu$ and their common variance be $\sigma^2$.
Now the sample mean is $X_b=\sum_i X_i/n$. Linearity of expectation shows that the mean of $X_b$ is also $\... | How does the standard error work? | Suppose $X_1, X_2, \ldots, X_n$ are independent and identically distributed. This is the situation I am pretty sure you are referring to. Let their common mean be $\mu$ and their common variance be $ | How does the standard error work?
Suppose $X_1, X_2, \ldots, X_n$ are independent and identically distributed. This is the situation I am pretty sure you are referring to. Let their common mean be $\mu$ and their common variance be $\sigma^2$.
Now the sample mean is $X_b=\sum_i X_i/n$. Linearity of expectation shows... | How does the standard error work?
Suppose $X_1, X_2, \ldots, X_n$ are independent and identically distributed. This is the situation I am pretty sure you are referring to. Let their common mean be $\mu$ and their common variance be $ |
12,495 | How does the standard error work? | +1 to both @JoelW. & @MichaelChernick. I want to add a detail to @JoelW.'s answer. He notes that "we almost never have a direct estimate of the SEM", which is essentially true, but it's worth explicitly recognizing a caveat to that statement. Specifically, when a study compares multiple groups / treatments (for exam... | How does the standard error work? | +1 to both @JoelW. & @MichaelChernick. I want to add a detail to @JoelW.'s answer. He notes that "we almost never have a direct estimate of the SEM", which is essentially true, but it's worth explic | How does the standard error work?
+1 to both @JoelW. & @MichaelChernick. I want to add a detail to @JoelW.'s answer. He notes that "we almost never have a direct estimate of the SEM", which is essentially true, but it's worth explicitly recognizing a caveat to that statement. Specifically, when a study compares mult... | How does the standard error work?
+1 to both @JoelW. & @MichaelChernick. I want to add a detail to @JoelW.'s answer. He notes that "we almost never have a direct estimate of the SEM", which is essentially true, but it's worth explic |
12,496 | Typical set concept | I know you have explicitly asked for an intuitive explanation and to leave out the formal definition, but I think they are rather related, so let me recall the definition of typical set:
$X_1, X_2 ,... $ are i.i.d. random variables $\sim $ $p(x)$ then the typical set $A_\epsilon^{(n)} $ with respect to $p(x)$ is the se... | Typical set concept | I know you have explicitly asked for an intuitive explanation and to leave out the formal definition, but I think they are rather related, so let me recall the definition of typical set:
$X_1, X_2 ,.. | Typical set concept
I know you have explicitly asked for an intuitive explanation and to leave out the formal definition, but I think they are rather related, so let me recall the definition of typical set:
$X_1, X_2 ,... $ are i.i.d. random variables $\sim $ $p(x)$ then the typical set $A_\epsilon^{(n)} $ with respect... | Typical set concept
I know you have explicitly asked for an intuitive explanation and to leave out the formal definition, but I think they are rather related, so let me recall the definition of typical set:
$X_1, X_2 ,.. |
12,497 | Typical set concept | Diegobatt's answer does a good job of explaining intuitively what the typical set is. This answer will address the OP's other question, echoed by @tomwesolowski: why would you define the typical set in a way that can exclude the most probable elements?
The short answer is that the typical set is primarily a theoretical... | Typical set concept | Diegobatt's answer does a good job of explaining intuitively what the typical set is. This answer will address the OP's other question, echoed by @tomwesolowski: why would you define the typical set i | Typical set concept
Diegobatt's answer does a good job of explaining intuitively what the typical set is. This answer will address the OP's other question, echoed by @tomwesolowski: why would you define the typical set in a way that can exclude the most probable elements?
The short answer is that the typical set is pri... | Typical set concept
Diegobatt's answer does a good job of explaining intuitively what the typical set is. This answer will address the OP's other question, echoed by @tomwesolowski: why would you define the typical set i |
12,498 | Typical set concept | According to theorem 6.3 in these lecture notes no matter if we take subset of sequences with highest probability or those with probability close to $2^{-nH(X)}$ (from typical set) we have to take approximately $2^{nH}$ to make sure that chosen subset contains random sequence with high probability. We usually take typi... | Typical set concept | According to theorem 6.3 in these lecture notes no matter if we take subset of sequences with highest probability or those with probability close to $2^{-nH(X)}$ (from typical set) we have to take app | Typical set concept
According to theorem 6.3 in these lecture notes no matter if we take subset of sequences with highest probability or those with probability close to $2^{-nH(X)}$ (from typical set) we have to take approximately $2^{nH}$ to make sure that chosen subset contains random sequence with high probability. ... | Typical set concept
According to theorem 6.3 in these lecture notes no matter if we take subset of sequences with highest probability or those with probability close to $2^{-nH(X)}$ (from typical set) we have to take app |
12,499 | Typical set concept | The idea of a typical set implicitly treats the outcome sequences as multisets, ie it assumes you just care about the histogram of each sequence, eg you consider all 10 coin toss sequences with 7 heads and 3 tails as equivalent.
Imagine you have a very biased coin, say $p(H) = .9$. This is just the binomial distributio... | Typical set concept | The idea of a typical set implicitly treats the outcome sequences as multisets, ie it assumes you just care about the histogram of each sequence, eg you consider all 10 coin toss sequences with 7 head | Typical set concept
The idea of a typical set implicitly treats the outcome sequences as multisets, ie it assumes you just care about the histogram of each sequence, eg you consider all 10 coin toss sequences with 7 heads and 3 tails as equivalent.
Imagine you have a very biased coin, say $p(H) = .9$. This is just the ... | Typical set concept
The idea of a typical set implicitly treats the outcome sequences as multisets, ie it assumes you just care about the histogram of each sequence, eg you consider all 10 coin toss sequences with 7 head |
12,500 | Typical set concept | Consider a biased coin that has a 60% probability of landing heads. If this coin is flipped 100 times, the most likely sequence would be 100 heads, but that isn't "typical". Typical sequences would have about 60 heads. | Typical set concept | Consider a biased coin that has a 60% probability of landing heads. If this coin is flipped 100 times, the most likely sequence would be 100 heads, but that isn't "typical". Typical sequences would ha | Typical set concept
Consider a biased coin that has a 60% probability of landing heads. If this coin is flipped 100 times, the most likely sequence would be 100 heads, but that isn't "typical". Typical sequences would have about 60 heads. | Typical set concept
Consider a biased coin that has a 60% probability of landing heads. If this coin is flipped 100 times, the most likely sequence would be 100 heads, but that isn't "typical". Typical sequences would ha |
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