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bayesian inference | Probability measures at play in Bayesian inference | https://stats.stackexchange.com/questions/230252/probability-measures-at-play-in-bayesian-inference | <p>This might be a purely notational, but I'm confused about the probability measures at play when using Bayesian inference. It's sufficient to focus on the numerator here. Let's assume that I have a prior over hypotheses $P(H)$ and that these hypotheses are themselves distributions about some data. When some data $D$ ... | <p>First, note that a Bayesian model is a joint probability distribution of all unknowns modeled as random variables. Or, if you use Bayes rule in other contexts, it still requires all random variables / events to be defined in the same joint probability space. The $P$s or $p$s then refer to marginal and conditional pr... | 200 |
bayesian inference | Inferring sample size from proportions | https://stats.stackexchange.com/questions/257998/inferring-sample-size-from-proportions | <p>How would one infer the number of people that took a test based on the percentages of people that got particular questions correctly. </p>
<p>For example
<code>
1. 85%
2. 25%
3. 95%
4. 15%
5. 35%
</code>
$ n = 20 $</p>
<p>A caveat is that these percentages actually come with some noise, therefore you cannot be sur... | 201 | |
bayesian inference | How do I calculate a posterior distribution for a Poisson model with exponential prior distribution for the parameter? | https://stats.stackexchange.com/questions/26199/how-do-i-calculate-a-posterior-distribution-for-a-poisson-model-with-exponential | <p>Suppose:</p>
<ul>
<li>$N \sim {\rm Poisson}(\lambda)$</li>
<li>$\lambda$ is unknown, but we believe that it can be assumed $\sim \exp(1)$</li>
</ul>
<p>If I want to calculate $N | X$, i.e., $P(model | data)$, I need to use the Bayes theorem in the following way:</p>
<p>$P(model|data) \propto P(data|model)*P(model... | <p><span class="math-container">$\Pr(\text{data}|\text{model}) =\Pr(N=n|\lambda) = \frac{\lambda^n}{n!}e^{-\lambda}$</span>.</p>
<p><span class="math-container">$p(\text{model}) = p(\lambda) = e^{-\lambda}$</span>.</p>
<p><span class="math-container">$p(\lambda|N=n) = \dfrac{\frac{\lambda^n}{n!}e^{-\lambda}\cdot e^{-\l... | 202 |
bayesian inference | Should Bayesian inference be avoided with a small sample size and weakly informative priors? | https://stats.stackexchange.com/questions/621780/should-bayesian-inference-be-avoided-with-a-small-sample-size-and-weakly-informa | <p>When specifying a Bayesian model, one can specify weakly-informative priors. However, such priors may represent a concern to many researchers. After all, if they are weakly-informed, one may be concerned that the imprecision of such a prior may be biasing the results. It is also my understanding that the influence o... | <p>Rather than "Am I using a prior that is strong or weak?" (in the sense of concentrated or diffuse over the parameter space), it depends on why you are using that particular prior. Do you actually trust your prior, or was it chosen arbitrarily / for convenience?</p>
<p>On the one hand, we may have a well-ju... | 203 |
bayesian inference | Bayesian approach on games of chance with physical devices | https://stats.stackexchange.com/questions/71834/bayesian-approach-on-games-of-chance-with-physical-devices | <p>Suppose we alter side 6 of a die to appear more than 1/6th of the time. We do not know the actual proportion of the time each side of the die will appear because all or some of the other 5 sides do not have the 1/6 proportion too. What do we need to be 99% confidence of each of the 6 sides of this altered die? How d... | 204 | |
bayesian inference | Bayesian inference adaptable to not mutually exlclusive events? | https://stats.stackexchange.com/questions/76229/bayesian-inference-adaptable-to-not-mutually-exlclusive-events | <p>Say I have $n$ possible events that lead to $m$ observable effects.</p>
<p>The Bayesian inference hypothesis is that events are mutually exclusive and jointly exhaustive. Could I still be able to use Bayesian inference (modified, eventually) when more than one event occurs at some observation?</p>
<p>Real example:... | 205 | |
bayesian inference | Finding optimal parameter values using a Bayesian model | https://stats.stackexchange.com/questions/95081/finding-optimal-parameter-values-using-a-bayesian-model | <p>I have a problem with the following setup. I've been reading <a href="http://www.indiana.edu/~kruschke/DoingBayesianDataAnalysis/" rel="nofollow">"Doing Bayesian Data Analysis: A Tutorial with R and BUGS"</a> and it seems like the Bayesian approach is a good one, but I'm not entirely sure how to model it.</p>
<p>I... | 206 | |
bayesian inference | Marginal posterior and prior are similar (and flat!) | https://stats.stackexchange.com/questions/49496/marginal-posterior-and-prior-are-similar-and-flat | <p>I designed a Bayesian model and sampled the posterior using a MCMC algorithm.
My problem is that the posterior marginal distribution of a given latent intermediate variable appears to be uniform just as the prior I assigned to it. In practice this variable is supposed to have a substential importance on the model. ... | <p>When the likelihood surface contains flat ridges related to the parameters of interest, the prior information has a substantial impact on the shape of the corresponding posterior distribution. In these cases, it is important to employ reasonable priors since they will basically drive the inference. Therefore, if the... | 207 |
bayesian inference | Bayesian combination of expert opinion | https://stats.stackexchange.com/questions/535101/bayesian-combination-of-expert-opinion | <p>In a population of <span class="math-container">$N$</span>, <span class="math-container">$K$</span> experts pick <span class="math-container">$M_{k\in\{1, ..., K\}}$</span> individuals that will have a certain attribute. Note that <span class="math-container">$M$</span> can be different across experts (e.g., one exp... | <p>I understand that you worry that you don't know the historical performance of the experts, so you don't know how reliable are their responses. It's a valid worry, but think of it in terms of <a href="https://en.wikipedia.org/wiki/Wisdom_of_the_crowd" rel="nofollow noreferrer">wisdom of a crowd</a>, their aggregate o... | 208 |
bayesian inference | How do you deal with a "multiple choice" observation in Bayesian inference, when the choices are on a scale? | https://stats.stackexchange.com/questions/13352/how-do-you-deal-with-a-multiple-choice-observation-in-bayesian-inference-when | <p>Suppose I have a questionnaire and I ask respondents how often they eat at McDonalds:</p>
<ol>
<li>Never</li>
<li>Less than once a month</li>
<li>At least once a month but less than once a week</li>
<li>1-3 times a week</li>
<li>More than 3 times a week</li>
</ol>
<p>I then correlate these answers with whether the... | <p>Are you being confused by the correlation / regression distinction perhaps? When you say 'I correlate these answers' with shoe colour, you could mean either</p>
<ol>
<li>'I compute the conditional probability of shoe colour given eating habits', or</li>
<li>'I compute a measure of linear relatedness underlying the... | 209 |
bayesian inference | Bayesian Interval Estimates for Multinomial Probabilities | https://stats.stackexchange.com/questions/339357/bayesian-interval-estimates-for-multinomial-probabilities | <p>Can anyone point me to a reference for calculating Bayesian interval estimates for multinomial probabilities? I am familiar with conventional methods (i.e.: Quesenberry and Hurst (1964), Goodman (1965), Bailey (1980), Fitzpatrick and Scott (1987), and Glaz and Sison (1999)). I have found methods to calculate inter... | 210 | |
bayesian inference | Effect of event on average probabilities given different base rates | https://stats.stackexchange.com/questions/440155/effect-of-event-on-average-probabilities-given-different-base-rates | <p>I am trying to solve the following question with my very rusty stats expertise:</p>
<p>I have a data set of people of which some do exercise with different frequencies per month and other don’t exercise at all (base rates of exercise). My data contains all the dates when each person did exercise.
At some known poi... | <p>What you're looking for can be formulated as:</p>
<p><span class="math-container">$$Imp := \frac
{p(E|P,D=1)}
{p(E|P,D=0)}
$$</span></p>
<p>Where <span class="math-container">$Imp$</span> is Impact of the recommendation, <span class="math-container">$E$</span> is Exercising (to a minimum acceptable level at least)... | 211 |
bayesian inference | Problem faced by Bayes when developing his method for Bayesian Inference | https://stats.stackexchange.com/questions/519436/problem-faced-by-bayes-when-developing-his-method-for-bayesian-inference | <p>I am reading <em>Principles of Statistics</em> (MG Bulmer, 1965) and stumbled upon the problem that Bayes considered when developing his theorem. Bulmer makes use of <span class="math-container">$dP$</span> and I have no idea what it means. In his words:</p>
<blockquote>
<p>The problem that Bayes himself considered ... | 212 | |
bayesian inference | Bayesian updating of a constant probability using one data point | https://stats.stackexchange.com/questions/544034/bayesian-updating-of-a-constant-probability-using-one-data-point | <p>A reformulation of a question that came up in a model:</p>
<p>Imagine a toy store that sells <span class="math-container">$K$</span> toys, where our prior is that each toy has equal probability <span class="math-container">$1/K$</span> of being purchased by a customer. Then you have a customer come in and buy a tedd... | <p>This is how I read your question: when a customer comes to a store, they always buy one, and only one, of <span class="math-container">$K$</span> items. There’s an infinite supply of the items (i.e. they are sampled with replacement). With no other prior knowledge, you assume a prior probability for picking any of t... | 213 |
bayesian inference | Intuition behind posterior predictive distribution | https://stats.stackexchange.com/questions/438218/intuition-behind-posterior-predictive-distribution | <p>I've recently encountered the "posterior predictive distribution" <span class="math-container">$$p(\bar{x}|X)=E_\theta[p(\bar{x}|\theta)]=\int_\theta p(\bar{x}|\theta)\hspace{0.5mm}p(\theta|X)d\theta$$</span>
where <span class="math-container">$\bar{x}$</span> is a new point, <span class="math-container">$\theta$</s... | <p>Let <span class="math-container">$X$</span> denotes the <em>observations</em> and <span class="math-container">$\theta \in \Theta$</span> the <em>parameter</em>. In a Bayesian approach, both are considered random quantities.
The first step of <em>modeling</em> is to define a statistical model, i.e. the distribution... | 214 |
bayesian inference | Is it possible to infer both prior and posterior simultaneously? | https://stats.stackexchange.com/questions/313768/is-it-possible-to-infer-both-prior-and-posterior-simultaneously | <p>It seems that most Bayesian inference focuses on inferring the posterior. Is it possible to infer both the prior and the posterior.</p>
| <p>Your question is ill-posed, it doesn't make sense to "infer" a prior. </p>
<p>Let's say you have a likelihood $p(x|\theta)$, where $x$ is the data and $\theta$ are some parameters. In bayesian inference, the objective is to find the distribution of the parameters given the data, $p(\theta|x)$, which is the posterio... | 215 |
bayesian inference | Bayesian inference on a sum of iid real-valued random variables | https://stats.stackexchange.com/questions/24344/bayesian-inference-on-a-sum-of-iid-real-valued-random-variables | <p>Let $X_1$, $X_2$, ..., $X_n$ be iid RV's with range $[0,1]$ but
unknown distribution. (I'm OK with assuming that the distribution
is continuous, etc., if necessary.)</p>
<p>Define $S_n = X_1 + \cdots + X_n$.</p>
<p>I am given $S_k$, and ask: What can I infer, in a Bayesian manner, about
$S_n$?</p>
<p>That is, I ... | <p>Consider the following Bayesian nonparametric analysis.</p>
<p>Define $\mathscr{X}=[0,1]$ and let $\mathscr{B}$ be the Borel subsets of $\mathscr{X}$. Let $\alpha$ be a nonzero finite measure over $(\mathscr{X},\mathscr{B})$.</p>
<p>Let $Q$ be a Dirichlet process with parameter $\alpha$, and suppose that $X_1,\dot... | 216 |
bayesian inference | Bayesian inferential target | https://stats.stackexchange.com/questions/250793/bayesian-inferential-target | <p>In frequentist, i.e., sampling-based statistics, we envision a target population to which inference is made. Notwithstanding the fact that our so-called random samples from this population are usually more convenience-based samples, we try to infer from a sample to the population. For example in a randomized clini... | <p>"What is the exact statement of what we are inferring? Is it that the treatment was actually effective in the group of patients we analyzed? Some deeper inference?"</p>
<p>I think you are confusing the terms of art with the discussion. One of the challenges of talking about things in multiple paradigms is that the... | 217 |
bayesian inference | Bayesian counterpart to parameter estimate precision | https://stats.stackexchange.com/questions/333749/bayesian-counterpart-to-parameter-estimate-precision | <p>In maximum likelihood theory it is common to summarise parameter estimates by their maxium likelihood estimate $\theta_{\mathrm{MLE}}$ and the corresponding standard error $\sigma_{\mathrm{MLE}}$ or coefficient of variation
$$CV = \frac{\sigma_{\mathrm{MLE}}}{\theta_{\mathrm{MLE}}}.$$ This works since we assume tha... | <p>Given that $\theta_{\text{MLE}}$ is a point estimator, the obvious counterpart in Bayesian analysis would be the <a href="https://en.wikipedia.org/wiki/Maximum_a_posteriori_estimation" rel="nofollow noreferrer">posterior mode estimator</a> $\theta_{\text{MAP}} \equiv \arg \max_\theta f( \theta | x )$. Although this... | 218 |
bayesian inference | Does the posterior necessarily follow the same conditional dependence structure as the prior? | https://stats.stackexchange.com/questions/414045/does-the-posterior-necessarily-follow-the-same-conditional-dependence-structure | <p>One of the assumptions in a model is the conditional dependence between random variables in the joint prior distribution. Consider the following model,
<span class="math-container">$$p(a,b|X) \propto p(X|a,b)p(a,b)$$</span></p>
<p>Now suppose an independence assumption for the prior <span class="math-container">$p(... | <p>Your question can also be stated as: "<span class="math-container">$X$</span> is dependent on <span class="math-container">$a$</span> and <span class="math-container">$b$</span>. And <span class="math-container">$a$</span> and <span class="math-container">$b$</span> are independent. Does this imply that <span class=... | 219 |
bayesian inference | Probability of hitting X shots in N tries knowing that the P(hit) is the ratio of previous hits | https://stats.stackexchange.com/questions/592969/probability-of-hitting-x-shots-in-n-tries-knowing-that-the-phit-is-the-ratio-o | <p>Let <span class="math-container">$X$</span> be the number of hits in <span class="math-container">$N$</span> tries, I know that the probability of the next hit is <span class="math-container">$P(\text{Hit}) =X/N$</span>.</p>
<p>How can I get the generic expression for the probability distribution function of hitting... | <p>"There were 3 hits and 3 misses, and so the probability of the 7th shot hitting is 0.5. " Do you mean: "There were 3 hits and 3 misses, and so <em>my estimate of</em> the probability of the 7th shot hitting is 0.5. "</p>
<p>Some more information is needed before your question can be answered. Fir... | 220 |
bayesian inference | Bayesian inference from "extra" information - Beta-binomial case | https://stats.stackexchange.com/questions/541233/bayesian-inference-from-extra-information-beta-binomial-case | <p>Say we have two coins with unknown success probabilities <span class="math-container">$p_1$</span> and <span class="math-container">$p_2$</span>. To know more about the probabilities, say that we use Bayesian approach.</p>
<p>To do so, we first set our prior: <span class="math-container">$P_1\sim Beta(1,1)$</span> a... | <p>What you're referring to in the first part of the question, is the <a href="https://stats.stackexchange.com/questions/47771/what-is-the-intuition-behind-beta-distribution/47782#47782">beta-binomial model</a>. where binomial distribution is assumed as the likelihood and beta as a prior, hence by conjugacy posterior i... | 221 |
bayesian inference | why does posterior prediction involve integration over all parameter space? | https://stats.stackexchange.com/questions/608478/why-does-posterior-prediction-involve-integration-over-all-parameter-space | <p>The primary objective of Bayesian inference is to compute the posterior.</p>
<p>For instance, if the posterior <span class="math-container">$p(\theta | x)$</span> is known then the expectation <span class="math-container">$\mathbb{E}$</span> of the test function <span class="math-container">$\tau(x)$</span> under th... | <p>What you are looking at is the <a href="https://en.wikipedia.org/wiki/Law_of_total_probability" rel="nofollow noreferrer">law of total probability</a>, <a href="https://en.wikipedia.org/wiki/Law_of_total_expectation" rel="nofollow noreferrer">law of total expectation</a>, etc. These laws follow directly from the de... | 222 |
bayesian inference | Distribution of oddsratio after bayesian inference under binomial model | https://stats.stackexchange.com/questions/305815/distribution-of-oddsratio-after-bayesian-inference-under-binomial-model | <p>Let us have 2 groups - treatment group (1) and control group(2).
Each group has survival probability $p_1$ and $p_2$ respectively. Ofc each patient survives or dies independently given $p_i$ of his group. Let us define $y_i$ as number of survivors in group i, $n_i - y_i$ - number of deceased in group i.</p>
<p>So,... | 223 | |
bayesian inference | Is there an implied distribution given the below for where a transaction happens? | https://stats.stackexchange.com/questions/635804/is-there-an-implied-distribution-given-the-below-for-where-a-transaction-happens | <p>Assume you have a buyer and a seller.</p>
<p>You know that the buyer's probability of buying the good at different prices (ie smth like P(B|price) and similarly for the seller P(S|price).</p>
<p>Given you know these you know a transaction happens if both agree - in that case for any given price P(transaction|price) ... | 224 | |
bayesian inference | Why does continuous Bayesian analysis seem to give this contradictory result? | https://stats.stackexchange.com/questions/5453/why-does-continuous-bayesian-analysis-seem-to-give-this-contradictory-result | <p>Let's say you have a process that generates data according to r = sin(t) + epsilon, where epsilon ~ N(0,V) is Gaussian noise. The unconditional variance of r is 0.5 + V. </p>
<p>Let's say we're forecasting r with a model m, and that our forecast is "perfect" in that m = sin(t). Construct v = r - m, which is the for... | <p>Actually, the variances are zero: $V(v|r) = V(r|v) = 0$.$p(v|r)$ is a Dyrac function which has a peak at the right spot (where $r = v -m$).</p>
<p>If you know $v$ or $r$, the other one is a deterministic function of the one you know.</p>
| 225 |
bayesian inference | Understanding posterior probability (Bayesian inference) | https://stats.stackexchange.com/questions/445096/understanding-posterior-probability-bayesian-inference | <p>I'm reading this <a href="https://statswithr.github.io/book/the-basics-of-bayesian-statistics.html" rel="nofollow noreferrer">online book</a> and there is something unclear to me in <a href="https://statswithr.github.io/book/the-basics-of-bayesian-statistics.html#tab:RU-486prior" rel="nofollow noreferrer">this table... | <p>It's just a normalizing constant which makes the posterior a valid density. In practice, we don't care so much about it. It should be noted that </p>
<p><span class="math-container">$$p(x) = \int p(x\vert \theta) p(\theta) \, d\theta$$</span></p>
<p>So it is as if you are averaging the likelihood over the prior.... | 226 |
bayesian inference | Definition of Likelihood in Bayesian Statistics | https://stats.stackexchange.com/questions/444781/definition-of-likelihood-in-bayesian-statistics | <p>Can the likelihood be defined as the probability of the rate parameter given a range of data. Or as the probability of the data, given a range of rate parameters?</p>
| <p>I think I understand your confusion. Typically, Bayes' rule is written as:</p>
<p><span class="math-container">$$p(\theta |y) = \frac{ p(y|\theta) p(\theta)}{p(y)}$$</span></p>
<p>where <span class="math-container">$p(\theta |y)$</span> is the posterior for the observed data <span class="math-container">$y$</span... | 227 |
bayesian inference | On the Bayesian setup in inference | https://stats.stackexchange.com/questions/212866/on-the-bayesian-setup-in-inference | <p>I've been trying to get into the chapter 4 in Lehmann's <em>Theory of point estimation</em>, but I can't seem to understand his presentation of the Bayesian setup. He starts of by the introduction below and after a few examples of uses of Bayesian estimators he outlines the idea (after dots in my photo). I don't kno... | <p>This is Fubini's theorem in action: when minimising in $\delta$
$$\mathbb{E_{\Lambda}} \{\mathbb{E}_{\theta}[L(\theta,\delta)]\}=\int_\Theta\int_\mathcal{X} L(\theta,\delta(x))\text{d}P_\theta(x)\text{d}\Lambda(\theta)=\int_\mathcal{X} \int_\Theta L(\theta,\delta(x))\text{d}\Lambda_x(\theta)\text{d}P(x)$$where $\Lam... | 228 |
bayesian inference | Can the Bayes factor be negative? | https://stats.stackexchange.com/questions/571438/can-the-bayes-factor-be-negative | <p>This is what I saw in a source I am referring to:</p>
<p><a href="https://i.sstatic.net/uAubu.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/uAubu.png" alt="enter image description here" /></a></p>
<p>Since both the numerator and the denominator are probabilities (so they can only take any value betw... | <p>Not sure what your source is, but whoever it is seems to have botched <a href="https://en.wikipedia.org/wiki/Bayes_factor#Interpretation" rel="noreferrer">Harold Jeffreys' cutoffs</a>. The items in the table <em>do</em> match his recommendations, but with two problems. The first is that the cutoffs are intended to b... | 229 |
bayesian inference | The role of variance of the distribution plays in Bayesian inference | https://stats.stackexchange.com/questions/626327/the-role-of-variance-of-the-distribution-plays-in-bayesian-inference | <p>Given prior <span class="math-container">$ \mu \sim \mathcal{N}(\mu_0, \tau^2) $</span>, likelihood <span class="math-container">$ X_i | \mu \sim \mathcal{N}(\mu, \sigma^2) $</span>, we know the closed-form solution of posterior is <span class="math-container">$ \mu | X_1, X_2, \ldots, X_n \sim \mathcal{N}(\mu_n, \t... | <p>Think about a simpler problem: estimating the mean <span class="math-container">$\mu$</span> in a frequentist setting.</p>
<p>It is sensible to use the MLE, <span class="math-container">$\bar X$</span>, as our estimator. How accurate is this on average? We know that <span class="math-container">$\bar X \sim \mathcal... | 230 |
bayesian inference | Why is the posterior distribution in Bayesian Inference often intractable? | https://stats.stackexchange.com/questions/208176/why-is-the-posterior-distribution-in-bayesian-inference-often-intractable | <p>I have a problem understanding why Bayesian Inference leads to intractable problems. The problem is often explained like this:</p>
<p><a href="https://i.sstatic.net/fIwYu.png" rel="noreferrer"><img src="https://i.sstatic.net/fIwYu.png" alt="enter image description here"></a></p>
<p>What I don't understand is why t... | <blockquote>
<p>Why can one not simply calculate the posterior distribution as the numerator of the right-hand side and then infer this normalization constant by requiring that the integral over the posterior distribution has to be 1?</p>
</blockquote>
<p>This is precisely what is being done. The posterior distribut... | 231 |
bayesian inference | At the end of the day, what do you do with Bayesian Estimates? | https://stats.stackexchange.com/questions/547923/at-the-end-of-the-day-what-do-you-do-with-bayesian-estimates | <p>I have often heard that in certain instances, it can be more beneficial to use Bayesian based methods because they provide "a distribution of possible answers" (i.e. the posterior distribution) instead of a single answer (as done in the frequentist case). However, it seems that at the end of the day, the a... | <p>First of all, Frequentist methods also provide a distribution over possible answers. It is just that we do not call them distributions because of a philosophical point. Frequentists consider parameters of a distribution as a fixed quantity. It is not allowed to be random; therefore, you cannot talk about distributio... | 232 |
bayesian inference | Question about probability vs inferential statistics | https://stats.stackexchange.com/questions/477324/question-about-probability-vs-inferential-statistics | <p>I'm currently struggling with a question involving probability and statistics.</p>
<p>I have this dataset of sales, and I was trying to make a probability of sales based on that dataset and the data that it provides me of weeks, months and years back. I started using Bayes Theorem to do it, and after a conversation ... | 233 | |
bayesian inference | How to compute the variance for a bayesian estimator | https://stats.stackexchange.com/questions/481595/how-to-compute-the-variance-for-a-bayesian-estimator | <p>I can't figure out how to compute the variance of an estimator which is the mean of the posterior distribution let's say Gamma(<span class="math-container">$\sum x_i+3, n+a$</span>)
How to find out the variance of this mean ?</p>
| <p>In the Bayesian paradigm, distributions of interest are uncertainty distributions of unknown parameters. So if you have a posterior distribution <span class="math-container">$f(\theta)$</span> for parameter <span class="math-container">$\theta$</span> you can get an uncertainty (credible) interval for <span class="... | 234 |
bayesian inference | What are the factors that cause the posterior distributions to be intractable? | https://stats.stackexchange.com/questions/4417/what-are-the-factors-that-cause-the-posterior-distributions-to-be-intractable | <p>In Bayesian statistics, it is often mentioned that the posterior distribution is intractable and thus approximate inference must be applied. What are the factors that cause this intractability? </p>
| <p>I had the opportunity to ask <a href="https://scholar.google.com/citations?user=8OYE6iEAAAAJ&hl=en&oi=ao" rel="noreferrer">David Blei</a> this question in person, and he told me that <em>intractability</em> in this context means one of two things:</p>
<ol>
<li><p>The integral has no closed-form solution. Th... | 235 |
bayesian inference | We flip a coin 20 times and observe 12 heads. What is the probability that the coin is fair? | https://stats.stackexchange.com/questions/442512/we-flip-a-coin-20-times-and-observe-12-heads-what-is-the-probability-that-the-c | <p><a href="https://i.sstatic.net/ZbJUs.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/ZbJUs.jpg" alt="enter image description here"></a><a href="https://i.sstatic.net/k8NUI.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/k8NUI.jpg" alt="enter image description here"></a>im having some tr... | <p>You appear to be using a Beta(1,1) prior on <span class="math-container">$\theta$</span>. Since this is a continuous distribution, the prior (and posterior) probability of the event that the coin is exactly fair, <span class="math-container">$\theta=1/2$</span>, is zero.</p>
<p>What would perhaps be a more sensible ... | 236 |
bayesian inference | Parameter covariance in Bayesian regression of time series | https://stats.stackexchange.com/questions/660688/parameter-covariance-in-bayesian-regression-of-time-series | <p>My problem is thus: given set of time series data <span class="math-container">$D = \{t_m, x_m\}$</span> where <span class="math-container">$m$</span> is a label, <span class="math-container">$m=1,2,3,...,n$</span> I have a model <span class="math-container">$f(t,\mathbf{w})$</span> which generates an equivalent tim... | 237 | |
bayesian inference | Can you use the beta-binomial distribution instead of MCMC? | https://stats.stackexchange.com/questions/653748/can-you-use-the-beta-binomial-distribution-instead-of-mcmc | <p>So, I have a project to test the hypothesis that a marketing campaign with a new art generates more purchases than the old one, I have 2 samples of data, one using the standard ad and one using the new ad. So we have the total number of impressions and the number of purchases. We can estimate this as a <span class="... | <p>Yes - if the model is simple enough to calculate the posterior analytically, there is no reason you can't do that instead of sampling the posterior. In practice, this is rarely the case, hence MCMC, but theoretically the analytical solution should always give accurate possible answer.</p>
| 238 |
bayesian inference | How to find the support of the posterior distribution to apply Metropolis-Hastings MCMC algorithm? | https://stats.stackexchange.com/questions/74330/how-to-find-the-support-of-the-posterior-distribution-to-apply-metropolis-hastin | <p>I am trying to sample from a posterior distribution using a MCMC algorithm using the Metropolis-Hastings sampler.</p>
<p>How should I deal with the situations where I'm stuck in regions of the posterior with zero probability?</p>
<p>These regions are present because the posterior distribution is truncated and also... | <p>This is an implementation problem since, theoretically, MCMC has no problem with truncated distributions.</p>
<p>Let $D$ be the support of your posterior. Just define your log-posterior as $-\infty$ if $\theta\not\in D$ and choose a suitable value on $D$ as an initial point.</p>
<p>For example, suppose that $x_1,\... | 239 |
bayesian inference | Approximating distributions in expectation propagation | https://stats.stackexchange.com/questions/82348/approximating-distributions-in-expectation-propagation | <p>Can the approximating distributions for various factors in Expectation propagation be different distributions but still from the exponential family. For example, I have the following posterior form:</p>
<p>$$
p(w, \lambda, \phi) = p(\phi)p(\lambda)p(w|\lambda) \prod_{i}p(y_{i}|w, \phi, \lambda)
$$</p>
<p>So, I nee... | <p>Yes, you can use different exponential families to approximate the marginal for different variables. You only need all messages into a variable to have the same type, so that they can be multiplied together to get the marginal. In your example, $p(w|\lambda)$ (call it factor $a$) can be approximated by the two mess... | 240 |
bayesian inference | Global search operators for approximate MAP inference? | https://stats.stackexchange.com/questions/82946/global-search-operators-for-approximate-map-inference | <p>In complicated Bayesian models, like for instance a hierarchical nonparameteric one, often times it's intractable to do Gibbs or other MCMC sampling methods to convergence. Rather, people tend to do variational inference and use expectation maximization to find the approximate MAP parameters.</p>
<p>Is there a reas... | <p>Gibbs and MCMC methods are sampling methods. EM methods are optimisation methods. Two very different things. The first ones sample from the posterior distribution, the other one finds a maximum/minimum.</p>
<p>EM algorithms are used whenever you can conduct the E and M steps relatively easy. Otherwise, you usually ... | 241 |
bayesian inference | Expectation propagation for feature selection | https://stats.stackexchange.com/questions/121757/expectation-propagation-for-feature-selection | <p>I'm using Expectation propagation algorithm (<code>infer.net</code> library) for my feature selection problem. </p>
<p>I generate input data and test my model. The thing is that when I use different number of data points, I get very different results. </p>
<p>For example, in my current setting it really works well... | <ol>
<li>Maybe it hasn't converged.</li>
<li>Try initializing near the true answer.</li>
<li>Try a different inference algorithm.</li>
</ol>
| 242 |
bayesian inference | Points to keep in mind while implementing a nonparametric bayesian inference procedure from scratch | https://stats.stackexchange.com/questions/61580/points-to-keep-in-mind-while-implementing-a-nonparametric-bayesian-inference-pro | <p>I have been trying to implement a Bayesian inference procedure from scratch for a specific problem, but I have implemented the procedure, and it doesn't seem to work. </p>
<p>Since, I can't just post the code online and ask community to debug my code, I was wondering if someone could provide with a broader checklis... | 243 | |
bayesian inference | Bayesian inference - posterior in a simple model | https://stats.stackexchange.com/questions/232037/bayesian-inference-posterior-in-a-simple-model | <p>Suppose you are measuring $n$ quantities with error. Let $\beta_1,\ldots, \beta_n$ represent the true values and $X_1, \ldots, X_n$ represent the measured values of those quantities. Assume that the errors are centered normal. Let $\sigma_i^2\,, i=1, \ldots, n$ represent the <strong>known</strong> standard deviatio... | <p>I assume the intention is that the $\epsilon$s are independent of the $\beta$s (like in a typical measurement noise model). </p>
<p>Then, $\epsilon_i$ and $X_i=\beta_i+\epsilon_i$ are not independent. The error in the first approach is assuming they are and proceeding as if $\epsilon_i \mid X_i \sim N(0, \sigma_i^2... | 244 |
bayesian inference | Coin tossing posterior density calculation | https://stats.stackexchange.com/questions/553926/coin-tossing-posterior-density-calculation | <p>I know that my prior distribution is Beta(3,3) and that after tossing 12 coins, the number of 'heads' is less than 4 but I don't know the exact number.
How do I calculate the posterior density?</p>
<p>What I've tried to do is:</p>
<p>If <span class="math-container">$X=\#$</span> of heads in <span class="math-contain... | <p>What use is four posterior densities? I would have thought you wanted one. It would be a weighted average of them, but perhaps difficult to find the weights.</p>
<p>If <span class="math-container">$p$</span> is the probability of a head then, if my calculations are correct,</p>
<ul>
<li><p>The prior density is prop... | 245 |
bayesian inference | Can I interpret the p-value of a statistic test as a part in Bayesian Formula? | https://stats.stackexchange.com/questions/558394/can-i-interpret-the-p-value-of-a-statistic-test-as-a-part-in-bayesian-formula | <p>Suppose we have a hypothesis test: <span class="math-container">$$H_0: \theta≥\theta_0 ~~~ vs~~~ H_1:\theta<\theta_0$$</span></p>
<p>With the observation <span class="math-container">$X$</span>, the p-value is calculated by <span class="math-container">$p = P(X|H_0)$</span>. <em>Which means the sum of probabili... | <p>Welcome to Cross Validated and a +1 from me. I once wondered this, and I saw two issues.</p>
<ol>
<li>That definition of a p-value is not quite right.</li>
</ol>
<p><span class="math-container">$$p=P(X\ge x \vert H_o)$$</span></p>
<p>(Or something similar for a two-sided test)</p>
<p>With that in mind, you are not q... | 246 |
bayesian inference | Can a global sensitivity analysis be performed on Bayesian inference? | https://stats.stackexchange.com/questions/559157/can-a-global-sensitivity-analysis-be-performed-on-bayesian-inference | <p>My question is, is it possible to perform a Global Sensitivity Analysis on a Bayesian inference model (not just on the prior, the entire model)?</p>
<p>A bit of context:
I am fairly new to Bayesian statistics.
Being a research student in astrobiology, I have read a few papers using Bayesian approaches to try constra... | 247 | |
bayesian inference | Using indirect prior information in Bayesian inference | https://stats.stackexchange.com/questions/296682/using-indirect-prior-information-in-bayesian-inference | <p>Hi I am trying to estimate the posteriors of four calibration parameters namely $c_1, c_2, c_3$ and $c_4$ in the following equation using Bayesian inference</p>
<p>$$
F=c_1 \cdot (i^{c_2}) \cdot(s^{c_3}) \cdot (1-\exp(c_4 t))
$$</p>
<p>I have the observed data for the output $F$ and inputs $i$,$s$ and $t$. I know ... | <p>As a first try, I would suggest you building the multidimensional Jeffreys prior $\Pi(c_1,c_2,c_3,c_4)$ weighted by the support function returning 1 if your constraints are fullfilled and 0 else. This will give you a procedure to get a prior that will not depend from the way you choosed to parametrize $F$, which may... | 248 |
bayesian inference | Calculating Bayes' factor for 2 Gamma distributions | https://stats.stackexchange.com/questions/336127/calculating-bayes-factor-for-2-gamma-distributions | <p>I have 2 model $M_1$ and $M_2$ which both have a gamma distribution and the same priors</p>
<p>$H_0 : \quad x_i \sim M_1 \\ H_a: \quad x_i \sim M_2$</p>
<p>Both $M_1$ and $M_2$ have prior $\sim Ga(7,3000)$ but my posteriors are </p>
<p>$M_1 \sim Ga(191,116665.4) \\ M_2 \sim Ga(192, 116188.9)$</p>
<p>I get the va... | 249 | |
bayesian inference | Can you interpret this question? | https://stats.stackexchange.com/questions/341115/can-you-interpret-this-question | <p>I'm studying for past exam and I'm actually stumped on what a particular question is asking me. I've thought about it for days and I actually just don't know what are they asking. Can anyone interpret the question?</p>
<p>It's part (ii)</p>
<p>" The Hobbits living in the Shire are not known for being very tall: wh... | <p>In part (ii) you can use the <em>population</em> information -- that is, the distribution of all hobbits living in the Shire -- but not information about the hobbits who attended Bilbo's party, to elicit your prior parameters. So your prior should be a normal distribution such that 2.5% of its mass falls below 60 cm... | 250 |
bayesian inference | Estimate the mean and variance 95% HPD credible region using Bayesian inference | https://stats.stackexchange.com/questions/346378/estimate-the-mean-and-variance-95-hpd-credible-region-using-bayesian-inference | <p>I have the following data:</p>
<p>31.0, 30.5, 20.6, 27.2, 26.5, 28.1, 25.8, 29.6, 30.0, 25.8, 25.1, 27.9, 23.0, 29.4, 28.7, 25.0, 31.1, 24.8, 24.8, 27.0, 22.3, 29.5, 31.5, 26.2, 24.6, 23.2, 25.7, 24.2, 28.8, 27.4, 29.6, 23.5, 26.4, 28.7, 25.5, 18.6, 25.2, 24.5, 27.9, 33.0, 21.4, 34.4, 27.2, 23.3, 29.3, 31.4, 24.6, ... | <p>Letting $\tau = \frac{1}{\sigma^{2}}$, be the precision, the priors become:</p>
<p>$\tau \sim Gamma(3, 36)$</p>
<p>$\mu \space | \space \tau \sim \mathcal{N}(26, n_{0}\tau)$</p>
<p>Then the posterior distributions have the form:</p>
<p>$\mu \space | \space \tau, x \sim \mathcal{N}(\frac{n\tau}{n\tau + n_{0}\tau}... | 251 |
bayesian inference | Bayesian updating for coin toss | https://stats.stackexchange.com/questions/367227/bayesian-updating-for-coin-toss | <p>I have used Bayesian reasoning in my research work and it has been extremely useful. The book I have read is E.T. Jayne's <em>Probability theory</em>. The idea is to formulate propositions and then probability theory tells how to assign numbers (viz. probability) to those propositions, conditional on one's informati... | 252 | |
bayesian inference | Truncated count model -- including information about the number of unobserved realisations | https://stats.stackexchange.com/questions/374316/truncated-count-model-including-information-about-the-number-of-unobserved-re | <h2>Background</h2>
<p>Suppose we have a model such that <span class="math-container">$Y \sim \mathcal{M}(\theta)$</span> is a discrete random variable taking values in <span class="math-container">$[0, 1, \ldots]$</span>. We would like to make inference about <span class="math-container">$\theta$</span> from a collec... | 253 | |
bayesian inference | Trouble with MLE | https://stats.stackexchange.com/questions/387298/trouble-with-mle | <p>I have a random sample <span class="math-container">$X_1, X_2, ..., X_n$</span> with <span class="math-container">$X_i$</span> having a pdf</p>
<p><span class="math-container">$$
f(x;\theta) = 2\theta^2x^{-2}
$$</span></p>
<p>I'd like to find the MLE of <span class="math-container">$\theta$</span>.</p>
<p>First,... | 254 | |
bayesian inference | Posterior predictive: what happens to integral over parameters? | https://stats.stackexchange.com/questions/402552/posterior-predictive-what-happens-to-integral-over-parameters | <h3>Question</h3>
<p>I don't understand how when integrating over the parameters in the posterior predictive, the integration "disappears". It's hard for me to ask simply because I am confused, so here is an example.</p>
<h3>Example</h3>
<p>Imagine we have a Gaussian model with unknown mean <span class="math-contain... | <p>If I understand correctly the question, it seems to me (and others before me in the comment section) that the derivation
<span class="math-container">$$\mathcal{N}(D' \mid \mu, \sigma^2) \mathcal{N}(\mu \mid \mu_N, \sigma_N^2) = \mathcal{N}(D' \mid \mu_N, \sigma_N^2 + \sigma^2)$$</span>
or equivalently
<span class="... | 255 |
bayesian inference | Best sampling method within the normal family | https://stats.stackexchange.com/questions/417198/best-sampling-method-within-the-normal-family | <p>Suppose that we want to make the best Bayesian inference about some value <span class="math-container">$\mu$</span> we have some normal prior about it. I.e. <span class="math-container">$\mu\sim N(\mu_0, \sigma_0^2)$</span> with known parameters. To do so, we can choose parameters <span class="math-container">$(\mu_... | 256 | |
bayesian inference | Posterior probability of hypothesis distributions | https://stats.stackexchange.com/questions/448987/posterior-probability-of-hypothesis-distributions | <p>Suppose I have <span class="math-container">$K$</span> classes with distribution <span class="math-container">$\theta$</span> over <span class="math-container">$\{1,...,K\}$</span> and an underlying domain <span class="math-container">$D$</span> on which each class defines a categorical distribution <span class="mat... | <p>You seem to be talking about <a href="https://en.wikipedia.org/wiki/Posterior_predictive_distribution" rel="nofollow noreferrer">posterior predictive distribution</a>, i.e. the <em>a posteriori</em> distribution of the data. You don't see <span class="math-container">$\theta$</span>, because it is <em>marginalized o... | 257 |
bayesian inference | confidence intervals for probabilities of a biased die | https://stats.stackexchange.com/questions/455919/confidence-intervals-for-probabilities-of-a-biased-die | <p>Given a biased die with d faces, you are given results of n die rolls. I need to calculate the confidence intervals of the probabilities of each of the d outcomes of the die.</p>
<p>A solution in R! - even better.</p>
| 258 | |
bayesian inference | Applying Bayesian Reasoning to Estimate the Type of a Feature | https://stats.stackexchange.com/questions/511886/applying-bayesian-reasoning-to-estimate-the-type-of-a-feature | <p>Suppose I have a set of strings <span class="math-container">$S$</span> and I want to find out whether these strings have a certain type. To be more specific, I want to find out whether these strings are surnames. Suppose I have a large list <span class="math-container">$L$</span> of surnames from many regions in th... | <p>What you are missing is a likelihood function. For that, you will need to go linguists. My mother's maiden name, for example, contains two changes. One letter was dropped and another letter substituted when my great grandparents arrived in America. The original spelling has survived in other branches of the fami... | 259 |
bayesian inference | Bayesian hypothesis test: Type I and II errors | https://stats.stackexchange.com/questions/387974/bayesian-hypothesis-test-type-i-and-ii-errors | <p>In a Bayesian hypothesis test between two alternatives A and B, what is the probability of making a type I and type II error?</p>
<p>This question has been asked many times on this forum in various formats: Is Bayesian hypothesis testing immune to peaking? What is the optimal stopping point? If the Bayes factor is ... | 260 | |
bayesian inference | In Bayesian inference, it is said that for large samples, the posterior density is dominated by the likelihood. What does this mean? | https://stats.stackexchange.com/questions/519250/in-bayesian-inference-it-is-said-that-for-large-samples-the-posterior-density | <p>In Bayesian inference, it is said that
<em><strong>for large samples, the posterior density is dominated by the likelihood. Furthermore, in the region where the likelihood is large, the posterior density is nearly constant.</strong></em>
Could you kindly explain the logic behind such as a statement?
I would really a... | 261 | |
bayesian inference | How to optimise waterfall questions of purchase value | https://stats.stackexchange.com/questions/434205/how-to-optimise-waterfall-questions-of-purchase-value | <p>Imagine I have an item I want to sell to a person. I know for sure that the person is not willing to pay more than \$X for this item, but I don't know which value between \$0 and \$X they <em>are</em> willing to pay for it, and I'm equally uncertain about all of them. So if I call how much they're willing to pay for... | <p><strong>Short answer:</strong></p>
<ol>
<li><p>Indeed, when the same customer may be approached at most <span class="math-container">$n$</span> times, it is optimal to start with offer <span class="math-container">$y_1=\frac{n-1}{n}x$</span> and decrease the price by <span class="math-container">$\frac{x}{n}$</span... | 262 |
bayesian inference | Fitting a single model to different datasets that include different variables | https://stats.stackexchange.com/questions/541925/fitting-a-single-model-to-different-datasets-that-include-different-variables | <p>Suppose we have two datasets <em>df_1</em> with variables {A,B,C} and <em>df_2</em> with variables {A,C,D} (A & C are the only mutual variable in the two datasets). Our aim is to predict A using B & C or C & D (depending on which pair is given). The simplest approach is to model A using <em>df_1</em> and... | 263 | |
bayesian inference | How to find the likelihood probability of an exponential data model | https://stats.stackexchange.com/questions/563991/how-to-find-the-likelihood-probability-of-an-exponential-data-model | <p>I have a very basic knowledge in statistics, so I am struggling a bit with the ideas of Bayesian inference.</p>
<p>My data model looks like this,</p>
<p><span class="math-container">$$ z(t) = \sum_{n = 1}^{N} e^{j 4\pi/\lambda \sqrt{(x_{n, t -1} + u_n.dt)^2 + (y_{n, t -1} + v_n.dt)^2}} + \mathcal{N}(0, \sigma^2) $$<... | 264 | |
bayesian inference | Understanding convergence in Bayesian inference of coin tossing | https://stats.stackexchange.com/questions/432710/understanding-convergence-in-bayesian-inference-of-coin-tossing | <p>When we are uncertain about the probability of head, <span class="math-container">$p_H$</span>, in a coin tossing, we often model it using a Beta prior as follows:
<span class="math-container">$$p_H\sim \text{Beta}(a_0,b_0),$$</span>
for some parameters <span class="math-container">$a_0,b_0$</span>. </p>
<p>When we... | <h4>Yes, it does converge to the "true distribution" (suitably defined)</h4>
<p>First of all, it is worth noting that it is a little strange to refer to the "true distribution" of the parameter as something aside from the prior and posterior. If you proceed under the operational Bayesian approach t... | 265 |
bayesian inference | Bayesian inference of Pr(X > Y) where X and Y each have an approximate posterior distribution | https://stats.stackexchange.com/questions/552919/bayesian-inference-of-prx-y-where-x-and-y-each-have-an-approximate-posterior | <p>I am developing a Bayesian system in which I would like to quantify the evidence for or against the conclusion that one data-generating process (X, for which we observe X = x) will produce a more extreme result than another process (Y, for which we observe Y = y).</p>
<p>For my purposes, by "more extreme" ... | <p>Yes you "could simply draw pairs from <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> an arbitrary number of times, and report the proportion of instances in which <span class="math-container">$x > y$</span>." What you wish to estimate is not a random variable; hence... | 266 |
bayesian inference | Bayesian Model(Write out likelihood and prior) | https://stats.stackexchange.com/questions/576071/bayesian-modelwrite-out-likelihood-and-prior | <p>I am working with a dataset regarding transmission rate for a disease spreading among cattle at different farms during a 5-month period.</p>
<p>The goal is to estimate the transmission parameter <span class="math-container">$\alpha$</span> using a Bayesian model.</p>
<p>I have a dataset with 12 entries(different far... | <p>There are two parameters in your model, <span class="math-container">$\beta_0$</span> and <span class="math-container">$\beta_1$</span>, which are real parameters that can vary from <span class="math-container">$-\infty$</span> to <span class="math-container">$\infty$</span>, so the most <a href="https://en.wikipedi... | 267 |
bayesian inference | Compute the Maximum A Posteriori (MAP) estimate of θ | https://stats.stackexchange.com/questions/578414/compute-the-maximum-a-posteriori-map-estimate-of-%ce%b8 | <p>How can I compute the Maximum A Posteriori (MAP) estimate of <span class="math-container">$\theta$</span> with those informations:
a discrete random variable y with values in {1, 2, . . . , N} has a Binomial distribution depending on the unknown probability <span class="math-container">$\theta \in (0,1)$</span> of t... | 268 | |
bayesian inference | Bayesian Prior definition | https://stats.stackexchange.com/questions/578510/bayesian-prior-definition | <p>The prior of an inference problem where we try to infer <span class="math-container">$x$</span> from observations <span class="math-container">$y$</span> is defined as <span class="math-container">$P(X)$</span>. Often (<a href="https://arxiv.org/pdf/1010.5141.pdf" rel="nofollow noreferrer">e.g.</a>) I see another de... | 269 | |
bayesian inference | Reasonable to incorporate sample size into beta-binomial? | https://stats.stackexchange.com/questions/581139/reasonable-to-incorporate-sample-size-into-beta-binomial | <p><strong>Setup:</strong></p>
<p>The relationship between the beta and binomial distributions is well known.</p>
<p><span class="math-container">$$\frac{\pi^{\alpha - 1} (1 - \pi)^{\beta - 1}}{B(\alpha, \beta)} \leftrightarrow {{n}\choose{x}}\pi^{x} (1 - \pi)^{n-x}$$</span></p>
<p>By comparing the two, one can see:</p... | 270 | |
bayesian inference | How to go about selecting an algorithm for approximate Bayesian inference | https://stats.stackexchange.com/questions/32214/how-to-go-about-selecting-an-algorithm-for-approximate-bayesian-inference | <p>I am wondering if there are any good rules of thumb for how to go about selecting an approximate inference algorithm for a problem/model (specifically when exact inference is intractable)? When you are faced with a problem, what are the things you consider when selecting an approach for inference (e.g. MCMC, belief ... | <p>At first you have to decide what amount of time you can afford. </p>
<p>In case you have a large amount of time for your numerical experiments you can try MCMC method, also in this case it is possible to avoid complex integrations in some cases. </p>
<p>In case you have a strong background in statistics and you wa... | 271 |
bayesian inference | Comparison of Variational Bayes and Expectation Maximization algorithms | https://stats.stackexchange.com/questions/82184/comparison-of-variational-bayes-and-expectation-maximization-algorithms | <p>I need to learn both the VB and EM methods for Bayesian Networks. Before going into detail of both algorithms, which I am a bit aware of, I need to EXACTLY understand the basic motivations behind them. Different resources use the terms "inference, parameters, estimation, learning" so intermingled that I easily lose ... | 272 | |
bayesian inference | Estimating total number of people from an observed sample | https://stats.stackexchange.com/questions/109166/estimating-total-number-of-people-from-an-observed-sample | <p>The well known "German tank problem" shows how to answer the question: "If I have tanks which have an increasing serial number, and I see a sample of tanks and record their serial numbers, what is the likely total number of tanks". This question is analogous but is where there is no ordering to the observations, eg... | <p>The model $p(m|N) \propto 1/{N \choose m}$ does not make sense. Once the company has decided to show $m$ people, then there are indeed ${N \choose m}$ sets of people that they could show. But this doesn't tell you anything about why the number was $m$. </p>
| 273 |
bayesian inference | Inferring prior distribution | https://stats.stackexchange.com/questions/114139/inferring-prior-distribution | <p>Suppose that we take a sample ($X_1, X_2, ... X_n$) from a distribution where we assume that $X_i $~$ Bin(n_i, p_i)$ and $n_i$ is known for every $i$. We also assume that $p_i$'s are independent and identically distributed, $p_i$ ~ $D$, where $D$ is some unknown distribution.
$n_i$ cannot be assumed to be large.</... | <p>I hope you like Python! I'll recite my comment here:</p>
<p>This sounds like a hierarchical model. If I wanted to recreate the dataset, here's what I'd do: Let $D$ be a $Beta(\alpha, \beta)$ distribution (reasonable since we are dealing with probabilities). We don't know $\alpha, \beta$, we assign priors to them, ... | 274 |
bayesian inference | Is the posterior distribution on means in a Bayesian Gaussian mixture model with symmetric priors Gaussian? | https://stats.stackexchange.com/questions/179882/is-the-posterior-distribution-on-means-in-a-bayesian-gaussian-mixture-model-with | <p>I am reading through a document on <a href="http://research.microsoft.com/en-us/um/cambridge/projects/infernet/docs/Mixture%20of%20Gaussians%20tutorial.aspx" rel="nofollow">learning Gaussian mixture models in Infer.NET</a>. They assume the data is generated from 2 Gaussians where the prior distribution on means is... | <p>The paper <a href="http://link.springer.com/chapter/10.1007/978-3-662-01131-7_26#page-1" rel="nofollow">Bayesian Inference for Mixture: The Label Switching Problem</a> says</p>
<blockquote>
<p>A K-Component mixture distribution is invariant to permutations of the labels of the components. As a consequence, in a... | 275 |
bayesian inference | How is prior knowledge possible under a purely Bayesian framework? | https://stats.stackexchange.com/questions/201686/how-is-prior-knowledge-possible-under-a-purely-bayesian-framework | <p>This is more of a philosophical question, but from a purely Bayesian standpoint how does one actually form prior knowledge? If we need prior information to carry out valid inferences then there seems to be a problem if we have to appeal to past experience in justifying today's priors. We're apparently left with th... | <p>Speaking of <em>prior knowledge</em> can be misleading, that is why you often see people speaking rather about <em>prior beliefs</em>. You do not need to have any prior knowledge to set up a prior. If you needed one, how would Longley-Cook manage with his problem?</p>
<blockquote>
<p>Here is an example from the 1... | 276 |
bayesian inference | How do I perform Bayesian Updating for a function of multiple parameters, each with its own distribution? | https://stats.stackexchange.com/questions/216164/how-do-i-perform-bayesian-updating-for-a-function-of-multiple-parameters-each-w | <p>I have a variable that is a recursive function involving other variables with known distributions (see problem below). </p>
<ul>
<li>Let $b(t+1) = b(t) + C \sqrt{b(t)}$ where I know $C \sim N(1.82, .0298)$ and the initial value of $b$ [$b_{initial} \sim N(.02,0.0036)$].</li>
</ul>
<p>My observation for updating is... | <p>Probabilistically, the right way to do this is a posterior <strong>joint</strong> distribution for C and binitial--if you know b(t) exactly, then you'll get a line of sorts, where if binitial is .022 that implies that C was 1.81, but it binitial was instead .023 then that implies that C was 1.76. Each of those point... | 277 |
bayesian inference | If I do the same experiment many times then does a 95% credible interval mean 95% of the time the true value lies within that range? | https://stats.stackexchange.com/questions/286761/if-i-do-the-same-experiment-many-times-then-does-a-95-credible-interval-mean-95 | <p>Its common misinterpretation of a 95% confidence interval to say that that 95% of the time the true value lies within that interval.</p>
<p>However, in Bayesian statistics, the 95% credible interval contains 95% of the probability from the probability density function. And if I repeat the experiment many times, I'... | 278 | |
bayesian inference | Intractable posterior - why not use kernel density for the data distribution? | https://stats.stackexchange.com/questions/300296/intractable-posterior-why-not-use-kernel-density-for-the-data-distribution | <p>In the Bayes rule, it is said that the posterior
$$
P(\theta|D) = \frac{P(D|\theta)P(\theta)}{P(D)}
$$
is <em>intractable</em>, because
$$
P(D) = \int P(D,\theta) d\theta
$$
and the latter is often a high-dimensional integral.</p>
<p>See <a href="https://stats.stackexchange.com/questions/208176/why-is-the-p... | <p>There are certainly lots of ways to try to numerically estimate high-dimensional definite integrals. The entire field of <a href="https://www.mathematik.hu-berlin.de/~romisch/papers/Rutg13.pdf" rel="nofollow noreferrer">high-dimensional numerical integration</a> is devoted to this problem, and it suffers from the d... | 279 |
bayesian inference | What Bayesian test to conduct with one independent variable and two dependent variables? | https://stats.stackexchange.com/questions/323062/what-bayesian-test-to-conduct-with-one-independent-variable-and-two-dependent-va | <p>In my current study I am looking at the effects of creatine monohydrate ingestion on ground reaction force and repeated sprint times. With CM as the independent variable, and ground reaction force and repeated sprint times as the dependent variables, what Bayesian inferential test would be the best to conduct?</p>
| <p>Bayesian inference would, in theory, involve determining the joint probability distribution (JPD) $P(R,S,C)$ that expresses the probability of a certain combination of reaction force ($R$), sprint time ($S$) and creatine monohydrate ingestion ($C$). The JPD over all variables can later be used to infer all other pos... | 280 |
bayesian inference | Bayesian inference of a coin's bias when we don't directly observe the flips | https://stats.stackexchange.com/questions/355646/bayesian-inference-of-a-coins-bias-when-we-dont-directly-observe-the-flips | <p>Consider a coin with bias $p$. We generate a random sample $x_1, \dots, x_n \sim \text{Bernoulli}(p)$, but <strong>we do not observe results of these coin tosses</strong>. Instead, for each $x_i$, we observe a set of features $y_{i1}, \dots, y_{im}$ about the flip, e.g. the height of the toss, the coin's rotational ... | 281 | |
bayesian inference | Bayesian Hypothesis Tests with continuous priors | https://stats.stackexchange.com/questions/421269/bayesian-hypothesis-tests-with-continuous-priors | <p>I am new to the Bayesian world, and I'm trying to understand how hypotheses tests are performed here (as opposed to the frequentist framework).</p>
<p>I am aware that likelihoods, priors and posteriors can be discrete or continuous. And once we have calculated posteriors, we can build a lot of things like credible ... | <p>The case is well-covered in <a href="https://amzn.to/31vg5C7" rel="nofollow noreferrer">Bayesian textbooks</a>, including <a href="http://amzn.to/2kxykkw" rel="nofollow noreferrer">ours</a>!, and can be summarised by the constraint that one can only test hypotheses for which the prior has a positive probability mass... | 282 |
bayesian inference | How to start coding for posterior inference | https://stats.stackexchange.com/questions/428676/how-to-start-coding-for-posterior-inference | <p>I am trying to implement the model given in <a href="http://proceedings.mlr.press/v84/andersen18a/andersen18a.pdf" rel="nofollow noreferrer">http://proceedings.mlr.press/v84/andersen18a/andersen18a.pdf</a> where they have used mean-field variational inference for posterior inference, but I want to use MCMC for infer... | 283 | |
bayesian inference | Real-time Bayesian updating. How to link posteriors? | https://stats.stackexchange.com/questions/461473/real-time-bayesian-updating-how-to-link-posteriors | <p>I have a general question about Bayesian inference which may help me solve a problem I have. It is best to illustrate this with an example. Inspired from this great post by AllenDowney:</p>
<p><a href="https://github.com/AllenDowney/BiteSizeBayes/blob/master/08_soccer_soln.ipynb" rel="nofollow noreferrer">https://g... | <p>Ok, so from what I understand from the blog post...</p>
<ul>
<li><p>The likelihood for the number of goals scored is Poisson. Each team has a goal scoring rate, <span class="math-container">$\lambda$</span> measured in units per game. We can divide this by 18 to yield the goal scoring rate per 5 minute increments... | 284 |
bayesian inference | (Bayesian) estimation of the underlying population size knowing its upper bound after $x$ draws | https://stats.stackexchange.com/questions/472669/bayesian-estimation-of-the-underlying-population-size-knowing-its-upper-bound | <p>Consider you have an initial bag of unique and identifiable items <span class="math-container">$(1.. K)$</span>. From this bag, someone used an arbitrary criteria to tag <span class="math-container">$N$</span> items. You don't know the chosen criteria (which can be anything, from odd numbers, to just the item 65) bu... | <p>Let's say that you take a sample of <span class="math-container">$s$</span> elements, with replacement, out of the <span class="math-container">$K$</span> items. Then the number of tagged item, <span class="math-container">$t$</span>, that you get follow a binomial distribution <span class="math-container">$\mathcal... | 285 |
bayesian inference | How to perform a sensitivity analysis in Bayesian statistics? | https://stats.stackexchange.com/questions/178502/how-to-perform-a-sensitivity-analysis-in-bayesian-statistics | <p>Bayesian inference is drawn from the posterior distribution or - in case we are interested in forecasting - from the predictive posterior distribution. However, these values are heavily affected by the choice of the prior, even if you have decided to go for an uninformed one (which can be implemented in many differe... | <p>A fairly standard approach to showing that your results were not heavily influenced by your choice of prior is simply to show that your results hold when choosing a different prior. For example, if you have an informed prior that suggests a certain result is more likely, you might want to also show your results also... | 286 |
bayesian inference | Bayesian Inference in the presence of multiple hypotheses | https://stats.stackexchange.com/questions/437456/bayesian-inference-in-the-presence-of-multiple-hypotheses | <blockquote>
<p>"Because [Bayesian Inference] respects the forward flow of time or information, there's
no need for nor availability of methods for correcting for
multiplicity ... The evidence of one question
is not tilted by whether other questions are being asked."</p>
</blockquote>
<p><a href="https://www.y... | <p>Arguments that Bayesians do not need to worry about type I errors are starting from the premise that the type I error rate does not matter/is not a relevant concept* and simply adhere to the likelihood principle**. </p>
<p>I don't think this kind of Bayesian viewpoint is compatible with coercing an inferential thre... | 287 |
bayesian inference | Bayesian inference - a use case | https://stats.stackexchange.com/questions/330554/bayesian-inference-a-use-case | <p>I've been recently studying Bayesian inference with PyMC3. I understand the flexibility that comes with multiple possible options for initial distribution choices, yet I can't seem to understand why one would need the sampling part. I realize this is a very naive question, yet I cant seem to understand why does one ... | <p>Since it seems that you lack some basic understanding of the process behind Bayesian modeling work, let me give you a short summary of the usual workflow: </p>
<ol>
<li><p>You define the <em>likelihood</em> function for your model, for example: you assume the <a href="https://en.wikipedia.org/wiki/Bernoulli_distrib... | 288 |
bayesian inference | Bayesian inference on a sum of iid random variables with known distribution | https://stats.stackexchange.com/questions/273473/bayesian-inference-on-a-sum-of-iid-random-variables-with-known-distribution | <p>Let $X_1$, $X_2$, ..., $X_n$ be iid RV's following a mixture distribution of two lognormals such that the pdf of each $X_i$ is $f_{mix}(x)=pf_1(x) + (1-p)f_2(x)$ where $f_1(x)$ and $f_2(x)$ are lognormal pdfs with parameters $\mu_1,\sigma$ and $\mu_2,\sigma$, respectively.</p>
<p>Define $S_i$ as a sum of 10 $X$s, e... | <p>Each $X_i$ comes from either one of the two lognormals with probabilities $p$ and $1-p$. Let $Z_j$ be the number of $X_i$'s in the sum $S_j=\sum_{i=10j-9}^{10j}X_i$ that comes from the first lognormal. Clearly $Z_j \sim \mbox{bin}(10,p)$. Conditional on $Z_j$, each $S_j$ is a sum of $Z_j$ lognormals with paramete... | 289 |
bayesian inference | Correctness of product of densities representing parts of information as prior density in Bayes inference | https://stats.stackexchange.com/questions/600027/correctness-of-product-of-densities-representing-parts-of-information-as-prior-d | <p>suppose I've got data <span class="math-container">$X$</span> from a model driven by parameter <span class="math-container">$\theta$</span>. Model of data is represented by conditional density (likelihood function)
<span class="math-container">$$f(x|\theta).$$</span>
Suppose the prior density of <span class="math-co... | 290 | |
bayesian inference | Question about the Bayesian Inference of a parameter | https://stats.stackexchange.com/questions/112322/question-about-the-bayesian-inference-of-a-parameter | <p>In order to understand the difference between the Frequentist and Bayesian inference, I was reading the presentation at: <a href="http://www.stat.ufl.edu/archived/casella/Talks/BayesRefresher.pdf" rel="nofollow">http://www.stat.ufl.edu/archived/casella/Talks/BayesRefresher.pdf</a> . In order to explain the differenc... | <p>I think the slides are a bit ambiguous. For the Bayesian approach you can (among many other things) assume that:</p>
<ol>
<li><p>All protocols share the same $p$. Then there's a single prior over what that $p$ might be. </p></li>
<li><p>Each study $i$ has its own $p_i$. Then you can either </p>
<p>2.1. Assume t... | 291 |
bayesian inference | How are custom kernel functions in Gaussian processes statistically justified? | https://stats.stackexchange.com/questions/612663/how-are-custom-kernel-functions-in-gaussian-processes-statistically-justified | <p>I am confused about one aspect of the use of Gaussian processes for Bayesian inference. I understand that it relies on the assumption that your train and test data points form a multivariate normal distribution where you define a prior mean and covariance for the distribution. What I don't understand is that I belie... | <p>In <a href="https://stats.stackexchange.com/questions/502531/elementary-explanation-of-gaussian-processes">Gaussian Process</a>, your task is to learn the <em>distribution over functions</em> <span class="math-container">$$f(\mathbf{x}) = [f(x_1), f(x_2), \dots, f(x_n)]'$$</span> This distribution is modeled by Gaus... | 292 |
bayesian inference | Why does the marginal likelihood integral have no closed-form solution? | https://stats.stackexchange.com/questions/430842/why-does-the-marginal-likelihood-integral-have-no-closed-form-solution | <p>In Bayesian inference we end up with the formula:</p>
<p><span class="math-container">$$
P(\mathbf{w|t,X)}= \frac{P(\mathbf{t|w,X)}P(\mathbf{w)}}{\int P(\mathbf{t|w,X}) P(\mathbf{w}) d\mathbf{w}}$$</span></p>
<p>Assume the prior <span class="math-container">$P(w)$</span> is a Gaussian distribution with 0 mean and ... | <p>Yes, the marginal likelihood has a closed-form for all polynomial models of the form <span class="math-container">$\mathbf{t} = X\mathbf{w} + \boldsymbol{\varepsilon}$</span>, where,
<span class="math-container">\begin{aligned}
X &=
\begin{bmatrix}
\mathbf{1}^T & \mathbf{(x^1)}^T & (\mathbf{x}^2)^T ... (... | 293 |
bayesian inference | Inference in Bayesian networks with hidden variables | https://stats.stackexchange.com/questions/617132/inference-in-bayesian-networks-with-hidden-variables | <p>Suppose I have the Bayesian network in the figure and the corresponding conditional probability table for each node, where A and B are the hidden variables, and C and D are the observed variables. What probabilistic inference algorithm can I use to get all the conditional probabilities in Table - 1? can I use likeli... | <p>Well I don't think sampling is needed here (unless I misunderstand your question / diagram). I believe what is intended is to expand the probabilities using something like the product rule, so that:</p>
<p><span class="math-container">\begin{align}
P(c1,d1\mid a1,b1) &= P(d1 \mid c1, a1, b1)\cdot P(c1\mid a1,b1)... | 294 |
bayesian inference | Finding values a and b to get PMF with certain mean and standard deviation | https://stats.stackexchange.com/questions/325864/finding-values-a-and-b-to-get-pmf-with-certain-mean-and-standard-deviation | <p>Suppose that the proportion θ of defective items in a large manufactured lot is known to be either 0.05 or 0.15, and the prior pmf of θ is as follows: ξ(0.05) = a and ξ(0.15) = b. Suppose also that when n = 10 items are selected at random from the lot, it is found that X = 5 of them are defective.</p>
<p>(a) Determ... | 295 | |
bayesian inference | What's the problem with model identifiability? | https://stats.stackexchange.com/questions/60446/whats-the-problem-with-model-identifiability | <p>I understand that in a decision perspective, identifiability of a model is needed to ensure the convergence (with increasing number of observations) of the parameters to estimate through a single value. But, if the non-identifiability of a given model is not a modeling artifacts but clearly characterises some "inacc... | <p>I recommend you read Andrew Gelman's blog post <a href="http://andrewgelman.com/2014/02/12/think-identifiability-bayesian-inference/" rel="noreferrer">Think identifiability Bayesian inference</a>. </p>
<p>Right off the bat, I can tell you that identifiability does not have to do with a model by itself (as in "an un... | 296 |
bayesian inference | How to construct the highest posterior density (HPD) interval | https://stats.stackexchange.com/questions/304957/how-to-construct-the-highest-posterior-density-hpd-interval | <p>Please, anybody could explain the steps to compute the highest posterior density (HPD) interval, when the posterior distribution is known? For instance, when the posterior distribution is Beta distributed.</p>
<p>When the posteriori distribution is simulated, the <a href="https://www.jstor.org/stable/1390921?seq=1#... | <p>An HPD region is defined as$$\mathfrak{h}_\tau \stackrel{\text{def}}{=} \{\theta;\ \pi(\theta|x)>\tau\}$$and it is an interval only when the parameter is unidimensional and the posterior is unimodal. Assuming this is the case and the posterior $\pi(\cdot|x)$ is available up to a multiplicative constant, finding a... | 297 |
bayesian inference | Change of variable in posterior distribution | https://stats.stackexchange.com/questions/247677/change-of-variable-in-posterior-distribution | <p>I am working in a Bayesian framework: I have some observations $y$, for which I assume a statistical model. The model depends on parameters $\theta \in \Theta$ ($\Theta$ is the parameters space). I assume a probability distribution $q$ on $\Theta$. The parameters of this model can be estimated in a <em>maximum a pos... | <p>The change of variable in the posterior density is a standard change of variable, involving the Jacobian. The impact on the maximum a posteriori estimator is thus significant in that the MAP of the transform is not the transform of the MAP. (There are <a href="https://xianblog.wordpress.com/2009/09/12/map-estimators... | 298 |
bayesian inference | difference between the linear predictor with uncertainty and predictive distribution for a new observation | https://stats.stackexchange.com/questions/609894/difference-between-the-linear-predictor-with-uncertainty-and-predictive-distribu | <p>I was reading an extract from the book "regression and other stories" and at chapter 9 the author distinguish between 3 cases</p>
<p>"After fitting a regression, <span class="math-container">$y = a + bx + error$</span>, we can use it to predict a new data point, or a set of
new data points, with pred... | <p>The difficulty might be that <span class="math-container">$y = f(x, a, b, c)$</span> is too abstract of a notation and doesn't fully specify a particular model.</p>
<p>So let's take simple linear regression as an example: <span class="math-container">$y = a + bx + e$</span> with <span class="math-container">$e \sim ... | 299 |
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