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Section: Applications > In classification. The bias–variance decomposition was originally formulated for least-squares regression. For the case of classification under the 0-1 loss (misclassification rate), it is possible to find a similar decomposition, with the caveat that the variance term becomes dependent on the t... | Wikipedia - Bias–variance tradeoff - Applications > In classification | 170 | 898 | null |
Section: Applications > In human learning. While widely discussed in the context of machine learning, the bias–variance dilemma has been examined in the context of human cognition, most notably by Gerd Gigerenzer and co-workers in the context of learned heuristics. They have argued (see references below) that the human... | Wikipedia - Bias–variance tradeoff - Applications > In human learning | 229 | 1,126 | null |
Article: Binary classification. Binary classification is the task of classifying the elements of a set into one of two groups (each called class). Typical binary classification problems include: Medical testing to determine if a patient has a certain disease or not; Quality control in industry, deciding whether a speci... | Wikipedia - Binary classification - Summary | 197 | 1,020 | null |
Section: Evaluation > The eight basic ratios. A common approach to evaluation is to begin by computing two ratios of a standard pattern. There are eight basic ratios of this form that one can compute from the contingency table, which come in four complementary pairs (each pair summing to 1). These are obtained by divid... | Wikipedia - Binary classification - Evaluation > The eight basic ratios | 321 | 1,298 | null |
These are the proportion of the population with a given test result for which the test is correct. with complement the false discovery rate (FDR) (FP/(TP+FP)) negative predictive value (NPV) (TN/(TN+FN)) with complement the false omission rate (FOR) (FN/(TN+FN)), also called dependence on prevalence. In diagnostic test... | Wikipedia - Binary classification - Evaluation > The eight basic ratios | 347 | 1,490 | null |
Section: Statistical binary classification. Statistical classification is a problem studied in machine learning in which the classification is performed on the basis of a classification rule. It is a type of supervised learning, a method of machine learning where the categories are predefined, and is used to categorize... | Wikipedia - Binary classification - Statistical binary classification | 172 | 989 | null |
Section: Converting continuous values to binary. Binary classification may be a form of dichotomization in which a continuous function is transformed into a binary variable. Tests whose results are of continuous values, such as most blood values, can artificially be made binary by defining a cutoff value, with test res... | Wikipedia - Binary classification - Converting continuous values to binary | 348 | 1,708 | null |
Section: Company history. BioSerenity was founded in 2014, by Pierre-Yves Frouin. The company was initially hosted at the ICM Institute (Institute du Cerveau et de la Moëlle épinière), in Paris, France. Fund Raising June 8, 2015 : The company raises a $4 million seed round with Kurma Partners and IdInvest Partners Sept... | Wikipedia - Bioserenity - Company history | 257 | 1,083 | null |
Article: Bradley–Terry model. The Bradley–Terry model is a probability model for the outcome of pairwise comparisons between items, teams, or objects. Given a pair of items i and j drawn from some population, it estimates the probability that the pairwise comparison i > j turns out true, as where pi is a positive real-... | Wikipedia - Bradley–Terry model - Summary | 326 | 1,451 | null |
In a survey of wine preferences, for instance, it might be difficult for respondents to give a complete ranking of a large set of wines, but relatively easy for them to compare sample pairs of wines and say which they feel is better. Based on a set of such pairwise comparisons, the Bradley–Terry model can then be used ... | Wikipedia - Bradley–Terry model - Summary | 163 | 795 | null |
Section: Definition. The Bradley–Terry model can be parametrized in various ways. Equation (1) is perhaps the most common, but there are a number of others. Bradley and Terry themselves defined exponential score functions p i = e β i {\displaystyle p_{i}=e^{\beta _{i}}} , so that Pr ( i > j ) = e β i e β i + e β j . {\... | Wikipedia - Bradley–Terry model - Definition | 339 | 797 | null |
logit p = log p 1 − p {\textstyle \operatorname {logit} p=\log {\frac {p}{1-p}}} for 0 < p < 1. {\textstyle 0<p<1.} This formulation highlights the similarity between the Bradley–Terry model and logistic regression. Both employ essentially the same model but in different ways. In logistic regression one typically k... | Wikipedia - Bradley–Terry model - Definition | 308 | 877 | null |
In the same notation as BT model: Pr ( y 1 > ⋯ > y N ) = ∏ i = 1 N p y i ∑ k = i N p y k = p y 1 p y 1 + ⋯ + p y N p y 2 p y 2 + ⋯ + p y N ⋯ p y N p y N {\displaystyle \Pr(y_{1}>\cdots >y_{N})=\prod _{i=1}^{N}{\frac {p_{y_{i}}}{\sum _{k=i}^{N}p_{y_{k}}}}={\frac {p_{y_{1}}}{p_{y_{1}}+\dots +p_{y_{N}}}}{\frac {p_{y_{2}}}... | Wikipedia - Bradley–Terry model - Plackett–Luce model | 317 | 561 | null |
\Pr(y_{1}>y_{2})=p_{y_{1}}/(p_{y_{1}}+p_{y_{2}})} . This can be imagined as drawing from an urn with replacement. The urn contains balls colored in proportion to p 1 , p 2 , … , p N {\displaystyle p_{1},p_{2},\dots ,p_{N}} , and one draws from the urn with replacement. If a ball has a new color, then that ball is place... | Wikipedia - Bradley–Terry model - Plackett–Luce model | 347 | 983 | null |
In this interpretation, it is immediately clear that the PL model satisfies Luce's choice axiom (from the same Luce). Therefore, for any two y , z {\displaystyle y,z} , Pr ( y > z ) = p y p y + p z {\displaystyle \Pr(y>z)={\frac {p_{y}}{p_{y}+p_{z}}}} reduces to the BT model, and in general, for any subset y 1 , … , y ... | Wikipedia - Bradley–Terry model - Plackett–Luce model | 330 | 688 | null |
Section: Inference. The most common application of the Bradley–Terry model is to infer the values of the parameters p i {\displaystyle p_{i}} given an observed set of outcomes i > j {\displaystyle i>j} , such as wins and losses in a competition. The simplest way to estimate the parameters is by maximum likelihood estim... | Wikipedia - Bradley–Terry model - Inference | 308 | 1,013 | null |
Then the likelihood of this set of outcomes within the Bradley–Terry model is ∏ i j [ Pr ( i > j ) ] w i j {\displaystyle \prod _{ij}[\Pr(i>j)]^{w_{ij}}} and the log-likelihood of the parameter vector p = [p1, ..., pn] is l ( p ) = ln ∏ i j [ Pr ( i > j ) ] w i j = ∑ i = 1 n ∑ j = 1 n ln [ ( p i p i + p j ) w i j ]... | Wikipedia - Bradley–Terry model - Inference | 521 | 1,090 | null |
{\displaystyle {\begin{aligned}{\mathcal {l}}(\mathbf {p} )&=\ln \prod _{ij}{{\bigl [}\Pr(i>j){\bigr ]}}^{w_{ij}}=\sum _{i=1}^{n}\sum _{j=1}^{n}\ln {\biggl [}\left({\frac {p_{i}}{p_{i}+p_{j}}}\right)^{w_{ij}}{\biggr ]}\\[6pt]&=\sum _{ij}w_{ij}\ln {\biggl (}{\frac {p_{i}}{p_{i}+p_{j}}}{\biggr )}=\sum _{ij}{\bigl [}w_{ij... | Wikipedia - Bradley–Terry model - Inference | 382 | 825 | null |
Starting from any convenient set of (positive) initial values for the p i {\displaystyle p_{i}} , one iteratively performs the update for all i in turn. The resulting parameters are arbitrary up to an overall multiplicative constant, so after computing all of the new values they should be normalized by dividing by thei... | Wikipedia - Bradley–Terry model - Inference | 282 | 978 | null |
Section: Inference > Worked example of solution procedure. Consider a sporting competition between four teams, who play a total of 22 games among themselves. Each team's wins are given in the rows of the table below and the opponents are given as the columns: For example, Team A has beat Team B twice and lost to team B... | Wikipedia - Bradley–Terry model - Inference > Worked example of solution procedure | 270 | 966 | null |
{\displaystyle p_{1}={\frac {\sum _{j(\neq 1)}w_{1j}p_{j}/(p_{1}+p_{j})}{\sum _{j(\neq 1)}w_{j1}/(p_{1}+p_{j})}}={\frac {2{\frac {1}{1+1}}+0{\frac {1}{1+1}}+1{\frac {1}{1+1}}}{3{\frac {1}{1+1}}+0{\frac {1}{1+1}}+4{\frac {1}{1+1}}}}=0.429.} Now, we apply (5) again to update p 2 {\displaystyle p_{2}} , making sure to use... | Wikipedia - Bradley–Terry model - Inference > Worked example of solution procedure | 345 | 590 | null |
_{j(\neq 2)}w_{2j}p_{j}/(p_{2}+p_{j})}{\sum _{j(\neq 2)}w_{j2}/(p_{2}+p_{j})}}={\frac {3{\frac {0.429}{1+0.429}}+5{\frac {1}{1+1}}+0{\frac {1}{1+1}}}{2{\frac {1}{1+0.429}}+3{\frac {1}{1+1}}+0{\frac {1}{1+1}}}}=1.172} Similarly for p 3 {\displaystyle p_{3}} and p 4 {\displaystyle p_{4}} we get p 3 = ∑ j ( ≠ 3 ) w 3 j p ... | Wikipedia - Bradley–Terry model - Inference > Worked example of solution procedure | 330 | 516 | null |
3)}w_{3j}p_{j}/(p_{3}+p_{j})}{\sum _{j(\neq 3)}w_{j3}/(p_{3}+p_{j})}}={\frac {0{\frac {0.429}{1+0.429}}+3{\frac {1.172}{1+1.172}}+1{\frac {1}{1+1}}}{0{\frac {1}{1+0.429}}+5{\frac {1}{1+1.172}}+3{\frac {1}{1+1}}}}=0.557} p 4 = ∑ j ( ≠ 4 ) w 4 j p j / ( p 4 + p j ) ∑ j ( ≠ 4 ) w j 4 / ( p 4 + p j ) = 4 0.429 1 + 0.429 + ... | Wikipedia - Bradley–Terry model - Inference > Worked example of solution procedure | 349 | 498 | null |
4)}w_{j4}/(p_{4}+p_{j})}}={\frac {4{\frac {0.429}{1+0.429}}+0{\frac {1.172}{1+1.172}}+3{\frac {0.557}{1+0.557}}}{1{\frac {1}{1+0.429}}+0{\frac {1}{1+1.172}}+1{\frac {1}{1+0.557}}}}=1.694} Then we normalize all the parameters by dividing by their geometric mean ( 0.429 × 1.172 × 0.557 × 1.694 ) 1 / 4 = 0.830 {\displayst... | Wikipedia - Bradley–Terry model - Inference > Worked example of solution procedure | 279 | 527 | null |
To improve the estimates further, we repeat the process, using the new p values. For example, p 1 = 2 ⋅ 1.413 0.516 + 1.413 + 0 ⋅ 0.672 0.516 + 0.672 + 1 ⋅ 2.041 0.516 + 2.041 3 ⋅ 1 0.516 + 1.413 + 0 ⋅ 1 0.516 + 0.672 + 4 ⋅ 1 0.516 + 2.041 = 0.725. {\displaystyle p_{1}={\frac {2\cdot {\frac {1.413}{0.516+1.413}}+0\cdot... | Wikipedia - Bradley–Terry model - Inference > Worked example of solution procedure | 327 | 602 | null |
{\displaystyle p_{1}={\frac {2\cdot {\frac {1.413}{0.516+1.413}}+0\cdot {\frac {0.672}{0.516+0.672}}+1\cdot {\frac {2.041}{0.516+2.041}}}{3\cdot {\frac {1}{0.516+1.413}}+0\cdot {\frac {1}{0.516+0.672}}+4\cdot {\frac {1}{0.516+2.041}}}}=0.725.} Repeating this process for the remaining parameters and normalizing, we get ... | Wikipedia - Bradley–Terry model - Inference > Worked example of solution procedure | 308 | 759 | null |
Section: Variations > Crowd-BT. The Crowd-BT model, developed in 2013 by Chen et al, attempts to extend the standard Bradley–Terry model for crowdsourced settings while reducing the number of comparisons needed by taking into account the reliability of each judge. In particular, it identifies and excludes judges presum... | Wikipedia - Bradley–Terry model - Variations > Crowd-BT | 153 | 765 | null |
Article: Category utility. Category utility is a measure of "category goodness" defined in Gluck & Corter (1985) and Corter & Gluck (1992). It attempts to maximize both the probability that two objects in the same category have attribute values in common, and the probability that objects from different categories have ... | Wikipedia - Category utility - Summary | 242 | 1,181 | null |
Section: Probability-theoretic definition of category utility. The probability-theoretic definition of category utility given in Fisher (1987) and Witten & Frank (2005) is as follows: C U ( C , F ) = 1 p ∑ c j ∈ C p ( c j ) [ ∑ f i ∈ F ∑ k = 1 m p ( f i k | c j ) 2 − ∑ f i ∈ F ∑ k = 1 m p ( f i k ) 2 ] {\displaystyle C... | Wikipedia - Category utility - Probability-theoretic definition of category utility | 347 | 743 | null |
The probability-theoretic definition of category utility given in Fisher (1987) and Witten & Frank (2005) is as follows: C U ( C , F ) = 1 p ∑ c j ∈ C p ( c j ) [ ∑ f i ∈ F ∑ k = 1 m p ( f i k | c j ) 2 − ∑ f i ∈ F ∑ k = 1 m p ( f i k ) 2 ] {\displaystyle CU(C,F)={\tfrac {1}{p}}\sum _{c_{j}\in C}p(c_{j})\left[\sum _{f_... | Wikipedia - Category utility - Probability-theoretic definition of category utility | 477 | 1,120 | null |
The term p ( f i k ) {\displaystyle p(f_{ik})\ } designates the marginal probability that feature f i {\displaystyle f_{i}\ } takes on value k {\displaystyle k\ } , and the term p ( f i k | c j ) {\displaystyle p(f_{ik}|c_{j})\ } designates the category-conditional probability that feature f i {\displaystyle f_{i}\ } t... | Wikipedia - Category utility - Probability-theoretic definition of category utility | 199 | 654 | null |
The motivation and development of this expression for category utility, and the role of the multiplicand 1 p {\displaystyle \textstyle {\tfrac {1}{p}}} as a crude overfitting control, is given in the above sources. Loosely (Fisher 1987), the term p ( c j ) ∑ f i ∈ F ∑ k = 1 m p ( f i k | c j ) 2 {\displaystyle \textsty... | Wikipedia - Category utility - Probability-theoretic definition of category utility | 302 | 977 | null |
The information-theoretic definition of category utility for a set of entities with size- n {\displaystyle n\ } binary feature set F = { f i } , i = 1 … n {\displaystyle F=\{f_{i}\},\ i=1\ldots n} , and a binary category C = { c , c ¯ } {\displaystyle C=\{c,{\bar {c}}\}} is given in Gluck & Corter (1985) as follows: C ... | Wikipedia - Category utility - Information-theoretic definition of category utility | 346 | 707 | null |
p(c)\ } is the prior probability of an entity belonging to the positive category c {\displaystyle c\ } (in the absence of any feature information), p ( f i | c ) {\displaystyle p(f_{i}|c)\ } is the conditional probability of an entity having feature f i {\displaystyle f_{i}\ } given that the entity belongs to category ... | Wikipedia - Category utility - Information-theoretic definition of category utility | 344 | 1,111 | null |
The intuition behind the above expression is as follows: The term p ( c ) ∑ i = 1 n p ( f i | c ) log p ( f i | c ) {\displaystyle p(c)\textstyle \sum _{i=1}^{n}p(f_{i}|c)\log p(f_{i}|c)} represents the cost (in bits) of optimally encoding (or transmitting) feature information when it is known that the objects to be ... | Wikipedia - Category utility - Information-theoretic definition of category utility | 280 | 838 | null |
Section: Information-theoretic definition of category utility > Category utility and mutual information. Gluck & Corter (1985) and Corter & Gluck (1992) mention that the category utility is equivalent to the mutual information. Here is a simple demonstration of the nature of this equivalence. Assume a set of entities e... | Wikipedia - Category utility - Information-theoretic definition of category utility > Category utility and mutual information | 339 | 1,251 | null |
For the purposes of this demonstration, without loss of generality, feature set F {\displaystyle F} can be replaced with a single aggregate variable F a {\displaystyle F_{a}} that has cardinality m n {\displaystyle m^{n}} , and adopts a unique value v i , i = 1 … m n {\displaystyle v_{i},\ i=1\ldots m^{n}} correspondin... | Wikipedia - Category utility - Information-theoretic definition of category utility > Category utility and mutual information | 345 | 1,221 | null |
From the definition of mutual information for discrete variables, the mutual information I ( F a ; C ) {\displaystyle I(F_{a};C)} between the aggregate feature variable F a {\displaystyle F_{a}} and the category variable C {\displaystyle C} is given by: I ( F a ; C ) = ∑ v i ∈ F a ∑ c j ∈ C p ( v i , c j ) log p ( v ... | Wikipedia - Category utility - Information-theoretic definition of category utility > Category utility and mutual information | 350 | 871 | null |
In terms of the conditional probabilities this can be re-written (or defined) as I ( F a ; C ) = ∑ v i ∈ F a ∑ c j ∈ C p ( v i , c j ) log p ( v i | c j ) p ( v i ) = ∑ v i ∈ F a ∑ c j ∈ C p ( v i | c j ) p ( c j ) [ log p ( v i | c j ) − log p ( v i ) ] = ∑ v i ∈ F a ∑ c j ∈ C p ( v i | c j ) p ( c j ) log p (... | Wikipedia - Category utility - Information-theoretic definition of category utility > Category utility and mutual information | 344 | 781 | null |
{\begin{aligned}I(F_{a};C)&=\sum _{v_{i}\in F_{a}}\sum _{c_{j}\in C}p(v_{i},c_{j})\log {\frac {p(v_{i}|c_{j})}{p(v_{i})}}\\&=\sum _{v_{i}\in F_{a}}\sum _{c_{j}\in C}p(v_{i}|c_{j})p(c_{j})\left[\log p(v_{i}|c_{j})-\log p(v_{i})\right]\\&=\sum _{v_{i}\in F_{a}}\sum _{c_{j}\in C}p(v_{i}|c_{j})p(c_{j})\log p(v_{i}|c_{j})-\... | Wikipedia - Category utility - Information-theoretic definition of category utility > Category utility and mutual information | 342 | 424 | null |
_{c_{j}\in C}p(v_{i}|c_{j})p(c_{j})\log p(v_{i}|c_{j})-\sum _{v_{i}\in F_{a}}\sum _{c_{j}\in C}p(v_{i},c_{j})\log p(v_{i})\\&=\sum _{v_{i}\in F_{a}}\sum _{c_{j}\in C}p(v_{i}|c_{j})p(c_{j})\log p(v_{i}|c_{j})-\sum _{v_{i}\in F_{a}}\log p(v_{i})\sum _{c_{j}\in C}p(v_{i},c_{j})\\&=\sum _{v_{i}\in F_{a}}\sum _{c_{j}\in C}p... | Wikipedia - Category utility - Information-theoretic definition of category utility > Category utility and mutual information | 350 | 428 | null |
original definition of the category utility from above is rewritten with C = { c , c ¯ } {\displaystyle C=\{c,{\bar {c}}\}} , C U ( C , F ) = ∑ f i ∈ F ∑ c j ∈ C p ( f i | c j ) p ( c j ) log p ( f i | c j ) − ∑ f i ∈ F p ( f i ) log p ( f i ) {\displaystyle CU(C,F)=\sum _{f_{i}\in F}\sum _{c_{j}\in C}p(f_{i}|c_{j}... | Wikipedia - Category utility - Information-theoretic definition of category utility > Category utility and mutual information | 350 | 868 | null |
Section: Insensitivity of category utility to ordinality. Like the mutual information, the category utility is not sensitive to any ordering in the feature or category variable values. That is, as far as the category utility is concerned, the category set {small,medium,large,jumbo} is not qualitatively different from t... | Wikipedia - Category utility - Insensitivity of category utility to ordinality | 220 | 1,024 | null |
Section: Category "goodness": models and philosophy > What makes a good category?. At least since the time of Aristotle there has been a tremendous fascination in philosophy with the nature of concepts and universals. What kind of entity is a concept such as "horse"? Such abstractions do not designate any particular in... | Wikipedia - Category utility - Category "goodness": models and philosophy > What makes a good category? | 332 | 1,693 | null |
Section: Category "goodness": models and philosophy > What purpose do concepts serve?. One approach to answering such questions is to investigate the "role" or "purpose" of concepts in cognition. Thus the answer to "What are concepts good for in the first place?" by Mill (1843, p. 425) and many others is that classific... | Wikipedia - Category utility - Category "goodness": models and philosophy > What purpose do concepts serve? | 332 | 1,587 | null |
From this base, Mill reaches the following conclusion, which foreshadows much subsequent thinking about category goodness, including the notion of category utility: The ends of scientific classification are best answered when the objects are formed into groups respecting which a greater number of general propositions c... | Wikipedia - Category utility - Category "goodness": models and philosophy > What purpose do concepts serve? | 300 | 1,552 | null |
Section: Category "goodness": models and philosophy > Attempts at formalization. A variety of different measures have been suggested with an aim of formally capturing this notion of "category goodness," the best known of which is probably the "cue validity". Cue validity of a feature f i {\displaystyle f_{i}\ } with re... | Wikipedia - Category utility - Category "goodness": models and philosophy > Attempts at formalization | 281 | 935 | null |
Clearly, these measures quantify only inference from feature to category (i.e., cue validity), but not from category to feature, i.e., the category validity p ( f i | c j ) {\displaystyle p(f_{i}|c_{j})\ } . Also, while the cue validity was originally intended to account for the demonstrable appearance of basic categor... | Wikipedia - Category utility - Category "goodness": models and philosophy > Attempts at formalization | 305 | 1,238 | null |
Section: Overview. The CIML community portal was created to facilitate an online virtual scientific community wherein anyone interested in CIML can share research, obtain resources, or simply learn more. The effort is currently led by Jacek Zurada (principal investigator), with Rammohan Ragade and Janusz Wojtusiak, aid... | Wikipedia - CIML community portal - Overview | 210 | 1,055 | null |
Section: Training > Constitutional AI. Constitutional AI is an approach developed by Anthropic for training AI systems, particularly language models like Claude, to be harmless and helpful without relying on extensive human feedback. The method, detailed in the paper "Constitutional AI: Harmlessness from AI Feedback" i... | Wikipedia - Claude (language model) - Training > Constitutional AI | 231 | 1,218 | null |
Section: Models > Claude 3 > Claude 3.5. On June 20, 2024, Anthropic released Claude 3.5 Sonnet, which demonstrated significantly improved performance on benchmarks compared to the larger Claude 3 Opus, notably in areas such as coding, multistep workflows, chart interpretation, and text extraction from images. Released... | Wikipedia - Claude (language model) - Models > Claude 3 > Claude 3.5 | 329 | 1,458 | null |
Section: Data configuration decay. Another important area is software engineering, where three types of data drift affecting data fidelity may be recognized. Changes in the software environment ("infrastructure drift") may invalidate software infrastructure configuration. "Structural drift" happens when the data schema... | Wikipedia - Concept drift - Data configuration decay | 261 | 1,436 | null |
Section: Examples. The behavior of the customers in an online shop may change over time. For example, if weekly merchandise sales are to be predicted, and a predictive model has been developed that works satisfactorily. The model may use inputs such as the amount of money spent on advertising, promotions being run, and... | Wikipedia - Concept drift - Examples | 229 | 1,169 | null |
Section: Possible remedies. To prevent deterioration in prediction accuracy because of concept drift, reactive and tracking solutions can be adopted. Reactive solutions retrain the model in reaction to a triggering mechanism, such as a change-detection test, to explicitly detect concept drift as a change in the statist... | Wikipedia - Concept drift - Possible remedies | 337 | 1,856 | null |
Article: Conditional random field. Conditional random fields (CRFs) are a class of statistical modeling methods often applied in pattern recognition and machine learning and used for structured prediction. Whereas a classifier predicts a label for a single sample without considering "neighbouring" samples, a CRF can ta... | Wikipedia - Conditional random field - Summary | 209 | 1,150 | null |
Section: Description. CRFs are a type of discriminative undirected probabilistic graphical model. Lafferty, McCallum and Pereira define a CRF on observations X {\displaystyle {\boldsymbol {X}}} and random variables Y {\displaystyle {\boldsymbol {Y}}} as follows: Let G = ( V , E ) {\displaystyle G=(V,E)} be a graph such... | Wikipedia - Conditional random field - Description | 182 | 508 | null |
Lafferty, McCallum and Pereira define a CRF on observations X {\displaystyle {\boldsymbol {X}}} and random variables Y {\displaystyle {\boldsymbol {Y}}} as follows: Let G = ( V , E ) {\displaystyle G=(V,E)} be a graph such that Y = ( Y v ) v ∈ V {\displaystyle {\boldsymbol {Y}}=({\boldsymbol {Y}}_{v})_{v\in V}} , so th... | Wikipedia - Conditional random field - Description | 461 | 1,132 | null |
Then ( X , Y ) {\displaystyle ({\boldsymbol {X}},{\boldsymbol {Y}})} is a conditional random field when each random variable Y v {\displaystyle {\boldsymbol {Y}}_{v}} , conditioned on X {\displaystyle {\boldsymbol {X}}} , obeys the Markov property with respect to the graph; that is, its probability is dependent only on... | Wikipedia - Conditional random field - Description | 415 | 1,082 | null |
Section: Description > Inference. For general graphs, the problem of exact inference in CRFs is intractable. The inference problem for a CRF is basically the same as for an MRF and the same arguments hold. However, there exist special cases for which exact inference is feasible: If the graph is a chain or a tree, messa... | Wikipedia - Conditional random field - Description > Inference | 164 | 828 | null |
Section: Description > Examples. In sequence modeling, the graph of interest is usually a chain graph. An input sequence of observed variables X {\displaystyle X} represents a sequence of observations and Y {\displaystyle Y} represents a hidden (or unknown) state variable that needs to be inferred given the observation... | Wikipedia - Conditional random field - Description > Examples | 243 | 1,010 | null |
inference, determining the most likely label sequence Y {\displaystyle Y} given X {\displaystyle X} . The conditional dependency of each Y i {\displaystyle Y_{i}} on X {\displaystyle X} is defined through a fixed set of feature functions of the form f ( i , Y i − 1 , Y i , X ) {\displaystyle f(i,Y_{i-1},Y_{i},X)} , whi... | Wikipedia - Conditional random field - Description > Examples | 334 | 1,483 | null |
Section: Variants > Higher-order CRFs and semi-Markov CRFs. CRFs can be extended into higher order models by making each Y i {\displaystyle Y_{i}} dependent on a fixed number k {\displaystyle k} of previous variables Y i − k , . . . , Y i − 1 {\displaystyle Y_{i-k},...,Y_{i-1}} . In conventional formulations of higher ... | Wikipedia - Conditional random field - Variants > Higher-order CRFs and semi-Markov CRFs | 317 | 1,389 | null |
Section: Variants > Latent-dynamic conditional random field. Latent-dynamic conditional random fields (LDCRF) or discriminative probabilistic latent variable models (DPLVM) are a type of CRFs for sequence tagging tasks. They are latent variable models that are trained discriminatively. In an LDCRF, like in any sequence... | Wikipedia - Conditional random field - Variants > Latent-dynamic conditional random field | 318 | 1,014 | null |
Instead of directly modeling P(y|x) as an ordinary linear-chain CRF would do, a set of latent variables h is "inserted" between x and y using the chain rule of probability: P ( y | x ) = ∑ h P ( y | h , x ) P ( h | x ) {\displaystyle P(\mathbf {y} |\mathbf {x} )=\sum _{\mathbf {h} }P(\mathbf {y} |\mathbf {h} ,\mathbf {... | Wikipedia - Conditional random field - Variants > Latent-dynamic conditional random field | 221 | 785 | null |
Article: Confusion matrix. In the field of machine learning and specifically the problem of statistical classification, a confusion matrix, also known as error matrix, is a specific table layout that allows visualization of the performance of an algorithm, typically a supervised learning one; in unsupervised learning i... | Wikipedia - Confusion matrix - Summary | 202 | 1,011 | null |
Section: Example. Given a sample of 12 individuals, 8 that have been diagnosed with cancer and 4 that are cancer-free, where individuals with cancer belong to class 1 (positive) and non-cancer individuals belong to class 0 (negative), we can display that data as follows: Assume that we have a classifier that distinguis... | Wikipedia - Confusion matrix - Example | 329 | 1,722 | null |
Fourth, if the actual classification is negative and the predicted classification is negative (0,0), this is called a true negative result because the negative sample gets correctly identified by the classifier. We can then perform the comparison between actual and predicted classifications and add this information to ... | Wikipedia - Confusion matrix - Example | 350 | 1,637 | null |
Section: Table of confusion. In predictive analytics, a table of confusion (sometimes also called a confusion matrix) is a table with two rows and two columns that reports the number of true positives, false negatives, false positives, and true negatives. This allows more detailed analysis than simply observing the pro... | Wikipedia - Confusion matrix - Table of confusion | 272 | 1,357 | null |
Section: Algorithm. The CLIP method trains a pair of models contrastively. One model takes in a piece of text as input and outputs a single vector representing its semantic content. The other model takes in an image and similarly outputs a single vector representing its visual content. The models are trained so that th... | Wikipedia - Contrastive Language-Image Pre-training - Algorithm | 221 | 911 | null |
Two vectors are considered "similar" if their dot product is large. The loss incurred on this batch is the multi-class N-pair loss, which is a symmetric cross-entropy loss over similarity scores: − 1 N ∑ i ln e v i ⋅ w i / T ∑ j e v i ⋅ w j / T − 1 N ∑ j ln e v j ⋅ w j / T ∑ i e v i ⋅ w j / T {\displaystyle -{\frac... | Wikipedia - Contrastive Language-Image Pre-training - Algorithm | 303 | 722 | null |
The loss incurred on this batch is the multi-class N-pair loss, which is a symmetric cross-entropy loss over similarity scores: − 1 N ∑ i ln e v i ⋅ w i / T ∑ j e v i ⋅ w j / T − 1 N ∑ j ln e v j ⋅ w j / T ∑ i e v i ⋅ w j / T {\displaystyle -{\frac {1}{N}}\sum _{i}\ln {\frac {e^{v_{i}\cdot w_{i}/T}}{\sum _{j}e^{v_{... | Wikipedia - Contrastive Language-Image Pre-training - Algorithm | 359 | 880 | null |
The parameter T > 0 {\displaystyle T>0} is the temperature, which is parameterized in the original CLIP model as T = e − τ {\displaystyle T=e^{-\tau }} where τ ∈ R {\displaystyle \tau \in \mathbb {R} } is a learned parameter. Other loss functions are possible. For example, Sigmoid CLIP (SigLIP) proposes the following l... | Wikipedia - Contrastive Language-Image Pre-training - Algorithm | 286 | 715 | null |
Section: CLIP models > Image model. The image encoding models used in CLIP are typically vision transformers (ViT). The naming convention for these models often reflects the specific ViT architecture used. For instance, "ViT-L/14" means a "vision transformer large" (compared to other models in the same series) with a p... | Wikipedia - Contrastive Language-Image Pre-training - CLIP models > Image model | 307 | 1,283 | null |
For example, in the original OpenAI model, the ResNet models have embedding dimensions ranging from 512 to 1024,: Table 19 and for the ViTs, from 512 to 768.: Table 20 Its implementation of ViT was the same as the original one, with one modification: after position embeddings are added to the initial patch embeddings, ... | Wikipedia - Contrastive Language-Image Pre-training - CLIP models > Image model | 247 | 1,015 | null |
Section: CLIP models > Text model. The text encoding models used in CLIP are typically Transformers. In the original OpenAI report, they reported using a Transformer (63M-parameter, 12-layer, 512-wide, 8 attention heads) with lower-cased byte pair encoding (BPE) with 49152 vocabulary size. Context length was capped at ... | Wikipedia - Contrastive Language-Image Pre-training - CLIP models > Text model | 235 | 964 | null |
Section: Dataset > WebImageText. The CLIP models released by OpenAI were trained on a dataset called "WebImageText" (WIT) containing 400 million pairs of images and their corresponding captions scraped from the internet. The total number of words in this dataset is similar in scale to the WebText dataset used for train... | Wikipedia - Contrastive Language-Image Pre-training - Dataset > WebImageText | 189 | 826 | null |
Section: Dataset > WebImageText > Data preprocessing. For the CLIP image models, the input images are preprocessed by first dividing each of the R, G, B values of an image by the maximum possible value, so that these values fall between 0 and 1, then subtracting by [0.48145466, 0.4578275, 0.40821073], and dividing by [... | Wikipedia - Contrastive Language-Image Pre-training - Dataset > WebImageText > Data preprocessing | 289 | 989 | null |
Section: Training. In the original OpenAI CLIP report, they reported training 5 ResNet and 3 ViT (ViT-B/32, ViT-B/16, ViT-L/14). Each was trained for 32 epochs. The largest ResNet model took 18 days to train on 592 V100 GPUs. The largest ViT model took 12 days on 256 V100 GPUs. All ViT models were trained on 224x224 im... | Wikipedia - Contrastive Language-Image Pre-training - Training | 191 | 647 | null |
Section: Applications > Applications in other domains. CLIP has been used in various domains beyond its original purpose: Image Featurizer: CLIP's image encoder can be adapted as a pre-trained image featurizer. This can then be fed into other AI models. Text-to-Image Generation: Models like Stable Diffusion use CLIP's ... | Wikipedia - Contrastive Language-Image Pre-training - Applications > Applications in other domains | 164 | 739 | null |
Section: Cost Matrix. The cost matrix is a crucial element within cost-sensitive modeling, explicitly defining the costs or benefits associated with different prediction errors in classification tasks. Represented as a table, the matrix aligns true and predicted classes, assigning a cost value to each combination. For ... | Wikipedia - Cost-sensitive machine learning - Cost Matrix | 336 | 1,331 | null |
Section: CPL description > Sharing among same-arity predicates. Each predicate P in the ontology has a list of other same-arity predicates with which P is mutually exclusive. If A is mutually exclusive with predicate B, A’s positive instances and patterns become negative instances and negative patterns for B. For examp... | Wikipedia - Coupled pattern learner - CPL description > Sharing among same-arity predicates | 160 | 671 | null |
Section: Meta-Bootstrap Learner. Meta-Bootstrap Learner (MBL) was also proposed by the authors of CPL. Meta-Bootstrap learner couples the training of multiple extraction techniques with a multi-view constraint, which requires the extractors to agree. It makes addition of coupling constraints on top of existing extracti... | Wikipedia - Coupled pattern learner - Meta-Bootstrap Learner | 227 | 1,065 | null |
Article: Cross-entropy method. The cross-entropy (CE) method is a Monte Carlo method for importance sampling and optimization. It is applicable to both combinatorial and continuous problems, with either a static or noisy objective. The method approximates the optimal importance sampling estimator by repeating two phase... | Wikipedia - Cross-entropy method - Summary | 162 | 884 | null |
Section: Estimation via importance sampling. Consider the general problem of estimating the quantity ℓ = E u [ H ( X ) ] = ∫ H ( x ) f ( x ; u ) d x {\displaystyle \ell =\mathbb {E} _{\mathbf {u} }[H(\mathbf {X} )]=\int H(\mathbf {x} )\,f(\mathbf {x} ;\mathbf {u} )\,{\textrm {d}}\mathbf {x} } , where H {\displaystyle H... | Wikipedia - Cross-entropy method - Estimation via importance sampling | 173 | 467 | null |
Consider the general problem of estimating the quantity ℓ = E u [ H ( X ) ] = ∫ H ( x ) f ( x ; u ) d x {\displaystyle \ell =\mathbb {E} _{\mathbf {u} }[H(\mathbf {X} )]=\int H(\mathbf {x} )\,f(\mathbf {x} ;\mathbf {u} )\,{\textrm {d}}\mathbf {x} } , where H {\displaystyle H} is some performance function and f ( x ; u ... | Wikipedia - Cross-entropy method - Estimation via importance sampling | 348 | 806 | null |
Using importance sampling this quantity can be estimated as ℓ ^ = 1 N ∑ i = 1 N H ( X i ) f ( X i ; u ) g ( X i ) {\displaystyle {\hat {\ell }}={\frac {1}{N}}\sum _{i=1}^{N}H(\mathbf {X} _{i}){\frac {f(\mathbf {X} _{i};\mathbf {u} )}{g(\mathbf {X} _{i})}}} , where X 1 , … , X N {\displaystyle \mathbf {X} _{1},\dots ,\m... | Wikipedia - Cross-entropy method - Estimation via importance sampling | 339 | 881 | null |
Generate a random sample X 1 , … , X N {\displaystyle \mathbf {X} _{1},\dots ,\mathbf {X} _{N}} from f ( ⋅ ; v ( t − 1 ) ) {\displaystyle f(\cdot ;\mathbf {v} ^{(t-1)})} Solve for v ( t ) {\displaystyle \mathbf {v} ^{(t)}} , where v ( t ) = argmax v 1 N ∑ i = 1 N H ( X i ) f ( X i ; u ) f ( X i ; v ( t − 1 ) ) log ... | Wikipedia - Cross-entropy method - Generic CE algorithm | 342 | 739 | null |
In several cases, the solution to step 3 can be found analytically. Situations in which this occurs are When f {\displaystyle f\,} belongs to the natural exponential family When f {\displaystyle f\,} is discrete with finite support When H ( X ) = I { x ∈ A } {\displaystyle H(\mathbf {X} )=\mathrm {I} _{\{\mathbf {x} \i... | Wikipedia - Cross-entropy method - Generic CE algorithm | 233 | 622 | null |
Section: Continuous optimization—example. The same CE algorithm can be used for optimization, rather than estimation. Suppose the problem is to maximize some function S {\displaystyle S} , for example, S ( x ) = e − ( x − 2 ) 2 + 0.8 e − ( x + 2 ) 2 {\displaystyle S(x)={\textrm {e}}^{-(x-2)^{2}}+0.8\,{\textrm {e}}^{-(x... | Wikipedia - Cross-entropy method - Continuous optimization—example | 335 | 903 | null |
To apply CE, one considers first the associated stochastic problem of estimating P θ ( S ( X ) ≥ γ ) {\displaystyle \mathbb {P} _{\boldsymbol {\theta }}(S(X)\geq \gamma )} for a given level γ {\displaystyle \gamma \,} , and parametric family { f ( ⋅ ; θ ) } {\displaystyle \left\{f(\cdot ;{\boldsymbol {\theta }})\right\... | Wikipedia - Cross-entropy method - Continuous optimization—example | 329 | 837 | null |
Hence, for a given γ {\displaystyle \gamma \,} , the goal is to find θ {\displaystyle {\boldsymbol {\theta }}} so that D K L ( I { S ( x ) ≥ γ } ‖ f θ ) {\displaystyle D_{\mathrm {KL} }({\textrm {I}}_{\{S(x)\geq \gamma \}}\|f_{\boldsymbol {\theta }})} is minimized. This is done by solving the sample version (stochastic... | Wikipedia - Cross-entropy method - Continuous optimization—example | 258 | 972 | null |
Section: Continuous optimization—example > Pseudocode. // Initialize parameters μ := −6 σ2 := 100 t := 0 maxits := 100 N := 100 Ne := 10 // While maxits not exceeded and not converged while t < maxits and σ2 > ε do // Obtain N samples from current sampling distribution X := SampleGaussian(μ, σ2, N) // Evaluate objectiv... | Wikipedia - Cross-entropy method - Continuous optimization—example > Pseudocode | 200 | 650 | null |
Article: Cross-validation (statistics). Cross-validation, sometimes called rotation estimation or out-of-sample testing, is any of various similar model validation techniques for assessing how the results of a statistical analysis will generalize to an independent data set. Cross-validation includes resampling and samp... | Wikipedia - Cross-validation (statistics) - Summary | 341 | 1,663 | null |
The MSE for given estimated parameter values a and β on the training set (xi, yi) 1 ≤ i ≤ n is defined as: MSE = 1 n ∑ i = 1 n ( y i − y ^ i ) 2 = 1 n ∑ i = 1 n ( y i − a − β T x i ) 2 = 1 n ∑ i = 1 n ( y i − a − β 1 x i 1 − ⋯ − β p x i p ) 2 {\displaystyle {\begin{aligned}{\text{MSE}}&={\frac {1}{n}}\sum _{i=1}^{n}(y_... | Wikipedia - Cross-validation (statistics) - Motivation > Example: linear regression | 350 | 722 | null |
the MSE for the validation set (the expected value is taken over the distribution of training sets). Thus, a fitted model and computed MSE on the training set will result in an optimistically biased assessment of how well the model will fit an independent data set. This biased estimate is called the in-sample estimate ... | Wikipedia - Cross-validation (statistics) - Motivation > Example: linear regression | 198 | 933 | null |
Section: Types > Exhaustive cross-validation > Leave-p-out cross-validation. Leave-p-out cross-validation (LpO CV) involves using p observations as the validation set and the remaining observations as the training set. This is repeated on all ways to cut the original sample on a validation set of p observations and a t... | Wikipedia - Cross-validation (statistics) - Types > Exhaustive cross-validation > Leave-p-out cross-validation | 255 | 966 | null |
Section: Types > Exhaustive cross-validation > Leave-one-out cross-validation. Leave-one-out cross-validation (LOOCV) is a particular case of leave-p-out cross-validation with p = 1. The process looks similar to jackknife; however, with cross-validation one computes a statistic on the left-out sample(s), while with jac... | Wikipedia - Cross-validation (statistics) - Types > Exhaustive cross-validation > Leave-one-out cross-validation | 198 | 747 | null |
However, n {\displaystyle n} passes may still require quite a large computation time, in which case other approaches such as k-fold cross validation may be more appropriate. Pseudo-code algorithm: Input: x, {vector of length N with x-values of incoming points} y, {vector of length N with y-values of the expected result... | Wikipedia - Cross-validation (statistics) - Types > Exhaustive cross-validation > Leave-one-out cross-validation | 282 | 770 | null |
Section: Types > Non-exhaustive cross-validation > k-fold cross-validation. In k-fold cross-validation, the original sample is randomly partitioned into k equal sized subsamples, often referred to as "folds". Of the k subsamples, a single subsample is retained as the validation data for testing the model, and the remai... | Wikipedia - Cross-validation (statistics) - Types > Non-exhaustive cross-validation > k-fold cross-validation | 335 | 1,470 | null |
Section: Types > Non-exhaustive cross-validation > Holdout method. In the holdout method, we randomly assign data points to two sets d0 and d1, usually called the training set and the test set, respectively. The size of each of the sets is arbitrary although typically the test set is smaller than the training set. We t... | Wikipedia - Cross-validation (statistics) - Types > Non-exhaustive cross-validation > Holdout method | 255 | 1,183 | null |
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