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Section: Methods > Convergent Cross Mapping. Convergent cross mapping (CCM) leverages a corollary to the Generalized Takens Theorem that it should be possible to cross predict or cross map between variables observed from the same system. Suppose that in some dynamical system involving variables X {\displaystyle X} and ... | Wikipedia - Empirical dynamic modeling - Methods > Convergent Cross Mapping | 294 | 1,167 | null |
Section: Background. The following situation is a general setting of many supervised learning problems. There are two spaces of objects X {\displaystyle X} and Y {\displaystyle Y} and we would like to learn a function h : X → Y {\displaystyle \ h:X\to Y} (often called hypothesis) which outputs an object y ∈ Y {\display... | Wikipedia - Empirical risk minimization - Background | 237 | 706 | null |
To do so, there is a training set of n {\displaystyle n} examples ( x 1 , y 1 ) , … , ( x n , y n ) {\displaystyle \ (x_{1},y_{1}),\ldots ,(x_{n},y_{n})} where x i ∈ X {\displaystyle x_{i}\in X} is an input and y i ∈ Y {\displaystyle y_{i}\in Y} is the corresponding response that is desired from h ( x i ) {\displaystyl... | Wikipedia - Empirical risk minimization - Background | 301 | 823 | null |
The assumption of a joint probability distribution allows for the modelling of uncertainty in predictions (e.g. from noise in data) because y {\displaystyle y} is not a deterministic function of x {\displaystyle x} , but rather a random variable with conditional distribution P ( y | x ) {\displaystyle P(y|x)} for a fix... | Wikipedia - Empirical risk minimization - Background | 238 | 865 | null |
The risk associated with hypothesis h ( x ) {\displaystyle h(x)} is then defined as the expectation of the loss function: R ( h ) = E [ L ( h ( x ) , y ) ] = ∫ L ( h ( x ) , y ) d P ( x , y ) . {\displaystyle R(h)=\mathbf {E} [L(h(x),y)]=\int L(h(x),y)\,dP(x,y).} A loss function commonly used in theory is the 0-1 loss ... | Wikipedia - Empirical risk minimization - Background | 308 | 758 | null |
The ultimate goal of a learning algorithm is to find a hypothesis h ∗ {\displaystyle h^{*}} among a fixed class of functions H {\displaystyle {\mathcal {H}}} for which the risk R ( h ) {\displaystyle R(h)} is minimal: h ∗ = a r g m i n h ∈ H R ( h ) . {\displaystyle h^{*}={\underset {h\in {\mathcal {H}}}{\operatorname ... | Wikipedia - Empirical risk minimization - Background | 161 | 481 | null |
Section: Formal definition. In general, the risk R ( h ) {\displaystyle R(h)} cannot be computed because the distribution P ( x , y ) {\displaystyle P(x,y)} is unknown to the learning algorithm. However, given a sample of iid training data points, we can compute an estimate, called the empirical risk, by computing the ... | Wikipedia - Empirical risk minimization - Formal definition | 345 | 1,098 | null |
Section: Properties. Guarantees for the performance of empirical risk minimization depend strongly on the function class selected as well as the distributional assumptions made. In general, distribution-free methods are too coarse, and do not lead to practical bounds. However, they are still useful in deriving asymptot... | Wikipedia - Empirical risk minimization - Properties | 243 | 1,052 | null |
Consider the risk L {\displaystyle L} defined over the hypothesis class C {\displaystyle {\mathcal {C}}} with growth function S ( C , n ) {\displaystyle {\mathcal {S}}({\mathcal {C}},n)} given a dataset of size n {\displaystyle n} . Then, for every ϵ > 0 {\displaystyle \epsilon >0} : P ( L ( ϕ n ) − L ( ϕ ∗ ) > ϵ ) ≤ 8... | Wikipedia - Empirical risk minimization - Properties | 246 | 710 | null |
Section: Properties > Impossibility results. It is also possible to show lower bounds on algorithm performance if no distributional assumptions are made. This is sometimes referred to as the No free lunch theorem. Even though a specific learning algorithm may provide the asymptotically optimal performance for any distr... | Wikipedia - Empirical risk minimization - Properties > Impossibility results | 348 | 1,410 | null |
Section: Properties > Computational complexity. Empirical risk minimization for a classification problem with a 0-1 loss function is known to be an NP-hard problem even for a relatively simple class of functions such as linear classifiers. Nevertheless, it can be solved efficiently when the minimal empirical risk is ze... | Wikipedia - Empirical risk minimization - Properties > Computational complexity | 200 | 936 | null |
Article: Energy-based model. An energy-based model (EBM) (also called Canonical Ensemble Learning or Learning via Canonical Ensemble – CEL and LCE, respectively) is an application of canonical ensemble formulation from statistical physics for learning from data. The approach prominently appears in generative artificial... | Wikipedia - Energy-based model - Summary | 208 | 1,064 | null |
Section: Description. For a given input x {\displaystyle x} , the model describes an energy E θ ( x ) {\displaystyle E_{\theta }(x)} such that the Boltzmann distribution P θ ( x ) = exp ( − β E θ ( x ) ) / Z ( θ ) {\displaystyle P_{\theta }(x)=\exp(-\beta E_{\theta }(x))/Z(\theta )} is a probability (density), and ty... | Wikipedia - Energy-based model - Description | 245 | 754 | null |
Since the normalization constant: Z ( θ ) := ∫ x ∈ X exp ( − β E θ ( x ) ) d x {\displaystyle Z(\theta ):=\int _{x\in X}\exp(-\beta E_{\theta }(x))dx} (also known as the partition function) depends on all the Boltzmann factors of all possible inputs x {\displaystyle x} , it cannot be easily computed or reliably estim... | Wikipedia - Energy-based model - Description | 333 | 1,044 | null |
However, for maximizing the likelihood during training, the gradient of the log-likelihood of a single training example x {\displaystyle x} is given by using the chain rule: ∂ θ log ( P θ ( x ) ) = E x ′ ∼ P θ [ ∂ θ E θ ( x ′ ) ] − ∂ θ E θ ( x ) ( ∗ ) {\displaystyle \partial _{\theta }\log \left(P_{\theta }(x)\right)... | Wikipedia - Energy-based model - Description | 243 | 778 | null |
Early energy-based models, such as the 2003 Boltzmann machine by Hinton, estimated this expectation via blocked Gibbs sampling. Newer approaches make use of more efficient Stochastic Gradient Langevin Dynamics (LD), drawing samples using: x 0 ′ ∼ P 0 , x i + 1 ′ = x i ′ − α 2 ∂ E θ ( x i ′ ) ∂ x i ′ + ϵ {\displaystyle ... | Wikipedia - Energy-based model - Description | 229 | 636 | null |
A replay buffer of past values x i ′ {\displaystyle x_{i}'} is used with LD to initialize the optimization module. The parameters θ {\displaystyle \theta } of the neural network are therefore trained in a generative manner via MCMC-based maximum likelihood estimation: the learning process follows an "analysis by synthe... | Wikipedia - Energy-based model - Description | 286 | 1,247 | null |
Section: Characteristics. EBMs demonstrate useful properties: Simplicity and stability–The EBM is the only object that needs to be designed and trained. Separate networks need not be trained to ensure balance. Adaptive computation time–An EBM can generate sharp, diverse samples or (more quickly) coarse, less diverse sa... | Wikipedia - Energy-based model - Characteristics | 227 | 1,156 | null |
Section: Applications. Target applications include natural language processing, robotics and computer vision. The first energy-based generative neural network is the generative ConvNet proposed in 2016 for image patterns, where the neural network is a convolutional neural network. The model has been generalized to vari... | Wikipedia - Energy-based model - Applications | 150 | 719 | null |
Section: Extensions > Joint energy-based models. Joint energy-based models (JEM), proposed in 2020 by Grathwohl et al., allow any classifier with softmax output to be interpreted as energy-based model. The key observation is that such a classifier is trained to predict the conditional probability p θ ( y | x ) = e f → ... | Wikipedia - Energy-based model - Extensions > Joint energy-based models | 340 | 804 | null |
The key observation is that such a classifier is trained to predict the conditional probability p θ ( y | x ) = e f → θ ( x ) [ y ] ∑ j = 1 K e f → θ ( x ) [ j ] for y = 1 , … , K and f → θ = ( f 1 , … , f K ) ∈ R K , {\displaystyle p_{\theta }(y|x)={\frac {e^{{\vec {f}}_{\theta }(x)[y]}}{\sum _{j=1}^{K}e^{{\vec {f}}_{... | Wikipedia - Energy-based model - Extensions > Joint energy-based models | 463 | 1,017 | null |
Without any change to the logits it was proposed to reinterpret the logits to describe a joint probability density: p θ ( y , x ) = e f → θ ( x ) [ y ] Z ( θ ) , {\displaystyle p_{\theta }(y,x)={\frac {e^{{\vec {f}}_{\theta }(x)[y]}}{Z(\theta )}},} with unknown partition function Z ( θ ) {\displaystyle Z(\theta )} and ... | Wikipedia - Energy-based model - Extensions > Joint energy-based models | 452 | 992 | null |
Article: Equalized odds. Equalized odds, also referred to as conditional procedure accuracy equality and disparate mistreatment, is a measure of fairness in machine learning. A classifier satisfies this definition if the subjects in the protected and unprotected groups have equal true positive rate and equal false posi... | Wikipedia - Equalized odds - Summary | 327 | 1,249 | null |
Article: Evaluation of binary classifiers. Evaluation of a binary classifier typically assigns a numerical value, or values, to a classifier that represent its accuracy. An example is error rate, which measures how frequently the classifier makes a mistake. There are many metrics that can be used; different fields have... | Wikipedia - Evaluation of binary classifiers - Summary | 185 | 981 | null |
Section: Contingency table. Given a data set, a classification (the output of a classifier on that set) gives two numbers: the number of positives and the number of negatives, which add up to the total size of the set. To evaluate a classifier, one compares its output to another reference classification – ideally a per... | Wikipedia - Evaluation of binary classifiers - Contingency table | 341 | 1,625 | null |
These numbers can then be totaled, yielding both a grand total and marginal totals. Totaling the entire table, the number of true positives, false negatives, true negatives, and false positives add up to 100% of the set. Totaling the columns (adding vertically) the number of true positives and false positives add up to... | Wikipedia - Evaluation of binary classifiers - Contingency table | 328 | 1,603 | null |
Section: Pairs of metrics > Sensitivity and specificity. The fundamental prevalence-independent statistics are sensitivity and specificity. Sensitivity or True Positive Rate (TPR), also known as recall, is the proportion of people that tested positive and are positive (True Positive, TP) of all the people that actually... | Wikipedia - Evaluation of binary classifiers - Pairs of metrics > Sensitivity and specificity | 339 | 1,598 | null |
In more practical, less contrived instances, however, there is usually a trade-off, such that they are inversely proportional to one another to some extent. This is because we rarely measure the actual thing we would like to classify; rather, we generally measure an indicator of the thing we would like to classify, ref... | Wikipedia - Evaluation of binary classifiers - Pairs of metrics > Sensitivity and specificity | 281 | 1,347 | null |
Section: Pairs of metrics > Positive and negative predictive values > Impact of prevalence on predictive values. Prevalence has a significant impact on prediction values. As an example, suppose there is a test for a disease with 99% sensitivity and 99% specificity. If 2000 people are tested and the prevalence (in the s... | Wikipedia - Evaluation of binary classifiers - Pairs of metrics > Positive and negative predictive values > Impact of prevalence on predictive values | 270 | 1,232 | null |
Section: Pairs of metrics > Relationships. There are various relationships between these ratios. If the prevalence, sensitivity, and specificity are known, the positive predictive value can be obtained from the following identity: PPV = ( sensitivity ) ( prevalence ) ( sensitivity ) ( prevalence ) + ( 1 − specificity )... | Wikipedia - Evaluation of binary classifiers - Pairs of metrics > Relationships | 295 | 943 | null |
Section: Unitary metrics. In addition to the paired metrics, there are also unitary metrics that give a single number to evaluate the test. Perhaps the simplest statistic is accuracy or fraction correct (FC), which measures the fraction of all instances that are correctly categorized; it is the ratio of the number of c... | Wikipedia - Evaluation of binary classifiers - Unitary metrics | 320 | 1,251 | null |
One diagnostic rule could be compared to another if the other's accuracy is known and substituted for p0 in calculating the z statistic. If not known and calculated from data, an accuracy comparison test could be made using "Two-proportion z-test, pooled for Ho: p1 = p2". Not used very much is the complementary statist... | Wikipedia - Evaluation of binary classifiers - Unitary metrics | 316 | 1,326 | null |
There is a one-parameter family of statistics, with parameter β, which determines the relative weights of precision and recall. The traditional or balanced F-score (F1 score) is the harmonic mean of precision and recall: F 1 = 2 ⋅ p r e c i s i o n ⋅ r e c a l l p r e c i s i o n + r e c a l l {\displaystyle F_{1}=2\cd... | Wikipedia - Evaluation of binary classifiers - Unitary metrics | 261 | 1,095 | null |
Section: Choosing the appropriate form of evaluation. Hand has highlighted the importance of choosing an appropriate method of evaluation. However, of the many different methods for evaluating the accuracy of a classifier, there is no general method for determining which method should be used in which circumstances. Di... | Wikipedia - Evaluation of binary classifiers - Choosing the appropriate form of evaluation | 229 | 1,260 | null |
Section: In information systems. Information retrieval systems, such as databases and web search engines, are evaluated by many different metrics, some of which are derived from the confusion matrix, which divides results into true positives (documents correctly retrieved), true negatives (documents correctly not retri... | Wikipedia - Evaluation of binary classifiers - In information systems | 324 | 1,713 | null |
Section: General framework. Let F n {\displaystyle F_{n}\,} and R n {\displaystyle R_{n}\,} be collections of functions on n {\displaystyle n\,} variables. Given an ideal function f ∈ F n {\displaystyle f\in F_{n}} , the goal is to find by local search a representation r ∈ R n {\displaystyle r\in R_{n}} that closely ap... | Wikipedia - Evolvability (computer science) - General framework | 319 | 973 | null |
However, this need not be the case. The goal then, is to find a representation that closely matches the phenotype of the ideal function, and the spirit of the local search is to allow only small changes in the genotype. Let the neighborhood N ( r ) {\displaystyle N(r)\,} of a representation r {\displaystyle r\,} be the... | Wikipedia - Evolvability (computer science) - General framework | 298 | 859 | null |
{\displaystyle \operatorname {Perf} (f,r)=\sum _{x\in X_{n}}f(x)r(x)D_{n}(x).} Note that Perf ( f , r ) = Prob ( f ( x ) = r ( x ) ) − Prob ( f ( x ) ≠ r ( x ) ) . {\displaystyle \operatorname {Perf} (f,r)=\operatorname {Prob} (f(x)=r(x))-\operatorname {Prob} (f(x)\neq r(x)).} In general, for non-Boolean function... | Wikipedia - Evolvability (computer science) - General framework | 339 | 933 | null |
The empirical performance is defined by Perf s ( f , r ) = 1 s ∑ x ∈ S f ( x ) r ( x ) , {\displaystyle \operatorname {Perf} _{s}(f,r)={\frac {1}{s}}\sum _{x\in S}f(x)r(x),} where S {\displaystyle S\,} is a multiset of s {\displaystyle s\,} independent selections from X n {\displaystyle X_{n}\,} according to D n {\di... | Wikipedia - Evolvability (computer science) - General framework | 325 | 879 | null |
Each r ′ ∈ N ( r ) {\displaystyle r'\in N(r)} is classified as beneficial, neutral, or deleterious, depending on its empirical performance. Specifically, r ′ {\displaystyle r'\,} is a beneficial mutation if Perf s ( f , r ′ ) − Perf s ( f , r ) ≥ t {\displaystyle \operatorname {Perf} _{s}(f,r')-\operatorname {Perf}... | Wikipedia - Evolvability (computer science) - General framework | 338 | 867 | null |
If there are any beneficial mutations, then Mut ( f , r , s , t ) {\displaystyle \operatorname {Mut} (f,r,s,t)} is equal to one of these at random. If there are no beneficial mutations, then Mut ( f , r , s , t ) {\displaystyle \operatorname {Mut} (f,r,s,t)} is equal to a random neutral mutation. In light of the si... | Wikipedia - Evolvability (computer science) - General framework | 301 | 948 | null |
Given r 0 ∈ R n {\displaystyle r_{0}\in R_{n}} , we define the sequence r 0 , r 1 , r 2 , … {\displaystyle r_{0},r_{1},r_{2},\ldots } by r i + 1 = Mut ( f , r i , s , t ) {\displaystyle r_{i+1}=\operatorname {Mut} (f,r_{i},s,t)} . Thus r g {\displaystyle r_{g}\,} is a random variable representing what r 0 {\displayst... | Wikipedia - Evolvability (computer science) - General framework | 222 | 570 | null |
We say that F {\displaystyle F\,} is evolvable by R {\displaystyle R\,} over D {\displaystyle D\,} if there exists polynomials p ( ⋅ , ⋅ ) {\displaystyle p(\cdot ,\cdot )} , s ( ⋅ , ⋅ ) {\displaystyle s(\cdot ,\cdot )} , t ( ⋅ , ⋅ ) {\displaystyle t(\cdot ,\cdot )} , and g ( ⋅ , ⋅ ) {\displaystyle g(\cdot ,\cdot )} suc... | Wikipedia - Evolvability (computer science) - General framework | 350 | 891 | null |
size is s ( n , 1 / ϵ ) {\displaystyle s(n,1/\epsilon )\,} , the tolerance is t ( 1 / n , ϵ ) {\displaystyle t(1/n,\epsilon )\,} , and the generation size is g ( n , 1 / ϵ ) {\displaystyle g(n,1/\epsilon )\,} . F {\displaystyle F\,} is evolvable over D {\displaystyle D\,} if it is evolvable by some R {\displaystyle R\,... | Wikipedia - Evolvability (computer science) - General framework | 168 | 451 | null |
Article: Expectation propagation. Expectation propagation (EP) is a technique in Bayesian machine learning. EP finds approximations to a probability distribution. It uses an iterative approach that uses the factorization structure of the target distribution. It differs from other Bayesian approximation approaches such ... | Wikipedia - Expectation propagation - Summary | 196 | 805 | null |
Variational Bayesian methods minimize K L ( q | | p ) {\displaystyle \mathrm {KL} (q||p)} instead. If q ( x ) {\displaystyle q(\mathbf {x} )} is a Gaussian N ( x | μ , Σ ) {\displaystyle {\mathcal {N}}(\mathbf {x} |\mu ,\Sigma )} , then K L ( p | | q ) {\displaystyle \mathrm {KL} (p||q)} is minimized with μ {\displayst... | Wikipedia - Expectation propagation - Summary | 199 | 536 | null |
Section: Details. An example of EBL using a perfect domain theory is a program that learns to play chess through example. A specific chess position that contains an important feature such as "Forced loss of black queen in two moves" includes many irrelevant features, such as the specific scattering of pawns on the boar... | Wikipedia - Explanation-based learning - Details | 281 | 1,408 | null |
Section: Application. An especially good application domain for an EBL is natural language processing (NLP). Here a rich domain theory, i.e., a natural language grammar—although neither perfect nor complete, is tuned to a particular application or particular language usage, using a treebank (training examples). Rayner ... | Wikipedia - Explanation-based learning - Application | 326 | 1,566 | null |
Section: Application in machine learning > Multi-armed bandit methods. The multi-armed bandit (MAB) problem was a classic example of the tradeoff, and many methods were developed for it, such as epsilon-greedy, Thompson sampling, and the upper confidence bound (UCB). See the page on MAB for details. In more complex RL ... | Wikipedia - Exploration–exploitation dilemma - Application in machine learning > Multi-armed bandit methods | 168 | 775 | null |
Section: Application in machine learning > Exploration problems. There are some problems that make exploration difficult. Sparse reward. If rewards occur only once a long while, then the agent might not persist in exploring. Furthermore, if the space of actions is large, then the sparse reward would mean the agent woul... | Wikipedia - Exploration–exploitation dilemma - Application in machine learning > Exploration problems | 157 | 820 | null |
Section: Application in machine learning > Exploration reward. This section based on. The exploration reward (also called exploration bonus) methods convert the exploration-exploitation dilemma into a balance of exploitations. That is, instead of trying to get the agent to balance exploration and exploitation, explorat... | Wikipedia - Exploration–exploitation dilemma - Application in machine learning > Exploration reward | 341 | 1,489 | null |
This is only possible in small and discrete state space. Density-based exploration extends count-based exploration by using a density model ρ n ( s ) {\displaystyle \rho _{n}(s)} . The idea is that, if a state has been visited, then nearby states are also partly-visited. In maximum entropy exploration, the entropy of t... | Wikipedia - Exploration–exploitation dilemma - Application in machine learning > Exploration reward | 178 | 544 | null |
Section: Application in machine learning > Prediction-based. This section based on. The forward dynamics model is a function for predicting the next state based on the current state and the current action: f : ( s t , a t ) ↦ s t + 1 {\displaystyle f:(s_{t},a_{t})\mapsto s_{t+1}} . The forward dynamics model is trained... | Wikipedia - Exploration–exploitation dilemma - Application in machine learning > Prediction-based | 344 | 1,051 | null |
That is, r t i = ‖ f ( s t , a t ) − ϕ ( s t + 1 ) ‖ 2 2 {\displaystyle r_{t}^{i}=\|f(s_{t},a_{t})-\phi (s_{t+1})\|_{2}^{2}} for some featurizer ϕ {\displaystyle \phi } . The featurizer can be the identity function (i.e. ϕ ( x ) = x {\displaystyle \phi (x)=x} ), randomly generated, the encoder-half of a variational aut... | Wikipedia - Exploration–exploitation dilemma - Application in machine learning > Prediction-based | 336 | 1,088 | null |
Random Network Distillation (RND) method attempts to solve this problem by teacher–student distillation. Instead of a forward dynamics model, it has two models f , f ′ {\displaystyle f,f'} . The f ′ {\displaystyle f'} teacher model is fixed, and the f {\displaystyle f} student model is trained to minimize ‖ f ( s ) − f... | Wikipedia - Exploration–exploitation dilemma - Application in machine learning > Prediction-based | 314 | 1,263 | null |
Section: Application in machine learning > Noise. For neural network–based agents, the NoisyNet method changes some of its neural network modules by noisy versions. That is, some network parameters are random variables from a probability distribution. The parameters of the distribution are themselves learnable. For exa... | Wikipedia - Exploration–exploitation dilemma - Application in machine learning > Noise | 228 | 735 | null |
Article: Fairness (machine learning). Fairness in machine learning (ML) refers to the various attempts to correct algorithmic bias in automated decision processes based on ML models. Decisions made by such models after a learning process may be considered unfair if they were based on variables considered sensitive (e.g... | Wikipedia - Fairness (machine learning) - Summary | 154 | 811 | null |
Section: Context. Discussion about fairness in machine learning is a relatively recent topic. Since 2016 there has been a sharp increase in research into the topic. This increase could be partly attributed to an influential report by ProPublica that claimed that the COMPAS software, widely used in US courts to predict ... | Wikipedia - Fairness (machine learning) - Context | 337 | 1,749 | null |
Section: Context > Language Bias. Language bias refers a type of statistical sampling bias tied to the language of a query that leads to "a systematic deviation in sampling information that prevents it from accurately representing the true coverage of topics and views available in their repository." Luo et al. show tha... | Wikipedia - Fairness (machine learning) - Context > Language Bias | 242 | 1,237 | null |
Section: Controversies. The use of algorithmic decision making in the legal system has been a notable area of use under scrutiny. In 2014, then U.S. Attorney General Eric Holder raised concerns that "risk assessment" methods may be putting undue focus on factors not under a defendant's control, such as their education ... | Wikipedia - Fairness (machine learning) - Controversies | 314 | 1,621 | null |
In 2020, an image cropping tool from Twitter was shown to prefer lighter skinned faces. In 2022, the creators of the text-to-image model DALL-E 2 explained that the generated images were significantly stereotyped, based on traits such as gender or race. Other areas where machine learning algorithms are in use that have... | Wikipedia - Fairness (machine learning) - Controversies | 175 | 842 | null |
Section: Limitations. Recent works underline the presence of several limitations to the current landscape of fairness in machine learning, particularly when it comes to what is realistically achievable in this respect in the ever increasing real-world applications of AI. For instance, the mathematical and quantitative ... | Wikipedia - Fairness (machine learning) - Limitations | 203 | 1,074 | null |
Section: Group fairness criteria > Independence. We say the random variables ( R , A ) {\textstyle (R,A)} satisfy independence if the sensitive characteristics A {\textstyle A} are statistically independent of the prediction R {\textstyle R} , and we write R ⊥ A . {\displaystyle R\bot A.} We can also express this notio... | Wikipedia - Fairness (machine learning) - Group fairness criteria > Independence | 341 | 1,105 | null |
Then ( R , A ) {\textstyle (R,A)} satisfy independence if I ( R , A ) = 0 {\textstyle I(R,A)=0} . A possible relaxation of the independence definition include introducing a positive slack ϵ > 0 {\textstyle \epsilon >0} and is given by the formula: P ( R = r | A = a ) ≥ P ( R = r | A = b ) − ϵ ∀ r ∈ R ∀ a , b ∈ A {\disp... | Wikipedia - Fairness (machine learning) - Group fairness criteria > Independence | 193 | 518 | null |
Section: Group fairness criteria > Separation. We say the random variables ( R , A , Y ) {\textstyle (R,A,Y)} satisfy separation if the sensitive characteristics A {\textstyle A} are statistically independent of the prediction R {\textstyle R} given the target value Y {\textstyle Y} , and we write R ⊥ A | Y . {\display... | Wikipedia - Fairness (machine learning) - Group fairness criteria > Separation | 258 | 804 | null |
{\displaystyle R\bot A\ |\ Y.} We can also express this notion with the following formula: P ( R = r | Y = q , A = a ) = P ( R = r | Y = q , A = b ) ∀ r ∈ R q ∈ Y ∀ a , b ∈ A {\displaystyle P(R=r\ |\ Y=q,A=a)=P(R=r\ |\ Y=q,A=b)\quad \forall r\in R\quad q\in Y\quad \forall a,b\in A} This means that all the dependence of... | Wikipedia - Fairness (machine learning) - Group fairness criteria > Separation | 441 | 1,260 | null |
Another equivalent expression, in the case of a binary target rate, is that the true positive rate and the false positive rate are equal (and therefore the false negative rate and the true negative rate are equal) for every value of the sensitive characteristics: P ( R = 1 | Y = 1 , A = a ) = P ( R = 1 | Y = 1 , A = b ... | Wikipedia - Fairness (machine learning) - Group fairness criteria > Separation | 341 | 1,203 | null |
The greater this separation coefficient is at a given score value, the more effective the model is at differentiating between the set of positives and negatives at a particular probability cut-off. According to Mayes: "It is often observed in the credit industry that the selection of validation measures depends on the ... | Wikipedia - Fairness (machine learning) - Group fairness criteria > Separation | 161 | 803 | null |
Section: Group fairness criteria > Sufficiency. We say the random variables ( R , A , Y ) {\textstyle (R,A,Y)} satisfy sufficiency if the sensitive characteristics A {\textstyle A} are statistically independent of the target value Y {\textstyle Y} given the prediction R {\textstyle R} , and we write Y ⊥ A | R . {\displ... | Wikipedia - Fairness (machine learning) - Group fairness criteria > Sufficiency | 249 | 797 | null |
Section: Group fairness criteria > Relationships between definitions. Finally, we sum up some of the main results that relate the three definitions given above: Assuming Y {\textstyle Y} is binary, if A {\textstyle A} and Y {\textstyle Y} are not statistically independent, and R {\textstyle R} and Y {\textstyle Y} are ... | Wikipedia - Fairness (machine learning) - Group fairness criteria > Relationships between definitions | 203 | 911 | null |
Section: Group fairness criteria > Mathematical formulation of group fairness definitions > Preliminary definitions. Most statistical measures of fairness rely on different metrics, so we will start by defining them. When working with a binary classifier, both the predicted and the actual classes can take two values: p... | Wikipedia - Fairness (machine learning) - Group fairness criteria > Mathematical formulation of group fairness definitions > Preliminary definitions | 282 | 1,495 | null |
It is usually referred to as precision, and represents the probability of a correct positive prediction. It is given by the following formula: P P V = P ( a c t u a l = + | p r e d i c t i o n = + ) = T P T P + F P {\displaystyle PPV=P(actual=+\ |\ prediction=+)={\frac {TP}{TP+FP}}} False discovery rate (FDR): the frac... | Wikipedia - Fairness (machine learning) - Group fairness criteria > Mathematical formulation of group fairness definitions > Preliminary definitions | 241 | 795 | null |
It represents the probability of an erroneous positive prediction, and it is given by the following formula: F D R = P ( a c t u a l = − | p r e d i c t i o n = + ) = F P T P + F P {\displaystyle FDR=P(actual=-\ |\ prediction=+)={\frac {FP}{TP+FP}}} Negative predicted value (NPV): the fraction of negative cases which w... | Wikipedia - Fairness (machine learning) - Group fairness criteria > Mathematical formulation of group fairness definitions > Preliminary definitions | 349 | 1,126 | null |
It represents the probability of an erroneous negative prediction, and it is given by the following formula: F O R = P ( a c t u a l = + | p r e d i c t i o n = − ) = F N T N + F N {\displaystyle FOR=P(actual=+\ |\ prediction=-)={\frac {FN}{TN+FN}}} True positive rate (TPR): the fraction of positive cases which were co... | Wikipedia - Fairness (machine learning) - Group fairness criteria > Mathematical formulation of group fairness definitions > Preliminary definitions | 249 | 821 | null |
It is given by the formula: T P R = P ( p r e d i c t i o n = + | a c t u a l = + ) = T P T P + F N {\displaystyle TPR=P(prediction=+\ |\ actual=+)={\frac {TP}{TP+FN}}} False negative rate (FNR): the fraction of positive cases which were incorrectly predicted to be negative out of all the positive cases. It represents ... | Wikipedia - Fairness (machine learning) - Group fairness criteria > Mathematical formulation of group fairness definitions > Preliminary definitions | 349 | 1,100 | null |
It represents the probability of the negative subjects to be classified correctly as such, and it is given by the formula: T N R = P ( p r e d i c t i o n = − | a c t u a l = − ) = T N T N + F P {\displaystyle TNR=P(prediction=-\ |\ actual=-)={\frac {TN}{TN+FP}}} False positive rate (FPR): the fraction of negative case... | Wikipedia - Fairness (machine learning) - Group fairness criteria > Mathematical formulation of group fairness definitions > Preliminary definitions | 311 | 1,175 | null |
To define these measures specifically, we will divide them into three big groups as done in Verma et al.: definitions based on a predicted outcome, on predicted and actual outcomes, and definitions based on predicted probabilities and the actual outcome. We will be working with a binary classifier and the following not... | Wikipedia - Fairness (machine learning) - Group fairness criteria > Mathematical formulation of group fairness definitions > Preliminary definitions | 188 | 878 | null |
Section: Group fairness criteria > Mathematical formulation of group fairness definitions > Definitions based on predicted outcome. The definitions in this section focus on a predicted outcome R {\textstyle R} for various distributions of subjects. They are the simplest and most intuitive notions of fairness. Demograph... | Wikipedia - Fairness (machine learning) - Group fairness criteria > Mathematical formulation of group fairness definitions > Definitions based on predicted outcome | 310 | 1,090 | null |
Section: Group fairness criteria > Mathematical formulation of group fairness definitions > Definitions based on predicted and actual outcomes. These definitions not only considers the predicted outcome R {\textstyle R} but also compare it to the actual outcome Y {\textstyle Y} . Predictive parity, also referred to as ... | Wikipedia - Fairness (machine learning) - Group fairness criteria > Mathematical formulation of group fairness definitions > Definitions based on predicted and actual outcomes | 333 | 1,087 | null |
A classifier satisfies this definition if the subjects in the protected and unprotected groups have equal FPR. This is, if the following formula is satisfied: P ( R = + | Y = − , A = a ) = P ( R = + | Y = − , A = b ) ∀ a , b ∈ A {\displaystyle P(R=+\ |\ Y=-,A=a)=P(R=+\ |\ Y=-,A=b)\quad \forall a,b\in A} Mathematically,... | Wikipedia - Fairness (machine learning) - Group fairness criteria > Mathematical formulation of group fairness definitions > Definitions based on predicted and actual outcomes | 268 | 751 | null |
A classifier satisfies this definition if the subjects in the protected and unprotected groups have equal FNR. This is, if the following formula is satisfied: P ( R = − | Y = + , A = a ) = P ( R = − | Y = + , A = b ) ∀ a , b ∈ A {\displaystyle P(R=-\ |\ Y=+,A=a)=P(R=-\ |\ Y=+,A=b)\quad \forall a,b\in A} Mathematically,... | Wikipedia - Fairness (machine learning) - Group fairness criteria > Mathematical formulation of group fairness definitions > Definitions based on predicted and actual outcomes | 250 | 670 | null |
This is, if the following formula is satisfied: P ( R = − | Y = + , A = a ) = P ( R = − | Y = + , A = b ) ∀ a , b ∈ A {\displaystyle P(R=-\ |\ Y=+,A=a)=P(R=-\ |\ Y=+,A=b)\quad \forall a,b\in A} Mathematically, if a classifier has equal FNR for both groups, it will also have equal TPR, satisfying the formula: P ( R = + ... | Wikipedia - Fairness (machine learning) - Group fairness criteria > Mathematical formulation of group fairness definitions > Definitions based on predicted and actual outcomes | 366 | 921 | null |
A classifier satisfies this definition if the subjects in the protected and unprotected groups have equal TPR and equal FPR, satisfying the formula: P ( R = + | Y = y , A = a ) = P ( R = + | Y = y , A = b ) y ∈ { + , − } ∀ a , b ∈ A {\displaystyle P(R=+\ |\ Y=y,A=a)=P(R=+\ |\ Y=y,A=b)\quad y\in \{+,-\}\quad \forall a,b... | Wikipedia - Fairness (machine learning) - Group fairness criteria > Mathematical formulation of group fairness definitions > Definitions based on predicted and actual outcomes | 323 | 916 | null |
A classifier satisfies this definition if the subject in the protected and unprotected groups have equal prediction accuracy, that is, the probability of a subject from one class to be assigned to it. This is, if it satisfies the following formula: P ( R = Y | A = a ) = P ( R = Y | A = b ) ∀ a , b ∈ A {\displaystyle P(... | Wikipedia - Fairness (machine learning) - Group fairness criteria > Mathematical formulation of group fairness definitions > Definitions based on predicted and actual outcomes | 237 | 659 | null |
Section: Group fairness criteria > Mathematical formulation of group fairness definitions > Definitions based on predicted probabilities and actual outcome. These definitions are based in the actual outcome Y {\textstyle Y} and the predicted probability score S {\textstyle S} . Test-fairness, also known as calibration ... | Wikipedia - Fairness (machine learning) - Group fairness criteria > Mathematical formulation of group fairness definitions > Definitions based on predicted probabilities and actual outcome | 226 | 839 | null |
A classifier satisfies this definition if individuals with the same predicted probability score S {\textstyle S} have the same probability of being classified in the positive class when they belong to either the protected or the unprotected group: P ( Y = + | S = s , A = a ) = P ( Y = + | S = s , A = b ) ∀ s ∈ S ∀ a , ... | Wikipedia - Fairness (machine learning) - Group fairness criteria > Mathematical formulation of group fairness definitions > Definitions based on predicted probabilities and actual outcome | 318 | 955 | null |
It states that when individuals inside or outside the protected group have the same predicted probability score S {\textstyle S} they must have the same probability of being classified in the positive class, and this probability must be equal to S {\textstyle S} : P ( Y = + | S = s , A = a ) = P ( Y = + | S = s , A = b... | Wikipedia - Fairness (machine learning) - Group fairness criteria > Mathematical formulation of group fairness definitions > Definitions based on predicted probabilities and actual outcome | 316 | 1,010 | null |
This means that the expected value of probability score for the protected and unprotected groups with positive actual outcome Y {\textstyle Y} is the same, satisfying the formula: E ( S | Y = + , A = a ) = E ( S | Y = + , A = b ) ∀ a , b ∈ A {\displaystyle E(S\ |\ Y=+,A=a)=E(S\ |\ Y=+,A=b)\quad \forall a,b\in A} Balanc... | Wikipedia - Fairness (machine learning) - Group fairness criteria > Mathematical formulation of group fairness definitions > Definitions based on predicted probabilities and actual outcome | 270 | 851 | null |
Section: Group fairness criteria > Equal confusion fairness. With respect to confusion matrices, independence, separation, and sufficiency require the respective quantities listed below to not have statistically significant difference across sensitive characteristics. Independence: (TP + FP) / (TP + FP + FN + TN) (i.e.... | Wikipedia - Fairness (machine learning) - Group fairness criteria > Equal confusion fairness | 327 | 981 | null |
Section: Individual fairness criteria. An important distinction among fairness definitions is the one between group and individual notions. Roughly speaking, while group fairness criteria compare quantities at a group level, typically identified by sensitive attributes (e.g. gender, ethnicity, age, etc.), individual cr... | Wikipedia - Fairness (machine learning) - Individual fairness criteria | 325 | 1,763 | null |
The problem of what variables correlated to sensitive ones are fairly employable by a model in the decision-making process is a crucial one, and is relevant for group concepts as well: independence metrics require a complete removal of sensitive information, while separation-based metrics allow for correlation, but onl... | Wikipedia - Fairness (machine learning) - Individual fairness criteria | 213 | 1,141 | null |
Section: Causality-based metrics. Causal fairness measures the frequency with which two nearly identical users or applications who differ only in a set of characteristics with respect to which resource allocation must be fair receive identical treatment. An entire branch of the academic research on fairness metrics is ... | Wikipedia - Fairness (machine learning) - Causality-based metrics | 337 | 1,366 | null |
The mathematical formulation reads: P ( R A ← a = 1 ∣ A = a , X = x ) = P ( R A ← b = 1 ∣ A = a , X = x ) , ∀ a , b ; {\displaystyle P(R_{A\leftarrow a}=1\mid A=a,X=x)=P(R_{A\leftarrow b}=1\mid A=a,X=x),\quad \forall a,b;} that is: taken a random individual with sensitive attribute A = a {\displaystyle A=a} and other f... | Wikipedia - Fairness (machine learning) - Causality-based metrics | 311 | 1,011 | null |
Machine learning models are often trained upon data where the outcome depended on the decision made at that time. For example, if a machine learning model has to determine whether an inmate will recidivate and will determine whether the inmate should be released early, the outcome could be dependent on whether the inma... | Wikipedia - Fairness (machine learning) - Causality-based metrics | 320 | 977 | null |
Plecko and Bareinboim propose a unified framework to deal with causal analysis of fairness. They suggest the use of a Standard Fairness Model, consisting of a causal graph with 4 types of variables: sensitive attributes ( A {\displaystyle A} ), target variable ( Y {\displaystyle Y} ), mediators ( W {\displaystyle W} ) ... | Wikipedia - Fairness (machine learning) - Causality-based metrics | 219 | 1,001 | null |
Section: Bias mitigation strategies > Preprocessing. Usually, the classifier is not the only problem; the dataset is also biased. The discrimination of a dataset D {\textstyle D} with respect to the group A = a {\textstyle A=a} can be defined as follows: d i s c A = a ( D ) = | { X ∈ D | X ( A ) ≠ a , X ( Y ) = + } | |... | Wikipedia - Fairness (machine learning) - Bias mitigation strategies > Preprocessing | 347 | 967 | null |
Algorithms correcting bias at preprocessing remove information about dataset variables which might result in unfair decisions, while trying to alter as little as possible. This is not as simple as just removing the sensitive variable, because other attributes can be correlated to the protected one. A way to do this is ... | Wikipedia - Fairness (machine learning) - Bias mitigation strategies > Preprocessing | 338 | 1,817 | null |
If the dataset D {\textstyle D} was unbiased the sensitive variable A {\textstyle A} and the target variable Y {\textstyle Y} would be statistically independent and the probability of the joint distribution would be the product of the probabilities as follows: P e x p ( A = a ∧ Y = + ) = P ( A = a ) × P ( Y = + ) = | {... | Wikipedia - Fairness (machine learning) - Bias mitigation strategies > Preprocessing > Reweighing | 350 | 845 | null |
For each X ∈ D {\textstyle X\in D} we get: W ( X ) = P e x p ( A = X ( A ) ∧ Y = X ( Y ) ) P o b s ( A = X ( A ) ∧ Y = X ( Y ) ) {\displaystyle W(X)={\frac {P_{exp}(A=X(A)\wedge Y=X(Y))}{P_{obs}(A=X(A)\wedge Y=X(Y))}}} When we have for each X {\textstyle X} a weight associated W ( X ) {\textstyle W(X)} we compute the w... | Wikipedia - Fairness (machine learning) - Bias mitigation strategies > Preprocessing > Reweighing | 345 | 705 | null |
Section: Bias mitigation strategies > Inprocessing. Another approach is to correct the bias at training time. This can be done by adding constraints to the optimization objective of the algorithm. These constraints force the algorithm to improve fairness, by keeping the same rates of certain measures for the protected ... | Wikipedia - Fairness (machine learning) - Bias mitigation strategies > Inprocessing | 189 | 1,013 | null |
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