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The Bayes risk of ^ is defined as ((, ^)), where the expectation is taken over the probability distribution of : this defines the risk function as a function of ^. An estimator θ ^ {\displaystyle {\widehat {\theta }}} is said to be a Bayes estimator if it minimizes the Bayes risk among all estimators.
For example, if the risk of developing health problems is known to increase with age, Bayes' theorem allows the risk to someone of a known age to be assessed more accurately by conditioning it relative to their age, rather than assuming that the person is typical of the population as a whole.
The Bayes classifier is a useful benchmark in statistical classification. The excess risk of a general classifier C {\displaystyle C} (possibly depending on some training data) is defined as R ( C ) − R ( C Bayes ) . {\displaystyle {\mathcal {R}}(C)-{\mathcal {R}}(C^{\text{Bayes}}).}
Utilizing Bayes' theorem, it can be shown that the optimal /, i.e., the one that minimizes the expected risk associated with the zero-one loss, implements the Bayes optimal decision rule for a binary classification problem and is in the form of
Bayes' theorem describes the conditional probability of an event based on data as well as prior information or beliefs about the event or conditions related to the event. [3] [4] For example, in Bayesian inference, Bayes' theorem can be used to estimate the parameters of a probability distribution or statistical model. Since Bayesian statistics ...
Bayesian inference (/ ˈ b eɪ z i ə n / BAY-zee-ən or / ˈ b eɪ ʒ ən / BAY-zhən) [1] is a method of statistical inference in which Bayes' theorem is used to calculate a probability of a hypothesis, given prior evidence, and update it as more information becomes available.
The minimum Bayes risk for the decision problem is therefore the smallest such that the line touches the risk set. [ 10 ] [ 11 ] This line may either touch only one extreme point of the risk set, i.e. correspond to a nonrandomised decision rule, or overlap with an entire side of the risk set, i.e. correspond to two nonrandomised decision rules ...
In this case, the Bayes risk is not even well-defined, nor is there any well-defined distribution over . However, the posterior π ( θ ∣ x ) {\displaystyle \pi (\theta \mid x)\,\!} —and hence the expected loss—may be well-defined for each x {\displaystyle x\,\!} , so that it is still possible to define a generalized Bayes rule.