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An estimator ^ is said to be a Bayes estimator if it minimizes the Bayes risk among all estimators. Equivalently, the estimator which minimizes the posterior expected loss E ( L ( θ , θ ^ ) | x ) {\displaystyle E(L(\theta ,{\widehat {\theta }})|x)} for each x {\displaystyle x} also minimizes the Bayes risk and therefore is a Bayes estimator.
The risk is constant, but the ML estimator is actually not a Bayes estimator, so the Corollary of Theorem 1 does not apply. However, the ML estimator is the limit of the Bayes estimators with respect to the prior sequence π n ∼ N ( 0 , n σ 2 ) {\displaystyle \pi _{n}\sim N(0,n\sigma ^{2})\,\!} , and, hence, indeed minimax according to ...
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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.
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.
In the latter equation, the integrand inside dx is known as the Posterior Risk, and minimising it with respect to decision a also minimizes the overall Bayes Risk. This optimal decision, a * is known as the Bayes (decision) Rule - it minimises the average loss over all possible states of nature θ, over all possible (probability-weighted) data ...
Bayesian linear regression is a type of conditional modeling in which the mean of one variable is described by a linear combination of other variables, with the goal of obtaining the posterior probability of the regression coefficients (as well as other parameters describing the distribution of the regressand) and ultimately allowing the out-of-sample prediction of the regressand (often ...
In general, the risk () cannot be computed because the distribution (,) 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 average of the loss function over the training set; more formally, computing the expectation with respect to the empirical measure: