Show that the Bayes estimator for the cost function C(g(𝐗), Θ)=|g(𝐗)-Θ|, is given by the median

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Show that the Bayes estimator for the cost function C(g(𝐗),...

Show that the Bayes estimator for the cost function $C(g(\mathbf{X}), \Theta)=|g(\mathbf{X})-\Theta|$, is given by the median of the a posteriori pdf $f_{\Theta}(\theta \mid \mathbf{X})$. Hint: Write the integral for the average cost as the sum of two integrals over the regions $g(\mathbf{X})>\theta$ and $g(\mathbf{X})<\theta$, and then differentiate with respect to $g(\mathbf{X})$.

Key Concepts

Minimization of Expected Loss

Minimization of the expected loss involves taking the derivative of the average cost, computed as an integral over the posterior, with respect to the estimator. By dividing the integration into regions where the estimator exceeds or falls below ?, one finds that the optimal point requires the cumulative probability on each side of the estimator to be equal, a condition that is met at the median.

Posterior Distribution

The posterior distribution f(? | X) represents the updated probability distribution for the parameter ? after observing data X. It is central in Bayesian analysis as it combines prior beliefs with the likelihood of the observed data to provide a comprehensive measure of uncertainty.

Median of the Posterior as Optimal Estimator

The median is the value that divides the posterior distribution such that half of the probability mass lies below it and half above it. In the case of minimizing the expected absolute error, this balancing of probabilities ensures that the loss is minimized, thereby making the posterior median the Bayes estimator.

Bayes Estimation

Bayes estimation is a method in statistical decision theory that minimizes the expected loss given a cost function and the posterior distribution of the parameter. In this context, it involves choosing an estimator that minimizes the average cost when the uncertainty about a parameter is updated through observed data.

Loss Function and Absolute Error

The loss function in question, defined as the absolute error |g(X) - ?|, quantifies the cost of deviating from the true parameter value. This type of loss leads the optimal estimator to be the median of the posterior distribution, as the median minimizes the expected absolute deviation.

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