This problem asks to show that the Bayesian estimator for a binomial likelihood $\text{Bin}(n, \theta)$, with a Kullback-Leibler (KL) divergence as the loss function, is the posterior expectation $\mathbb{E}[\theta | X]$.

Let's break it down step-by-step.

1. Bayesian Estimator with KL Loss

The Bayesian estimator minimizes the expected loss, which in this case is the KL divergence. The KL divergence between two distributions $P$ and $Q$ is:

$D_{\text{KL}}(P || Q) = \int P(x) \log \frac{P(x)}{Q(x)} \, dx$

When minimizing KL divergence in the context of Bayesian estimation, the loss function measures the difference between the posterior distribution (which we infer from the data) and the estimated parameter. For the problem at hand, we are trying to minimize:

$\mathbb{E}[D_{\text{KL}}(\theta || \hat{\theta})]$

Where $\hat{\theta}$ is the estimator of $\theta$.

2. Bayesian Setup

Given the binomial likelihood $X|\theta \sim \text{Bin}(n, \theta)$, we have the following:

• Likelihood function: $P(X | \theta) = \binom{n}{X} \theta^X (1 - \theta)^{n-X}$

• Prior: Suppose we have a prior distribution for $\theta$, say $\pi(\theta)$, which could be a conjugate prior like the Beta distribution, $\theta \sim \text{Beta}(\alpha, \beta)$.

• Posterior: By Bayes' theorem, the posterior is proportional to the likelihood times the prior: $P(\theta | X) \propto P(X | \theta) \pi(\theta)$

3. Posterior Expectation of $\theta$

For many loss functions, including KL divergence, the Bayesian estimator is the posterior mean of the parameter $\theta$, which is given by:

$\hat{\theta}_{\text{Bayes}} = \mathbb{E}[\theta | X]$

For a binomial likelihood with a Beta prior (the conjugate prior for the binomial distribution), the posterior is also Beta-distributed, with parameters updated based on the data $X$. Specifically, if $\theta \sim \text{Beta}(\alpha, \beta)$, then after observing $X$, the posterior distribution is:

$\theta | X \sim \text{Beta}(\alpha + X, \beta + n - X)$

The posterior mean is:

$\mathbb{E}[\theta | X] = \frac{\alpha + X}{\alpha + \beta + n}$

4. Conclusion: Why $\mathbb{E}[\theta | X]$ is the Bayesian Estimator

In this case, when the loss function is the KL divergence, the Bayesian estimator is simply the posterior mean $\mathbb{E}[\theta | X]$. This result comes from the fact that the KL divergence is minimized when the estimator is set to the expected value under the posterior distribution, which makes $\mathbb{E}[\theta | X]$ the optimal estimate.

Thus, the Bayesian estimator for $\theta$ with a Kullback-Leibler loss function is indeed $\mathbb{E}[\theta | X]$.

Would you like further details on any part of this explanation? Here are some related questions you might find helpful:

1. How do we derive the posterior distribution using Bayes' Theorem?
2. Why is the Beta distribution a conjugate prior for the Binomial distribution?
3. How does the choice of prior affect the Bayesian estimator?
4. What are other common loss functions used in Bayesian estimation?
5. Can we use a non-conjugate prior, and what happens in that case?

Tip: When solving Bayesian estimation problems, always check if the prior and likelihood form a conjugate pair. This simplifies the calculation of the posterior distribution.