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A uniform distribution represents a scenario where all outcomes within a specific range are equally likely. It's akin to rolling a fair die, where each side is equally probable. For the problem at hand, the parameter \(\mu\), which is the mean of the normal random variable, is uniformly distributed over the interval \([a, b]\). This implies that every value of \(\mu\) within this interval has the same probability density. The prior distribution for \(\mu\) takes the form \(f(\mu) = \frac{1}{b-a} \) for \(\mu \in [a, b]\), which ensures that the total probability sums up to 1 over \([a, b]\).

This constant probability density is what defines the uniform nature of the distribution. Outside this interval, the probability density is zero, indicating that we assume \(\mu\) cannot take any values beyond \([a, b]\).

• This uniform prior is non-informative—it does not suggest any particular value within \([a, b]\) is more likely than another.

The posterior distribution is the crux of Bayesian statistics. It blends prior beliefs with new evidence to form an updated belief about a parameter. Using Bayes' Theorem, we calculate this posterior distribution for \(\mu\) given the data observed.

The theorem posits that the posterior \( f(\mu|x) \) is proportional to the likelihood \( f(x|\mu) \) times the prior \( f(\mu) \). In our example:

• The likelihood function \( f(x|\mu) \) for a normal distribution is \( \frac{1}{\sqrt{2\pi \sigma^2}} e^{-\frac{(x-\mu)^2}{2\sigma^2}} \).
• The prior distribution, as we previously established, is uniform, \( \frac{1}{b-a} \).

Multiplying these provides the unnormalized posterior:
\( f(\mu|x) \propto e^{-\frac{(x-\mu)^2}{2\sigma^2}} \).

This expression resembles a Gaussian or normal distribution centered around the observed data point \(x\).

Moreover, because the normal distribution exponentially fades out from its mean, it aligns perfectly with how updated beliefs (posterior) around \(\mu\) emerge given new information.

A Bayes estimator minimizes expected loss, giving us a 'best guess' about a parameter based on our posterior beliefs. Here, under the uniform prior and the rule of squared error loss, the Bayes estimator for \(\mu\) is simply the mean of the posterior distribution.

For the case of a truncated normal distribution, this estimator usually corresponds to the sample mean \(x\), or the observed data itself. However, the condition here is that \(x\) must fall within our prior's bounds \([a, b]\). If \(x\) does not reside within this interval, the estimator defaults to the nearest boundary of the interval.

This ensures that our estimation remains consistent with our uniform prior assumptions. The Bayes estimator offers a reliable point estimation within the framework of Bayesian inference, accounting for data evidence and prior distributions.

• The simplicity of choosing the sample mean emphasizes the efficacy and intuitiveness of Bayesian techniques in providing practical solutions.