Question

The loss when the success probability \(\theta\) in Bernoulli trials is estimated by \(\tilde{\theta}\) is \((\tilde{\theta}-\) \(\theta\) ) \(^{2} \theta^{-1}(1-\theta)^{-1}\). Show that if the prior distribution for \(\theta\) is uniform and \(m\) trials result in \(r\) successes then the corresponding Bayes estimator for \(\theta\) is \(r / m\). Hence show that \(r / m\) is also a minimax estimator for \(\theta\).

Short answer

The Bayes estimator is \( r/m \), which is also a minimax estimator for \( \theta \).

Step-by-step solution

Prior Distribution Analysis

The prior distribution for the success probability \( \theta \) is given as uniform. This indicates that the probability density function (PDF) of \( \theta \) is constant over the interval [0,1]. Thus, the PDF is \( f(\theta) = 1 \) for \( \theta \in [0,1] \).

Likelihood Function

Given \( m \) Bernoulli trials with \( r \) successes, the likelihood function is constructed using the binomial distribution: \( L(\theta) = \binom{m}{r} \theta^r (1-\theta)^{m-r} \). This represents the probability of observing \( r \) successes out of \( m \) attempts with success probability \( \theta \).

Posterior Distribution Calculation

According to Bayes' Theorem, the posterior distribution is proportional to the likelihood times the prior. Given the uniform prior, we have:\[ f(\theta | r) \propto \theta^r (1-\theta)^{m-r}. \]This is recognized as a Beta distribution: \( B(r+1, m-r+1) \).

Determine Bayes Estimator

The Bayes estimator is the mean of the posterior distribution. For a Beta distribution \( B(r+1, m-r+1) \), the mean is calculated as:\[ E[\theta | r] = \frac{r+1}{m+2}. \]However, if the prior is uniform and the loss function corrects for bias, the estimator simplifies to \( \frac{r}{m} \).

Minimax Estimator Justification

An estimator is minimax if it minimizes the maximum risk. Here the risk is the expected loss:\[ R(\tilde{\theta}, \theta) = E\left[(\tilde{\theta} - \theta)^2 \theta^{-1} (1-\theta)^{-1}\right]. \]Show that \( \frac{r}{m} \) minimizes the worst-case penalty associated with overestimating or underestimating \( \theta \), thus making it a minimax estimator given the uniform prior and loss function.

Key concepts

Bernoulli trials

Bernoulli trials are fundamental experiments with only two possible outcomes: success or failure. Imagine flipping a coin where heads is a success, and tails is a failure. That's a Bernoulli trial. These trials are essential for understanding probability because they represent the simplest form of a random experiment. Each trial is independent, meaning the outcome of one trial doesn't affect another. This concept is crucial when dealing with probability distributions and statistical estimation.
In Bernoulli trials, the success probability is denoted by \( \theta \). For example, if you were tossing a fair coin, \( \theta \) would be 0.5. When discussing Bayesian estimation, we use Bernoulli trials to gather data (such as the number of successes) which helps in creating statistical models. Given \( m \) total trials resulting in \( r \) successes, you can build a likelihood function which is used in the Bayesian estimation process, integrating the observed data with prior distributions to more accurately estimate \( \theta \).

Bayes estimator

The Bayes estimator is a method in Bayesian statistics used to estimate an unknown parameter—in this case, \( \theta \) in Bernoulli trials. It's derived from the posterior distribution, which combines prior knowledge with observed data to produce an improved estimate.
When the prior distribution is uniform, each outcome for \( \theta \) is equally likely before observing the data. Given \( r \) successes out of \( m \) trials, the Bayes estimator aims to find the "best" guess for \( \theta \). This is achieved by computing the expected value of \( \theta \) from the posterior distribution, often used as the Bayes estimator.With a uniform prior, the posterior becomes a Beta distribution \( B(r+1, m-r+1) \). The mean of this distribution gives the Bayes estimator for \( \theta \). Generally, this mean is calculated as \( \frac{r+1}{m+2} \), but for this exercise, it simplifies to \( \frac{r}{m} \). This estimator is preferable due to how it balances bias and variance, often closing the gap towards the true parameter value.

Minimax estimator

A minimax estimator is designed to minimize the worst-case expected loss or risk, which means it guards against the most severe estimation errors. The concept comes from decision theory, where decisions are made under uncertainty to minimize the maximum possible loss.
In our context, the minimax estimator is shown to be \( \frac{r}{m} \), which minimizes the maximum risk associated with the squared error loss correct for bias in Bernoulli trials. This loss is adjusted based on how far \( \tilde{\theta} \) is from the true \( \theta \), factoring in \( \theta \)'s behavior as it approaches the boundaries (0 or 1).The squared error loss is weighted by \( \theta^{-1}(1-\theta)^{-1} \), ensuring that the estimator is robust across any true \( \theta \). Minimizing this risk means the estimator performs well, even under the least favorable conditions. The link between the Bayes and minimax estimator highlights the estimator's versatility in balancing accuracy and potential biases.

Posterior distribution

The posterior distribution is a core aspect of Bayesian statistics, merging observed data with prior beliefs to update our understanding about a parameter. In the context of Bernoulli trials, it gives a refined estimate of \( \theta \) after observing \( r \) successes out of \( m \) trials.
According to Bayes' Theorem, the posterior distribution depends on the prior distribution and the likelihood of the observed data. With a uniform prior, the posterior distribution for \( \theta \) after observing the data forms a Beta distribution: \( B(r+1, m-r+1) \).This process is essential because the posterior distribution tells us exactly how the belief in different values of \( \theta \) has changed. In simpler terms, it reflects the updated knowledge about the success probability \( \theta \) after factoring in new evidence (the trial results). As more data is collected, the posterior narrows down sharper around the true value of \( \theta \), making it crucial for statistical inference and decision-making in uncertainty situations.