Question

Consider the Bayes model \(X_{i} \mid \theta, i=1,2, \ldots, n \sim\) iid with distribution \(b(1, \theta), 0<\theta<1\) (a) Obtain the Jeffreys' prior for this model. (b) Assume squared-error loss and obtain the Bayes estimate of \(\theta\).

Short answer

The Jeffreys' prior derived is proportional to 1/θ(1-θ) and the Bayes estimate of θ under a squared error loss will be (α + Σ x_i) / (α + β + n - Σ x_i)

Step-by-step solution

Derive the Jeffreys' Prior

We start by deriving the likelihood function from our given model. From a Bernoulli distribution, we can express the likelihood function as \(L(\theta\)= \( \prod_{i=1}^{n} \theta^{x_i}(1-\theta)^{1-x_i}\). To identify Jeffreys' prior, we need to compute the second derivative of the natural logarithm of the likelihood function (a.k.a the observed Fisher information), and take its square root.

Calculate the Observed Fisher Information

We do this by taking the natural logarithm of our likelihood function which we denote as l(θ) = Σ x_i log(θ) + (n - Σ x_i) log(1-θ). Its second derivative is found as obtaned using these equations \(-E[\frac{\partial^2}{\partial\theta^2} l(θ))]\) = - ( Σ x_i / θ^2 + n - Σ x_i / (1-θ)^2). The square root of this gives us the Jeffreys' Prior.

Derive the Bayes Estimate

Assuming a squared-error loss, we need to compute the posterior mean to get the Bayes estimate. Given the conjugate prior for the Bernoulli distribution is a Beta distribution, and the Posterior distribution is also Beta, the Bayes estimate for θ will be the mean of the posterior distribution. The mean of a Beta distribution is given by alpha / (alpha + beta), where alpha and beta are the parameters of the prior distribution.

Key concepts

Jeffreys' prior

When dealing with Bayesian statistics, the choice of prior distribution is critical. A non-informative prior, which doesn't contribute any additional information, can sometimes be desired for inference purposes. Jeffreys' prior is a type of non-informative prior that is designed to be invariant under reparametrization of the parameter space. This means that our conclusions won't depend on the way we choose to describe the problem.

For a given parameter \(\theta\), Jeffreys' prior is proportional to the square root of the determinant of the Fisher information matrix of \(\theta\), or the square root of the observed Fisher information for a single parameter. This prior is named after Harold Jeffreys, who introduced it as a way to represent ignorance about a parameter's value in a principled way.

Observed Fisher Information

The observed Fisher information plays a significant role in statistical estimation and inference. It's a measure of the amount of information that an observable random variable carries about an unknown parameter upon which the likelihood depends. The higher the Fisher information, the lower the expected variance of the estimator for the parameter.

In practical terms, for a given likelihood function \(L(\theta)\), the Fisher information is calculated as the negative expected value of the second derivative of the log-likelihood function with respect to the parameter \(\theta\). This quantity helps in estimating the Jeffreys' prior, guiding us towards a more robust Bayesian analysis by focusing on information provided directly by the data.

Squared-error loss

In statistical estimation, different loss functions can be used to quantify the cost associated with estimation errors. The squared-error loss is one of the simplest and most commonly used loss functions. It measures the cost as the square of the difference between the estimator and the true parameter value, which in mathematical terms is \((\text{estimator} - \text{parameter})^2\).

The use of squared-error loss leads to estimators that have desirable properties, such as minimizing the mean squared error of the estimate. In the context of Bayesian estimation, the goal becomes finding the estimator that minimizes the expected squared-error loss, which typically equates to the posterior mean.

Posterior distribution

The posterior distribution is a cornerstone of Bayesian statistics, encapsulating what we know about an unknown parameter after accounting for both the prior distribution and the observed data. It's calculated using Bayes' theorem, which combines the likelihood of the observed data given the parameter and the prior belief about the parameter's distribution.

In practice, the posterior distribution is often the main result of a Bayesian analysis, providing a full probabilistic description of our uncertainty about the parameter after observing the data. When assuming a squared-error loss, the best point estimate of a parameter is given by the mean of the posterior distribution, which minimizes the expected loss.

Bernoulli distribution

The Bernoulli distribution is a discrete probability distribution for a random variable that takes the value 1 with success probability \(\theta\) and the value 0 with failure probability \(1-\theta\). It's the simplest case of a binomial distribution where the number of experiments, or trials, is 1.

Mathematically, if \(X\) is a random variable that follows a Bernoulli distribution, the probability mass function of \(X\) is: \[P(X = x) = \theta^x(1 - \theta)^{1 - x}\] for \(x = 0, 1\). The Bernoulli distribution is a convenient model when dealing with binary outcomes, such as coin tosses, and is fundamental in the study of binary data and binary response models in statistics.

Conjugate prior

In Bayesian statistics, when the posterior distributions are in the same family as the prior probability distributions, the prior and posterior are then said to be 'conjugate distributions,' and the prior is called a conjugate prior for the likelihood function. Conjugate priors simplify the process of finding the posterior distribution because the form remains consistent after incorporating the evidence.

For the Bernoulli likelihood, the conjugate prior is the Beta distribution. This compatibly allows us to easily update our beliefs in the presence of new data. The resulting posterior can then be used to derive estimates under different loss functions, like the squared-error loss.

Beta distribution

The Beta distribution is a versatile distribution that is defined on the interval \([0, 1]\) and parameterized by two positive shape parameters, \(\alpha\) and \(\beta\). It is commonly used as a prior distribution in Bayesian statistics, especially for parameters that are probabilities, due to its conjugate relationship with the Bernoulli distribution.

The probability density function of the Beta distribution is given by: \[f(x;\alpha,\beta) = \frac{x^{\alpha-1}(1-x)^{\beta-1}}{B(\alpha,\beta)}\] where \(B(\alpha,\beta)\) is the Beta function, which serves as a normalization constant making sure that the total probability integrates to one. The mean of the Beta distribution, which is used as the Bayes estimate under squared-error loss for a Bernoulli likelihood, is found by the ratio \(\frac{\alpha}{\alpha + \beta}\).