Question

Let \(X_{1}, X_{2}\) be a random sample from a Cauchy distribution with pdf $$ f\left(x ; \theta_{1}, \theta_{2}\right)=\left(\frac{1}{\pi}\right) \frac{\theta_{2}}{\theta_{2}^{2}+\left(x-\theta_{1}\right)^{2}}, \quad-\infty

Short answer

The pdf of \(X_{1}, X_{2}\) is computed, and then the posterior pdf is evaluated to give a set of 16 values. The maximum value amongst these gives the maximum of the posterior pdf. There are many computer programs like MATLAB, PYTHON or R, which could potentially be employed for the last step, but that requires programming knowledge and differs depending on software.

Step-by-step solution

Find the posterior pdf

Since the prior \(h\left(\theta_{1}, \theta_{2}\right) \propto 1\), the posterior is proportional to the likelihood. The joint density function of \(X_{1}, X_{2}\) is given by \(f\left(x_1, x_2; \theta_{1}, \theta_{2}\right) = f\left(x_1 ; \theta_{1}, \theta_{2}\right) \cdot f\left(x_2 ; \theta_{1}, \theta_{2}\right)\). So, you can compute the joint pdf by multiplying the given pdf by itself and then plug in \(x_{1}, x_{2}\) to get the posterior pdf apart from the constant of proportionality.

Evaluate the posterior pdf

For part (b), substitute \(x_{1}=1, x_{2}=4\) and then compute the values of the posterior pdf for all combinations of \(\theta_{1}=1,2,3,4\) and \(\theta_{2}=0.5,1.0,1.5,2.0\). This will give you a set of 16 values.

Find the maximum of the posterior pdf

From the 16 values computed, identify the maximum. This corresponds to the maximum of the posterior pdf, denoted as \((\theta_{1},\theta_{2})\).

Applying a Computer Program

Typically there are numerous software programs that can assist in finding maxima. For example, MATLAB, PYTHON or R (RStudio) could help solve step 3 analytically. However, for this particular task, knowledge of programming is necessary and the way to use a program will depend on the specific software chosen.

Key concepts

Posterior Probability Density Function

In Bayesian statistics, the posterior probability density function (posterior pdf) is a fundamental concept that represents the probability of the parameters being estimated given the data that has been observed. After seeing some evidence, in our case particular values of the random sample from a Cauchy distribution, we want to update our beliefs about the parameters \((\theta_1, \theta_2)\). The initial beliefs are expressed by the prior distribution, and the update is done by applying Bayes' Theorem. For our Cauchy distribution example, the posterior pdf is found by multiplying the likelihood function, representing the probability of observing the data under different parameter settings, by a prior distribution over the parameters. Specifically, when the prior is noninformative, meaning it does not favor any parameter values over others, the posterior pdf is directly proportional to the likelihood function.

Applying this to the given exercise, once you have the likelihood from the observed sample, you can use the noninformative prior to calculate the posterior pdf. This is done by multiplying the probability density functions of each observed data point, assuming independence, while considering the prior distribution. The result reveals how our beliefs about the parameters should change in light of the collected data.

Noninformative Prior

The noninformative prior plays a crucial role in the Bayesian estimation process. It is used when we do not have specific information about the distribution of our parameters before seeing data. Instead of imposing any strong assumptions about the parameters, a noninformative prior, like the one used in our example \((h(\theta_1, \theta_2) \propto 1)\), conveys equal weight to all possible values. It is important because it allows the data itself to drive the inferences about the parameters without the influence of prior beliefs.

This approach is often used in many real-world applications where prior information is either unavailable or intentionally excluded to obtain a purely data-driven analysis. In Bayesian statistics, using a noninformative prior is a way to express objectivity and allows the posterior distribution to reflect the information contained solely in the data.

Bayesian Statistics in Mathematical Analysis

The application of Bayesian statistics to mathematical analysis involves the use of probability distributions to model uncertainty in various parameters and to update these distributions as new data becomes available. Using Bayes' Theorem, we can systematically refine these probabilities, which is particularly useful when tackling complex problems where the underlying truth is hidden within random noise or incomplete information. It is a framework that allows us to incorporate prior knowledge, such as expert opinions or historical data, and to combine this with observed evidence to make more informed decisions.

In the context of the Cauchy distribution problem at hand, Bayesian statistics enables the calculation of the posterior distribution of the parameters \((\theta_1, \theta_2)\) after observing the sample data \((X_1, X_2)\). It gives a comprehensive view of what the parameters might be after incorporating both our prior beliefs (expressed through the noninformative prior) and the new evidence (the sample data).

Maximum Posterior Estimation

The concept of maximum posterior estimation is about finding the most probable values of the parameters given the data, which in Bayesian terms is the mode of the posterior distribution. This is akin to the method of maximum likelihood estimation in frequentist statistics but includes the prior distribution in the calculations. By locating the peak of the posterior density function, we can pinpoint the parameter values that are most consistent with the observed data and the prior information.

When applied to the given exercise, maximum posterior estimation entails computing the posterior pdf across different values of the parameters and identifying where its maximum lies. A range of computational tools such as Python, R, or MATLAB can help perform this optimization task on a larger scale. These tools can execute complex calculations quickly and can be crucial to identifying the point \((\theta_1, \theta_2)\) of maximum posterior, especially when dealing with more than two dimensions or when the analytical solution is not straightforward.