TABLE OF CONTENTS In the next three chapters, we will be primarily concerned with estimating model parameters when our state of knowledge leads us to assign a Gaussian sampling distribution when calculating the likelihood function. In this chapter, we start with a simple problem of computing the posterior probability of the mean of a data set. Initially, we assume the variance of the sampling distribution is known and then consider the case where the variance is unknown. We next look at the question of how to determine whether the signal present in the data is constant or variable. In the final section, we consider a Bayesian treatment of a fundamental problem that occurs in experimental science that of analyzing two independent measurements of the same physical quantity, one control and one trial.
Here we suppose that we have collected a set of N data values { d 1,..., d N} and we are assuming the following model is true:
where e i represents the noise component of the ith data value. For this one data set, and any prior information, we want to obtain the Bayesian estimate of . We will investigate three interesting cases. In all three cases, our prior information about e i leads us to adopt an independent Gaussian sampling distribution. [1] In Section 9.2.1, we analyze the case where the noise ? is the same for all e i. In...
