By M.L. Silverstein

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**Sample text**

The observations are speciﬁed to come from a distribution with parameters θ1 , . . , θJ where the θ’s may be scalars, vectors, or matrices. Prior We quantify available prior knowledge (before performing the experiment and obtaining the data) in the form of a joint prior distribution for the parameters p(θ1 , . . 1) where they are not necessarily independent. Likelihood With an independent sample of size n, the joint distribution (likelihood) of the observation vectors is the product of the individual distributions (likelihoods) and is given by n p(x1 , .

N ) . The rows of M are individual µ vectors. 9) Which is often refered to as Jeﬀreys invariant prior distribution. Note that this reduces to the scalar version when p = 1. 2 Conjugate Priors Conjugate prior distributions are informative prior distributions. Conjugate prior distributions follow naturally from classical statistics. It is well known that if a set of data were taken in two parts, then an analysis which takes the ﬁrst part as a prior for the second part is equivalent to an analysis which takes both parts together.

J |X1 , . . , Xn ) ∝ p(θ1 , . . , θJ )p(X1 , . . , Xn |θ1 , . . 5) in which the joint posterior distribution is proportional to the product of the prior distribution and the likelihood distribution. From the posterior distribution, estimates of the parameters are obtained. Estimation of the parameters is described later. © 2003 by Chapman & Hall/CRC Exercises 1. State Bayes’ rule for the probability of event B occurring given that event A has occurred. 2. Assume that we select a Beta prior distribution p( ) ∝ α−1 (1 − )β−1 for the probability of success in a Binomial experiment with likelihood p(x| ) ∝ x (1 − )n−x .