"Frontmatter". In: Analysis of Financial Time Series

ALTERNATIVE ALGORITHMS 403distribution of λ is a beta distribution with positive parameters α and β, then the posteriordistribution of λ is a beta distribution with parameters α+mn and β + ∑ ni=1 x i .Next we consider the case of a normal distribution with an unknown mean µand an unknown precision η. The two-dimensional prior distribution is partitioned asP(µ, η) = P(µ | η)P(η).Result 7: Suppose that x 1 ,...,x n form a random sample from a normal distributionwith an unknown mean µ and an unknown precision η. Suppose also that theconditional distribution of µ given η = η o is a normal distribution with mean µ oand precision τ o η o and the marginal distribution of η is a gamma distribution withpositive parameters α and β. Then the conditional posterior distribution of µ givenη = η o is a normal distribution with mean µ ∗ and precision η ∗ ,µ ∗ = τ oµ o + n ¯xτ o + nand η ∗ = (τ o + n)η o ,where ¯x = ∑ ni=1 x i /n is the sample mean, and the marginal posterior distribution ofη is a gamma distribution with parameters α + (n/2) and β ∗ ,whereβ ∗ = β + 1 2n∑(x i −¯x) 2 + τ on( ¯x − µ o ) 2.2(τ o + n)i=1When the conditional variance of a random variable is of interest, inverted chisquareddistribution (or inverse chi-squared) is often used. A random variable Yhas an inverted chi-squared distribution with v degrees of freedom if 1/Y followsa chi-squared distribution with the same degrees of freedom. The probability densityfunction of Y isf (y | v) = 2−v/2Ɣ(v/2) y−(v/2+1) e −1/(2y) , y > 0.For this distribution, we have E(Y ) = 1/(v − 2) if v>2andVar(Y ) = 2/[(v −2) 2 (v − 4)] if v>4.Result 8: Suppose that a 1 ,...,a n form a random sample from a normal distributionwith mean zero and variance σ 2 . Suppose also that the prior distribution of σ 2is an inverted chi-squared distribution with v degrees of freedom [i.e., (vλ)/σ 2 ∼ χv 2,where λ>0]. Then the posterior distribution of σ 2 is also an inverted chi-squareddistribution with v + n degrees of freedom—that is, (vλ + ∑ ni=1 ai 2)/σ 2 ∼ χv+n 2 .10.4 ALTERNATIVE ALGORITHMSIn many applications, there are no closed-form solutions for the conditional posteriordistributions. But many clever alternative algorithms have been devised in the statis-