Mathematical Methods for Physics and Engineering - Matematica.NET

STATISTICSBy comparing this result with that given towards the end of subsection 31.5.4, we see that,as we might expect, the Bayesian and classical confidence intervals differ somewhat. ◭The above discussion is generalised straightforwardly to the estimation ofseveral parameters a 1 ,a 2 ,...,a M simultaneously. The elements of the inverse ofthe covariance matrix of the ML estimators can be approximated by∣(V −1 ) ij = − ∂2 ln L ∣∣∣a=â. (31.86)∂a i ∂a jFrom (31.36), we see that (at least for unbiased estimators) the expectation valueof (31.86) is equal to the element F ij of the Fisher matrix.The construction of a multi-dimensional Bayesian confidence region is alsostraightforward. For a given confidence level 1 − α (say), it is most commonto construct the confidence region as the M-dimensional region R in a-space,bounded by the ‘surface’ L(x; a) = constant, for which∫L(x; a) d M a =1− α,Rwhere it is assumed that L(x; a) is normalised to unit volume. Moreover, wesee from (31.83) that (assuming a uniform prior probability) we may obtain themarginal posterior distribution for any parameter a i simply by integrating thelikelihood function L(x; a) over the other parameters:∫ ∫P (a i |x,H)= ··· L(x; a) da 1 ···da i−1 da i+1 ···da M .Here the integral extends over all possible values of the parameters, and againis∫it assumed that the likelihood function is normalised in such a way thatL(x; a) d M a = 1. This marginal distribution can then be used as above todetermine Bayesian confidence intervals on each a i separately.◮Ten independent sample values x i , i =1, 2,...,10, are drawn at random from a Gaussiandistribution with unknown mean µ and standard deviation σ. The sample values are asfollows (to two decimal places):2.22 2.56 1.07 0.24 0.18 0.95 0.73 −0.79 2.09 1.81Find the Bayesian 95% central confidence intervals on µ and σ separately.The likelihood function in this case is[L(x; µ, σ) =(2πσ 2 ) −N/2 exp − 1N]∑(x2σ 2 i − µ) 2 . (31.87)i=1Assuming uniform priors on µ and σ (over their natural ranges of −∞ → ∞ and 0 → ∞respectively), we may identify this likelihood function with the posterior probability, as in(31.83). Thus, the marginal posterior distribution on µ is given by∫ [∞1P (µ|x,H) ∝σ exp − 1N]∑(x N 2σ 2 i − µ) 2 dσ.01268i=1