Mathematical Methods for Physics and Engineering : A Comprehensive Guide

STATISTICS

Thus, a Bayesian statistician considers the ML estimatesaˆMLof the parameters

to be the values that maximise the posteriorP(a|x,H) under the assumption of

a uniform prior. More importantly, however, a Bayesian wouldnotcalculate the

standard error or confidence interval on this estimate using the (classical) method

employed in subsection 31.3.4. Instead, a far more straightforward approach is

adopted. Let us assume, for the moment, that one is estimating just a single

parametera. Using (31.83), we may determine the valuesa−anda+such that

Pr(a<a−|x,H)=

∫a−

−∞

L(x;a)da=α,

Pr(a>a+|x,H)=

∫∞

a+

L(x;a)da=β.

where it is assumed that the likelihood has been normalised in such a way that
∫
L(x;a)da= 1. Combining these equations gives

Pr(a−≤a<a+|x,H)=

∫a+

a−

L(x;a)da=1−α−β, (31.84)

and [a−,a+]istheBayesian confidence intervalon the value ofaat the confidence

level 1−α−β. As in the case of classical confidence intervals, one often quotes

the central confidence interval, for whichα=β. Another common choice (where

possible) is to use the two valuesa−anda+satisfying (31.84), for whichL(x;a−)=

L(x;a+).

It should be understood that a frequentist would consider the Bayesian confi-

dence interval as anapproximationto the (classical) confidence interval discussed

in subsection 31.3.4. Conversely, a Bayesian would consider the confidence inter-

val defined in (31.84) to be the more meaningful. In fact, the difference between

the Bayesian and classical confidence intervals is rather subtle. The classical con-

fidence interval is defined in such a way that if one took a large number of

samples each of sizeNand constructed the confidence interval in each case then

the proportion of cases in which the true value ofawould be contained within the

interval is 1−α−β. For the Bayesian confidence interval, one does not rely on the

frequentist concept of a large number of repeated samples. Instead, its meaning is

that, given the single samplex(and our hypothesisHfor the functional form of

the population), the probability thatalies within the interval [a−,a+]is1−α−β.

By adopting the Bayesian viewpoint, the likelihood functionL(x;a) may also

be used to obtain an approximationσˆaˆto the standard error in the ML estimator;

the approximation is given by

σˆaˆ=

(

−

∂^2 lnL

∂a^2

∣

∣

∣

∣

a=aˆ

)− 1 / 2

. (31.85)

Clearly, ifL(x;a) were a Gaussian centred ona=aˆthenσˆˆawould be its standard

deviation. Indeed, in this case, the resulting ‘one-sigma’ limits would constitute a