Past exam of the physics A course of the University of Cambridge / 2011 / Paper 2 / Section D / D12 / a

Bayes' theorem updates a parameter distribution using measured data:The likelihood describes the measurement process, the prior distribution encodes information before the present data, and the denominator normalizes the posterior distribution. Include instrumental noise and known selection effects in the likelihood. nuisance parameters, such as an uncertain background, should be jointly inferred and marginalized, rather than silently fixed.

For $n$ independent readings with a Gaussian distribution with known common variance ${\sigma }^2$ and prior distribution $\theta \sim N({\mu }_0,{\tau }_0^2)$, completing the square givesThe posterior distribution is $N({\mu }_n,{\tau }_n^2)$: prior distribution and data contribute in proportion to precision. A broad prior distribution recovers the usual estimate $\bar{x}$ with standard deviation $\sigma /\sqrt{n}$. Report a posterior distribution estimate and credible interval, specify the prior distribution, and examine sensitivity when the data are weak. A credible interval is a posterior distribution probability statement about the parameter conditional on the model; it is not defined by the repeated-sampling coverage of a confidence interval.