 Solution

Bayes' theorem updates a hypothesis using observed data. For mutually exclusive exhaustive hypotheses $H_j$ and data $D$ of nonzero probability,The prior $P(H_j)$ describes information before these data; the likelihood $P(D\mid H_j)$ describes the sampling model; the denominator is the evidence; and the posterior is the updated probability. likelihood and posterior are different conditional quantities. For a continuous parameter, sums become integrals and the posterior is a probability density.

Consider a counter registering independent events at unknown constant rate $\lambda$, observed for known exposure $T$ with negligible background and dead time. The observed count has a Poisson distribution:Take a proper gamma prior in shape-rate convention,Multiplying by the likelihood adds the count to the shape and the exposure to the rate. Normalization therefore givesHere $\beta$ has exposure units. This is Poisson-gamma conjugacy. Successive independent counting runs update the same posterior by adding their counts and exposures, so the result agrees with analysing all the data together.

A credible interval contains a specified posterior probability for $\lambda$, conditional on the chosen prior and model. For low counts, the posterior can be asymmetric and sensitive to the prior, and a symmetric Gaussian error bar can be misleading. For large $n$ and an exposure dominating prior information, the mean approaches $n/T$ and its standard deviation approaches $\sqrt{n}/T$. Background, missed events or time-dependent rates require a richer likelihood; Bayes' theorem cannot compensate for an incorrect physical counting model. The prior must be specified independently of the same data being used in the likelihood, unless its estimation is explicitly included in a larger model.