Binomial with unknown probability and sample size: some of the difficulties with setting prior distributions in multiparameter models can be illustrated with the simple binomial distribution. Consider data y1, . . . , yn modeled as independent Bin(N, ), with both N and unknown. Defining a convenient family of prior distributions on (N, ) is difficult, partly because of the discreteness of N. Raftery (1988) considers a hierarchical approach based on assigning the parameter N a Poisson distribution with unknown mean . To define a prior distribution on (, N), Raftery defines = and specifies a prior distribution on (, ). The prior distribution is specified in terms of rather than because 'it would seem easier to formulate prior information about , the unconditional expectation of the observations, than about , the mean of the unobserved quantity N.' (a) A suggested noninformative prior distribution is p(, ) 1 . What is a motivation for this noninformative distribution? Is the distribution improper? Transform to determine p(N, ).

(b) The Bayesian method is illustrated on counts of waterbuck obtained by remote pho- tography on five separate days in Kruger Park in South Africa. The counts were

53, 57, 66, 67, and 72. Perform the Bayesian analysis on these data and display a scatterplot of posterior simulations of (N, ). What is the posterior probability that N > 100? (c) Why not simply use a Poisson with fixed as a prior distribution for N?