Abstract
The conjugate binomial and Poisson models are commonly used for estimating proportions or rates. However, it is not well known that the conventional noninformative conjugate priors tend to shrink the posterior quantiles toward the boundary or toward the middle of the parameter space, making them thus appear excessively informative. The shrinkage is always largest when the number of observed events is small. This behavior persists for all sample sizes and exposures. The effect of the prior is therefore most conspicuous and potentially controversial when analyzing rare events. As alternative default conjugate priors, I introduce Beta(1/3, 1/3) and Gamma(1/3, 0), which I call ‘neutral’ priors because they lead to posterior distributions with approximately 50 per cent probability that the true value is either smaller or larger than the maximum likelihood estimate. This holds for all sample sizes and exposures as long as the point estimate is not at the boundary of the parameter space. I also discuss the construction of informative prior distributions. Under the suggested formulation, the posterior median coincides approximately with the weighted average of the prior median and the sample mean, yielding priors that perform more intuitively than those obtained by matching moments and quantiles.
Highlights
The focus of this paper is on the choice of a default or an informative prior distribution for the two well-known conjugate models: the binomialbeta model y ∼ Binomial(n, θ) with a prior distribution θ ∼ Beta(a, b), and the Poisson-gamma model y ∼ Poisson(λX) with the λ ∼ Gamma(c, d) conjugate prior, where X is the given exposure
The beta distribution is parameterized as being proportional to θa−1(1 − θ)b−1 and the gamma distribution proportional to λc−1 exp(−dλ)
The role of the noninformative gamma and beta distributions is, essentially, to regularize inferences by assigning little weight to extreme assumptions, avoiding extreme posterior statements that we find hard to believe a priori
Summary
The focus of this paper is on the choice of a default (noninformative) or an informative prior distribution for the two well-known conjugate models: the binomialbeta model y ∼ Binomial(n, θ) with a prior distribution θ ∼ Beta(a, b), and the Poisson-gamma model y ∼ Poisson(λX) with the λ ∼ Gamma(c, d) conjugate prior, where X is the given exposure. The role of the noninformative gamma and beta distributions is, essentially, to regularize inferences by assigning little weight to extreme assumptions, avoiding extreme posterior statements that we find hard to believe a priori The effect of these priors can be characterized as ‘default’ information, being most conspicuous in the case of y = 0 (or y = n), as the posterior quantiles, relative to the scale (n or X), depend only on the information provided by the prior. One must decide how much ‘default information’ about the background rate in the model is appropriate As it will be shown in this paper, this default information has a certain effect on the posterior inferences even if the underlying prior is commonly considered to be noninformative
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