Summary
A general and basic model for inference about characteristics of a finite population of distinguishable elements is presented from a subjectivistic–Bayesian point of view. A subjectivist analogue to simple random sampling, based on the notion of exchangeable random variables, is discussed and the inputs and assumptions underlying the model are shown to involve nothing more than is required for inference under Bayesian models for infinite populations. The model is illustrated by a number of particular examples including one based on the multinomial distribution which incorporates a prior distribution representing an extreme position of initial ignorance. Inferences under this particular model are shown to agree closely in several respects with usual “classical” results. Finally, an extension of the results is presented involving the use of concomitant measurements, and under this Bayesian model several common ratio and regression estimators are shown to arise as means of posterior distributions.
Summary
It is a well‐known result, see for example Lindley (1965) and Raiffa and Schlaifer (1961), that if x̄ is the mean of a sample of independent observations distributed N(μ, σ2) where σ2 is known, and if μ has been assigned a normal prior distribution, N(m, v), then the posterior expectation of μ, given the sufficient statistic x̄, has the form {x̄(n/σ2) + m/v}/{(n/σ2)+1/v}, that is, has the intuitively appealing form of a weighted average of the prior mean and sample mean with weights inversely proportional to the prior variance and the conditional sampling variance of X̄ respectively. In this note we present an extremely simple generalization of this result which seems not to have been noted explicitly before and which holds for a variety of distributions other than the normal.
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