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In this section $ x_1,x_2,\cdots,x_n $ are iid observations from the Binomial distribution with density:

$ f(x\mid p,n)\propto p^{x} (1-p)^{n-x} ,\,0<x<n,\,(0<p<1,n\in \mathbb{Z}^{+}). $

In the parameterization above $ p $ represent the probability of an exitus while $ n $ is the number of repetitions of the experiment and hence, the maximum value for $ x $.

Case 1: p unknown, n known Edit

Case 1: Recommended prior
$ \pi(p \mid n) = Beta(1/2,\, 1/2) $

Formal endorsements Edit

This Beta distribution was first proposed by [1] and is the reference as well as the Jeffreys prior for $ p $.

Posterior propriety Edit

The posterior distribution associated with this prior is always proper given that the prior it is so. In particular, the (marginal) posterior is

$ Beta(p\mid x+1/2,\,n-x+1/2) $

Other properties Edit

add if any.

Alternatives Edit

[2] propose an improper prior also known as the Haldane prior.

$ \pi(p\mid n)\propto p^{-1} (1-p)^{-1}. $

This is a limiting case of a $ Beta(k,k) $ when $ k\rightarrow 0 $

Case 2: Prior for n Edit

Case 2: Recommended prior
$ \pi(n)\propto 1/n $

Formal endorsements Edit

This prior is the reference prior discussed by [3].

Posterior propriety Edit

add

ReferencesEdit

  1. Bernardo, J. and Smith, A. (1994), Bayesian Theory, John Wiley and Sons, London.
  2. Novick, W.R. and Hall, W.J. (1965), A Bayesian indifference procedure. Journal of American Statistical Association, 60, 1104-1117.
  3. Alba, E.D. and Mendoza, M. (1995), A discrete model for Bayesian forecasting with stable seasonal patterns. Advances in Econometrics,11.