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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.

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