ABSTRACT
For the binomial case, we observe r successes in n trials with success probability θ . The likelihood is
L(θ | r, n) = (
n r
) θ r (1 − θ )n−r .
The conjugate beta(a, b) prior is
π(θ ) = 1 B(a, b)
θ a−1(1 − θ )b−1;
this is proper for a > 0, b > 0. For this prior, the posterior is beta(r +a, n−r +b); credible intervals for θ are available from the beta distribution percentiles. Common non-or weakly informative prior distributions for θ are versions of the conjugate prior:
• The uniform prior – beta(1,1): π(θ ) = 1 • The Jeffreys prior – beta( 12 ,
1 2 ): π(θ ) = 1π θ−1/2(1 − θ )−1/2
• The Haldane prior – beta(0,0): π(θ ) = c · θ−1(1 − θ )−1
The Haldane prior is uniform on the logit scale; this prior is improper and gives improper posteriors if r = 0 or n. The Haldane prior is therefore unsuitable when modeling rare-event data with small probabilities. The Jeffreys prior has infinite density at θ = 0 and 1, so when r = 0 or n this prior strongly reinforces the mode in the likelihood at the boundary. It thus reinforces the data evidence for an extreme value of θ . This may not be desirable when modeling rare events. We use the uniform prior throughout this book for binomial data.
