By Insua D.R., Ruggeri F., Wiper M.P.
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In case that different experts disagree, there may be considerable uncertainty about the appropriate prior distribution to apply. In such cases, it is important to assess the sensitivity of any posterior results to changes in the prior distribution. This is typically done by considering appropriate classes of prior distributions, close to the postulated expert prior distribution, and then assessing how the posterior results vary over such classes. 8: Assume that the gambler in the gambler’s ruin problem is not certain about her Be(5, 5) prior and wishes to consider the sensitivity of the posterior predictive ruin probability over a reasonable class of alternatives.
V. (1991) Statistical Inference in Stochastic Processes. New York: Marcel Dekker. S. (1996) Stochastic Processes and Statistical Inference. New Delhi: New Age International. M. (2000) Stochastic Processes: Inference Theory. Dordrecht: Kluwer. E. I. (2005) Gaussian Processes for Machine Learning. Cambridge, MA: The MIT Press. G. and Williams, D. (2000a) Diffusions, Markov Processes and Martingales: Volume 1 Foundations. Cambridge: Cambridge University Press. G. and Williams, D. (2000b) Diffusions, Markov Processes and Martingales: Volume 2 Ito Calculus.
Assume we test until we observe n failures, and we observe t = (t1 , t2 , · · · , tn ) as times between failures, that is, the first failure occurs at time s1 = t1 ; the second one occurs at time s2 = t1 + t2 , and so on, until sn = t1 + t2 + . . + tn . Assume gamma prior distributions for a and b a ∼ Ga(α1 , β1 ), b ∼ Ga(α2 , β2 ). After some computations, we find n i=1 si f (a, b|t) ∝ a α1 +n−1 e−β1 a bα2 −1 e−b(β2 + ) e− ab (1−e−bsn ) , from which we obtain f (a|b, t) ∝ a α1 +n−1 e−a [β1 + b (1−e 1 −bsn )] , which is a gamma distribution with parameters α1 + n and β1 + f (b|a, t) ∝ bα2 −1 e−b(β2 + n i=1 si 1 b 1 − e−bsn , and ) e− ab (1−e−bsn ) , which is a nonstandard distribution.
Bayesian analysis of stochastic process models by Insua D.R., Ruggeri F., Wiper M.P.