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17.2 Discrete models with continuous parameter space

For discrete model X ~ p(·|θ);θ∈ Θ where Θ is continuous with a priori density π(·) on Θ, and data D, Bayes’ theorem has the form

where l(.;D) is the likelihood function and ∝ stands for ‘proportional’ which means equal up to a multiplicative constant, i.e.

Example 17.1

Let X ~ Aθ;θ ∈ [0;1] have a Bernoulli distribution with point probabilities p(1|θ) = θ and p(0|θ) = 1 − θ. For a priori density π(·) = I[0;1](·) on Θ = [0;1] and observed sample x1,…,xn with xi ∈ {0, 1} the a posteriori density is obtained using the likelihood function

Therefore the a posteriori density π(·|x1,…,xn) is fulfilling

or

By π(·) = I[0;1](·) we obtain

which is proportional to the density of a beta distribution.

In Figure 17.1 the uniform a priori density on ...

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