Derivation of the Posterior Probability Pa c s

The derivation begins with Bayes' theorem, expressing the posterior probability in term of the likelihood, the prior probability, and the global likelihood:

Each of the terms on the right side is approximated in what follows, using p1_6 to denote positive constants (which can be ignored during the optimization process).

Prior Probabilities

It is assumed that a, c, s, and N are independent, so

Because the elements of a represent relative volumes, they are constrained to sum to 1 and are all positive:

TABLE 5 Probabilities, using Bayesian terminology from [3]

P(a, c,

s, N |h)

Posterior probability (maximized)

P(a, c,

s, N)

Prior probability

P(h|a,

c, s, N)

Likelihood

P(h)

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