Pareidolia, but underwater: What is under the green paint?

[Jorge_Stolfi]

Where do we get this prior probability, is it just by estimating the proportion of the pixels representing paint, ink or vellum?

That is the fundamental limitation of Bayesian inference: it does not give the probabilities of each possible cause, it only tell us how to change out a priori probabilities.  So you must specify those priors in order to use the formula.

When the evidence is strong enough, the results do not depend much on the priors.  These matter only when the evidence is ambiguous.  In the worst case, when the evidence carries no information about the cause, the results will be the same as the priors.

When it rains at night, streets are usually wet by the next morning.  If you have no special information about the night's weather, and you see the street wet in the morning, it is reasonable to assume that it rained.  Bayes's formula will say so, whether your prior for "it rained" is 1% or 99%.

That's because P(Wet|NoRain) is very small; say, 0.0001 (but not zero, because there may have been a flood or a street washing truck mat have showed up.  While P(Wet|Rain) is basically 1.  Therefore P(Rain)P(Wet|Rain)  is still much bigger than P(NoRain)P(Wet|NoRain) in any case, and these numbers will become ~1 and ~0 after normalization.

But  if you are sure that it did not rain, because you have been out stargazing in the garden all night, the wet street should not make you change your belief.  That's because your P(Rain) will be essentially zero, and then P(Rain)P(Wet|Rain) will become much smaller than P(NoRain)P(Wet|NoRain), even if P(Wet|NoRain) is only 0.00001. 

In the analysis reported in that document, the prior probability of "OTHER" was set arbitrarily at 0.05, and that of each of the three other classes was set to 0.95/3 = ~0.32.

All the best, --stolfi