Comments from Daniel Weise on determining Pr(k)

The first unanswered question was "why isn't the Pr(K) that they showed in
Figure x a posterior distribution?" The second unanswered question was "If
Pr(K) isn't a posterior distribution, then what is it?"  Pr(K) is not a
posterior distribution because it isn't computed based on X, the observed
genomes/loci.  Equation (1), which gives the definition of Pr(Z, P | X),
only defines the posterior distributions for Z and P.  Pr(K) ain't anywhere
in there, so it can't be posterior distribution.  Instead, the distributions
defined by Equation (1) are estimated separately for K = 1, K = 2, etc.

So what is Pr(K), the metric for declaring the distribution of Z and P for a
given K more likely than for a different K?  Pr(K) is an ad hoc metric
applied on top of the real posterior distributions for Z and P looking for
distributions with given shapes.  Simply put, distributions with clear peaks
are deemed more probable than distributions without clear peaks.  For
example, if the distribution for Z and P is flat for K = 1, indicating no
clear single population, but there are clear humps in the distribution for K
= 2, then the calculation for Pr(K) will declare K = 2 much more likely than
K = 1. On the other hand, if the distribution for K = 1 shows a clear peak,
and that of K = 2 shows no peaks, then the calculation for Pr(K) will
declare K = 1 much more likely than K = 2.

Here's the mathematics behind the above paragraph.  The first thing to
remember is that the purpose of the random walk (MCMC algorithm) is not to
find the top of the mountain, but to find the shape of the mountain, for we
are trying to approximate the probability for every point in the space, and
not simply find the most likely point.  The shape of the mountain is
approximated by the number of points sampled in a given area.  The more
samples in an area, the more likely the hypotheses in that area, and the
higher the mountain there.  This is the whole reason of ensuring that the
Markov Chain created by the random walk has a "stationary" distribution;
this requirement ensures that the distribution is approximated by the
elements of the chain.  So all the elements of the chain
approximate/estimate the distribution by summing up elements that fall in a
given area.

Equations (12), (13), and (14), which define Pr(X|K), look at the shape of
the mountain by computing the average (mean) and variance of how well each
sampled point in the distribution explains the data X (the set of
genomes/loci).  Equation (12) computes the average of the negative log of
how well each sampled hypothesis explains the data.  This means that given a
distribution D1 with a higher average than a distribution D2, D1 explains
the data less well than D2 and the equations for Pr(K) will declare D1 less
likely than D2).  (Note that the average is not over the points in the
distribution, i.e., the abscissa, but how well each point explains the data,
that is, the ordinate.)  Furthermore (now we get to Equation (14)), given
that two distributions have the same average (explain the data equally
well), one wants to prefer the distribution with sharper peaks, that is, the
one with less variance. So (12) includes a term for variance (computed by
(14)) -- the greater the variance of a distribution, the lower probably (12)
assigns to the the distribution.