To test different models, the user has to have control of
the relative probability of different descendant
rangesizes. The probability of each descendant rangesize
could be parameterized individually, but we have a
limited amount of observational data (essentially one
character), so efficient parameterizations should be
sought. One way to do this is with the Maximum Entropy
(Harte (2011)) discrete probability distribution
of a number of ordered states. Normally this is applied
(in examples) to the problem of estimation of the
relative probability of the different faces of a 6-sided
die. The input "knowledge" is the true mean of the dice
rolls. If the mean value is 3.5, then each face of the
die will have probability 1/6. If the mean value is
close to 1, then the die is severely skewed such that the
probability of rolling 1 is 99
other die rolls is very small. If the mean value is
close to 6, then the probability distribution is skewed
towards higher numbers.
Here in BioGeoBEARS, we use the same Maximum
Entropy function to specify the relative probability of
geographic ranges of a number of different rangesizes.
This is merely used so that a single parameter can
control the probability distribution -- there is no
MaxEnt estimation going on here. The user specifies a
value for the parameter maxent_constraint_01
between 0.0001 and 0.9999. This can then be applied to
all of the different ancestor-descendant range
combinations in the cladogenesis/speciation matrix.
Example values of maxent_constraint_01 would give
the following results:
maxent_constraint_01 = 0.0001 -- The smaller
descendant has rangesize 1 with 100
LAGRANGE) maxent_constraint_01 = 0.5 --
The smaller descendant can be any rangesize equal
probability. This is effectively what happens in
DIVA's version of vicariance speciation
maxent_constraint_01 = 0.9999 -- The smaller
descendant will take the largest possible rangesize for a
given type of speciation, and a given ancestral
rangesize. E.g., for sympatric/range-copying speciation
(the ancestor is simply copied to both descendants, as in
a continuous-time model with no cladogenesis effect), an
ancestor of size 3 would product two descendant lineages
of size 3. Such a model is implemented in the program
BayArea (Landis et al. (2013)).
LAGRANGE, on the other hand, would only allow
range-copying for ancestral ranges of size 1.
Note: In LAGRANGE-type models, at
speciation/cladogenesis events, one descendant daughter
branch ALWAYS has size 1, whereas the other descendant
daughter branch either (a) is the same (in
sympatric/range-copying speciation), (b) inherits the
complete ancestral range (in sympatric/subset speciation)
or (c) inherits the remainder of the range (in
vicariant/range-division speciation). LAGRANGE-type
behavior (the smaller descendant has rangesize 1 with
100
rangesize) can be achieved by setting the
maxent_constraint_01 parameter to 0.0001.
See also: Maximum Entropy probability distribution
for discrete variable with given mean (and discrete
uniform flat prior)
http://en.wikipedia.org/wiki/Maximum_entropy_probability_distribution
Currently, the function maxent from the
FD package is used to get the discrete
probability distribution, given the number of states and
the maxent_constraint_01 parameter. This could
also be done with get_probvals, which uses
calcZ_part, calcP_n,
following equations 6.3-6.4 of Harte (2011),
although this is not yet implemented.