Mathematical Models of Adipose Tissue Dynamics - The Mechanobiology of Obesity and Related Diseases - page 14

complex model is less likely to give robust predictions, especially since most
biological datasets are much smaller than the complexity of the underlying
processes would require for a complete fit. Model selection can be done using the
Bayes factor comparing two models, M 1 and M 2 :

P ð D j M 2 Þ ¼ R dx 1 P ð D j x 1 ; M 1 Þ P ð x 1 j M 1 Þ

P ð D j M 1 Þ

R dx 2 P ð D j x 2 ; M 2 Þ P ð x 2 j M 2 Þ :

ð 6 Þ

The Bayes factor can be easily computed if we introduce a temperature
parameter. We define a partition function in analogy with statistical physics
(generating function in probability theory):
Z ð b Þ ¼ Z

dx P ð x j M Þ exp½ bE ð x Þ; ð 7 Þ

where b represents the inverse temperature. Small values of b correspond to high
temperature, while large b corresponds to low temperature. As in statistical
physics, the parameter sets x that contribute significantly to Z at high temperature
are determined more by the volume of the parameter space and less by the
goodness of fit embodied in E : At low temperature, only configurations that fit the
data very well contribute to Z : Using the partition function, we can show that

ob ln Z ¼ R dx E ð x Þ P ð x j M Þ exp½ bE ð x Þ

o ¼ E b ;

R dx P ð x j M Þ exp½ bE ð x Þ

ð 8 Þ

relating the average cost at a given b with the logarithmic derivative of the par-
tition function. Therefore, by integrating this relation the partition function can be
obtained by computing the cost at different temperatures:
ln Z ¼ Z dbE b : ð 9 Þ

Thus, the model likelihood normalized by a summed probability for all models,

P ð D Þ ¼ P i P ð D j M i Þ , corresponds to the partition function for b ¼ 1:

¼ Z ð 1 Þ ¼exp Z 1
0

dbE b :

P ð D j M Þ
P ð D Þ

ð 10 Þ

This is practically very useful when we use MC methods. Such parallel tempering
MC methods are powerful tools for exploring the cost (or energy) landscape because
high temperature is suited to searching the global landscape, while low temperature
is good for searching fine landscape. Therefore, parallel tempering MC methods
can take both advantages to find a global minimum. In addition, the parallel tem-
pering MC using multiple temperatures allows to automatically compute the average
costs E b at different temperatures, and finally the temperature integration in Eq. ( 10 )
[ 21 ]. In summary, Bayesian inference can estimate likelihood values of parameters
of models given data, and quantitatively compare different models.