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# How many models are there?

There is only finite number of models, but that number can be very big. The number of models depends on the number of variables we have.

## Superexponentially many models

If we only have one variable we only have one possible "dependency" model, but that is of course overly simple situation, since dependency model are meant to capture dependencies between different variables. If we have two variables we have two possible models (either these two are dependent or they are not). But if we have three variables we have 11 different dependency models that can be presented by at least one Bayesian network. If we have four variables we have more than hundred models, but now the number of different models starts to be already awkward to calculate (that is why we did not give the exact number of models for four variables.)

Anyway, the number of models that can be presented as Bayesian networks grows rapidly as the number of variables increases. Since Bayesian networks with different skeletons always represent different models we can calculate a very underestimated approximation for the number of models when we have n variables. The formula for this approximation is simply 2 to the power of n*(n-1)/2. To get an idea that this formula that severely underestimates the number of models still grows pretty fast, we calculated it for n:s from one to 20.

Number of
variables
Underestimated number of models
11
22
38
464
51024
632768
72097152
8268435456
968719476736
1035184372088832
1136028797018963968
1273786976294838206464
13302231454903657293676544
142475880078570760549798248448
1540564819207303340847894502572032
161329227995784915872903807060280344576
1787112285931760246646623899502532662132736
1811417981541647679048466287755595961091061972992
192993155353253689176481146537402947624255349848014848
201569275433846670190958947355801916604025588861116008628224

The numbers are too big to be comprehended. Luckily there is no need to try to comprehend those numbers. It is enough to know that they are Biiiiig.

 B-Course, version 2.0.0 CoSCo 2002