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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.
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 |
---|---|
1 | 1 |
2 | 2 |
3 | 8 |
4 | 64 |
5 | 1024 |
6 | 32768 |
7 | 2097152 |
8 | 268435456 |
9 | 68719476736 |
10 | 35184372088832 |
11 | 36028797018963968 |
12 | 73786976294838206464 |
13 | 302231454903657293676544 |
14 | 2475880078570760549798248448 |
15 | 40564819207303340847894502572032 |
16 | 1329227995784915872903807060280344576 |
17 | 87112285931760246646623899502532662132736 |
18 | 11417981541647679048466287755595961091061972992 |
19 | 2993155353253689176481146537402947624255349848014848 |
20 | 1569275433846670190958947355801916604025588861116008628224 |
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 |