Multinomial Distribution |
(discrete probability dist. for equations) |
Usage: |
MultinomialDist (bc, n, k1, p1, k2, p2, ... km, pm) |
Required: |
n ⋝ 0 ki ⋝ 0 0 ⋜ pi ⋜ 1 sum pi != 0 bc boolean n, ki integer |
Support: |
n = sum ki
|
Mean: |
E[ki] = n pi |
Covariance: |
cov[ki,kj] = -n pi pj if i != j if i=j then cov[ki,ki] = var[ki] = n pi (1-pi) |
The multinomial distribution is a generalization of the binomial distribution to the situation where there are not just two outcomes (usually labeled "success" and "fail"), but rather m outcomes, each having probability pi (i=1..m), and we are interested in the number of occurrences of each outcome (ki), given that a total of n trials are performed.
To create a multinomial distribution between the ki and n nodes, first add to the net a new boolean node, in this example called bc. Then add links from the nodes of all the non-fixed parameters (usually n and all ki) to node bc. At node bc, put an equation with MultinomialDist, and convert the equation to a table. Finally, give node bc a finding of true.
Normally the sum of pi is one, but Netica will just normalize the pi if that is not the case.
If m is 2, then k2 is deterministically determined by k1 (i.e., k2 = n - k1), and k1 is distributed by BinomialDist.
Each of the ki separately has a binomial distribution with parameters n and pi, and because of the constraint that the sum of the ki's is n, they are negatively correlated.
The Dirichlet distribution is the conjugate prior of the multinomial in Bayesian statistics.
See also the multinomial function.