Dirichlet–Multinomial Predictive Counts

Dirichlet-Multinomial Predictive Counts

Integrate uncertain category probabilities to predict joint counts. The Dirichlet-multinomial distribution models overdispersed categorical counts by assigning a Dirichlet prior to the underlying multinomial probabilities.

Worked Example

Consider 3 trials with a Uniform concentration (1, 1, 1). We want to find the predictive probability of all counts being category A (3, 0, 0).

Alpha = (1, 1, 1), A = 3, n = 3
Counts = (3, 0, 0)
Coeff = n! / (n1! n2! n3!) = 3! / (3! 0! 0!) = 1
Numerator = rising(1, 3) * rising(1, 0) * rising(1, 0) = (1*2*3) * 1 * 1 = 6
Denominator = rising(A, n) = rising(3, 3) = 3 * 4 * 5 = 60
Predictive PMF = 1 * 6 / 60 = 0.1

Experiment

Explore the interactive module to see how different priors and counts affect probabilities:

Select Concentration to Favor A. Notice how the predictive distribution shifts.
Change the Trials to 4 and observe the expansion of possible outcomes.
Switch the Count pattern to Balanced and compare the predictive probability to the fixed multinomial probability.

Limitations

The interactive visualization enumerates the complete support space. This approach is computationally viable for small counts but does not scale to large numbers of trials due to combinatorial explosion. Real-world applications typically use efficient approximations or MCMC sampling.

References

For further reading on the Dirichlet-multinomial distribution and its applications, consult the following resources:

Dirichlet–Multinomial Predictive Counts