Editing Multinomial distribution (section)

=== Sampling using repeated conditional binomial samples ===
Given the parameters <math>p_1, p_2, \ldots, p_k</math> and a total for the sample <math>n</math> such that <math>\sum_{i=1}^k X_i = n </math>, it is possible to sample sequentially for the number in an arbitrary state <math>X_i </math>, by partitioning the state space into <math>i </math> and not-<math>i </math>, conditioned on any prior samples already taken, repeatedly.

==== Algorithm: Sequential conditional binomial sampling ====
<syntaxhighlight lang="bash">
S = n
rho = 1
for i in [1,k-1]:
    if rho != 0:
        X[i] ~ Binom(S,p[i]/rho)
    else 
        X[i] = 0
    S = S - X[i]
    rho = rho - p[i]
X[k] = S
</syntaxhighlight>Heuristically, each application of the binomial sample reduces the available number to sample from and the conditional probabilities are likewise updated to ensure logical consistency.<ref>{{Cite web |date=2020-05-05 |title=11.5: The Multinomial Distribution |url=https://stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_(Siegrist)/11%3A_Bernoulli_Trials/11.05%3A_The_Multinomial_Distribution |access-date=2023-09-13 |website=Statistics LibreTexts |language=en}}</ref>