Dirichlet-multinomial Distribution - Probability Mass Function - For A Multinomial Distribution Over Category Counts

For A Multinomial Distribution Over Category Counts

For a random vector of category counts, distributed according to a multinomial distribution, the marginal distribution is obtained by integrating out p:

which results in the following explicit formula:

\Pr(\mathbf{x}\mid\boldsymbol{\alpha})=\frac{N!} {\prod_{k}\left(n_{k}!\right)}\frac{\Gamma\left(A\right)} {\Gamma\left(N+A\right)}\prod_{k}\frac{\Gamma(n_{k}+\alpha_{k})}{\Gamma(\alpha_{k})}

where A is defined as the sum . Note that this differs crucially from the above formula in having an extra term at the front that looks like the factor at the front of a multinomial distribution. Another form for this same compound distribution, written more compactly in terms of the beta function, B, is as follows:

\Pr(\mathbf{x}\mid\boldsymbol{\alpha})=\frac{N B\left(A,N\right)} {\prod_{k:n_k>0} n_k B\left(\alpha_k,n_k \right)} .

Read more about this topic:  Dirichlet-multinomial Distribution, Probability Mass Function

Famous quotes containing the words for a, distribution, category and/or counts:

    “I confess that I have hitherto indulged very little in philanthropic enterprises.... While my townsmen and women are devoted in so many ways to the good of their fellows, I trust that one at least may be spared to other and less humane pursuits. You must have a genius for charity as well as for anything else. As for Doing-good, that is one of the professions which are full.”
    —Henry David Thoreau (1817–1862)

    “In this distribution of functions, the scholar is the delegated intellect. In the right state, he is, Man Thinking. In the degenerate state, when the victim of society, he tends to become a mere thinker, or, still worse, the parrot of other men’s thinking.”
    —Ralph Waldo Emerson (1803–1882)

    “The truth is, no matter how trying they become, babies two and under don’t have the ability to make moral choices, so they can’t be “bad.” That category only exists in the adult mind.”
    —Anne Cassidy (20th century)

    “The gentle serpent, green in the mulberry bush,
    Riots with his tongue through the hush—
    Sentinel of the grave who counts us all!”
    —Allen Tate (1899–1979)