Continuous Distributions as Conjugates of Discrete Distributions in likelihood testing
- From: "DMcG" <djmcgoldrick@xxxxxxxxxxx>
- Date: 5 Apr 2006 22:19:05 -0700
I have Dirichlet distribution which I know is a conjugate distribution
for a multinomial. Is it true that the actual value of the Dirichlet
probability at any integer value is lower than the Multinomial because
the Multinomial takes positive integers only - hence the density stacks
up on the integers? Both integrate to 1 and are always positive or
zero so the Dirichlet likelihood at any integer value has to be less
than the Multinomial? The probability of the modes of these
distributions are not equal numerically for example. I am performing a
likelihood analysis with a Dirichlet and a Multinomial model and
finding that the Test Statistic is lower for the Dirichlet than the
equivalent Multinomial presumably because of the continuity issue. The
log ratio of the modes relative to some other ratio along the two
functions is not equal so even a likelihood ratio test seems
conservative with the continuous conjugate (Dirichlet) vs the discrete.
Integrating the Dirichlet by dividing up the real number line is hard
so I though maybe look at the two likelihood ratio tests instead...
Can someone point me in the direction of some literature or papers
looking into the continuity issues of discrete and continuous
conjugates with regard to likelihood ratio testing? I am having a hard
time finding anyone who works or publishes on that.
.
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