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The necessary context is, I want to model the world as states, and a state is a frequency distribution of people's opinions. So I wonder if I can use a random variable, say $T$, to model the state the world is in and hence the frequency distribution. Thanks.

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A random variable has a distribution. Are you talking about modeling some observed data as arising from unobserved random variables? Some more clarity would be helpful. –  Macro Mar 6 '12 at 18:10
@Macro, excuse me for my lack of knowledge of the terminology here, but what I would like to do is to use a random variable to represent a frequency distribution, and I use an instantiation of this distribution to represent a state of world (the world can take one of many possible states, and each state corresponds to an instantiated frequency distribution). Of course, this random variable has a distribution, which would be like a prior distribution over all the possible states the world can be in. I hope that clarify my question a bit. –  Simon Hughes Mar 6 '12 at 18:21
It sounds like you want to generate outcomes from a specified distribution. Are you trying to do a simulation study? –  Macro Mar 7 '12 at 3:27

1 Answer 1

What you want is a hierarchical model. Basically, we draw the opinion-distribution's parameters from a higher-level distribution (prior). Your opinion-distribution is probably a Multinomial. The natural choice of the prior over the parameters is then the Dirichlet distribution.

In practical terms, say there are 3 possible opinions (Like, Dislike, Neutral). The distribution of the opinion is then parameterized by the individual probabilities, say Like: 0.49, Dislike: 0.35, Neutral: 0.16.

But we don't want to bake these numbers (0.49, 0.35, 0.16) into our model. We want them to be represented as random variables themselves.

This is where the Dirichlet distribution comes handy, since it is the joint distribution over a set of numbers that sum to one. Therefore, you can use it to model the parameters, i.e. the individual opinion-probabilities (which of course sum to one).

In other words, one draw from the Dirichlet distribution yields a set of opinion-probabilities. One can then generate the opinions of people independently using these probabilities.

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