[Swarm-Modelling] Re: Mixtures of Distributions, Vol 1 # 117

[Rick Lightburn]

Paul Johnson wants to know if there is an "easy" (my quotes, not his) 
distribution from which the first parameter from among a mixture of betas has a 
distribution.  Probably not, but that shouldn't really matter.  

Johnson is well to choose the beta for the proclivity of political orientation. 
  The resulting mixture distribution will have to be calculated by what used to 
be called 'brute-force' methods, and now might be called 'computer-intensive' 
ones.  I don't think that this group of modelers would be dissuaded from 
computer-intensive methods, or find more value inherent in a closed-form for 
the mixture.

I imagine that there is there some content-based motivation for assuming the 
distribution of the 'hyperparameter'.  (I'm not a political scientist, 
otherwise I could indicate something what it might be.)  I'd think the 
'natural' thing to do would be that a priori the distribution of the parameter 
in, say, 2000 would be the observed distribution in 1996.

Note that the mean of the 'state-level' beta is a/(a+b).  Therefore it might be 
useful to re-parameterize the 'state-level' distributions in terms of their 
means and something else (maybe b, but I haven't thought it through).  
Alternatively, it might make sense to fix a and b relatively, so that a/(a+b) = 
0.5, so that a=b, and then the family of distributions Johnson would be 
examining would be a one-parameter family.  If this common parameter were 
allowed to vary uniformly over the range (0.5, 1.5), say, then individual 
states would vary from highly polarized (when a=b~0.5) to reasonably cohesive 
and 'predictable' (when a=b~1.5).

As for 'estimating' the hyperparameter, or the individual parameters, there is 
going to be a problem with identification:  I think there will be more 
parameters than observations, and only a very devout Bayesian would even 
contemplate such a problem.  Gregg Allenby, at Ohio State, is the key guru on 
the Bayesian analysis of mixture distributions, and if it ends up in a 
very-high dimensional integral, well, Allenby would be the resource on the 
MonteCarlo integrals that do such things.  

But 'simulating' a mixtures of betas (even one with equal parameters), which is 
a very natural thing to do, shouldn't be all that difficult to code.  (One 
could probably do it in Excel in under an hour.  Sam Savage, at Stanford, has 
an Excel add-in that would substantially facilitate that.)

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Today's Topics:

   1. Can you help me with a probablility question about mixtures of 
distributions? (Paul Johnson)

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Message: 1
Date: Tue, 13 Apr 2004 11:30:27 -0500
From: Paul Johnson <address@hidden>
To: swarm-modelling <address@hidden>
Subject: [Swarm-Modelling] Can you help me with a probablility question about 
mixtures of distributions?
Reply-To: address@hidden

Suppose there are 50 collections of agents. Think of these collections 
as districts in a political system.   For each collection, we have 1000 
agents.  I want each agent to have a meaningul parameter in the 
left-right political scale, and I'm thinking of using a Beta 
distribution because it is bounded and displays a wide variety of 
shapes.  I vary the parameters so that not all districts are exactly the 
same in political composition.   As a first take, I have the assumption 
that the distribution within each cluster has a "cluster-specific" 
parameter, a_k, representing the first beta parameter.  So the 
observations are B(a_k,b) for districts k=1...50.

Question: is there a distribution from which to draw a_k so that the 
combined set of all agents has a known distribution?

I've stumbled around a while and I find plenty of literature on Bayesian 
statistics and the Beta as a prior to the Binomail distribution, but I 
can't find anything about the more mundane simulation question "if I 
generate cases like so, what do I have?"

What is the proper  literature to read?

pj

-- 
Paul E. Johnson                       email: address@hidden
Dept. of Political Science            http://lark.cc.ku.edu/~pauljohn
1541 Lilac Lane, Rm 504                              
University of Kansas                  Office: (785) 864-9086
Lawrence, Kansas 66044-3177           FAX: (785) 864-5700



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