Date: Jul 12, 2012 11:25 PM
Author: Steve Sparling
Subject: Re: [ap-stat] Question for the experts

I was hoping one of the Bayesian experts would come up with a Bayesian
method.

I am on my way to California (from BC) and the traffic through Seattle gave
me time to think. Would a Dirchlet prior do the trick? I am not sure how
to do it but it seems to me a Dirchlet prior with a multinomial would be
like a beta with binomial but on steroids. My question is: is it
possible?

Steve Sparling
Campbell River BC (currently in Centralia wa)

On Thu, Jul 12, 2012 at 7:50 AM, Bullard, Floyd <bullard@ncssm.edu> wrote:

> Jeff asks about using Bayesian inference for determining whether a
> 20-sided die is fair or not. (Question quoted below.)****
>
> ** **
>
> Floyd replies:****
>
> ** **
>
> Frequentist hypothesis testing is probably a better approach for you to
> use than Bayesian parameter estimation or model selection.****
>
> ** **
>
> Bayesian methods require the specification of all candidate models or
> parameter values, along with ?a priori? probabilities attached to them.
> One such model is that all the probabilities on your die are the same,
> p=0.05 for each face. But it?s not clear what other models you?d want to
> consider. If you want to consider *all* possible probabilities for all
> die faces (contingent upon them summing to 1), then you?re talking about a
> 19-dimensional parameter space. (It would be 20 dimensional, except the
> requirement that the probabilities add up to 1 reduces the dimensionality
> of the parameter space by 1 dimension.) While it?s possible to consider
> such a model, and a Bayesian approach is not out of the question,
> estimating the ?a posteriori? probabilities (i.e., the probabilities of the
> different models conditional upon the observed data) is fairly challenging,
> and I wouldn?t advise anyone to tackle this problem who isn?t already very
> comfortable applying Bayesian methods already?including not just parameter
> estimation, but model selection as well. (See postscript for the
> difference between these.)****
>
> ** **
>
> Since your question of interest is really simply: ?Are my data consistent
> with this die being fair?? then a chi-square test is probably the best way
> to go. If you?re concerned that your test has low power, that?s not going
> to be improved by using a Bayesian approach, unfortunately. ****
>
> ** **
>
> --Floyd****
>
> ** **
>
> P.S. Regarding the difference between parameter estimation and model
> selection. An example is this. If you randomly allocate 20 sick people
> into two groups of 10 and give one group a placebo and the other an
> experimental treatment for their sickness, then you might be interested in
> either (or both) of two things. First, you might want to estimate the
> chance of recovery under the placebo, and the chance of recovery under the
> treatment. Those are both parameter estimation problems. And if a
> ?credible interval? (similar to a confidence interval) estimating the
> difference in those probabilities should exclude zero, then you probably
> don?t really need to do model selection. But if the interval does include
> 0, then you might still want to address the question: ?What?s the
> probability that this drug gives a greater chance of recovery than does a
> placebo?? Then you?d want to perform a model selection problem. Model A:
> Both drug and placebo work equally well. This model has one parameter in
> it: the chance of recovery under either treatment. Model B:The drug works
> better than the placebo. This has two parameters in it: the two chances of
> recovery, of which one must be greater than the other. Model C: The drug
> works worse than the placebo. Again there are two parameters in the model,
> but now the inequality is reversed. ****
>
> ** **
>
> A Bayesian analysis could be used to determine, conditional upon the data,
> the probability that the drug is actually doing better than the placebo. *
> ***
>
> ** **
>
> In the case of your dice problem, you?d probably have Model A: The die is
> fair. This model has no parameters at all to estimate. Model B:The die is
> not fair. This model has 19 parameters to estimate. But the primary
> question of interest may not be estimates of the face probabilities, but
> simply the a posteriori probabilities of the two models. Nevertheless,
> parameter estimation is required as a step in the model selection process,
> and estimating 19 parameters is not easy.****
>
> ** **
>
> Model selection can also be used to help determine whether a given
> bivariate quantitative data set is more consistent with a linear model or
> an exponential model; or whether such is more consistent with a linear
> model or a constant model; etc.****
>
> ** **
>
> ** **
>
> ?I have a dataset of 1000 rolls for a few 20 sided dice, and I'm trying to
> determine if they are likely fair or not. First, I tried a Pearson's
> chi-square test, but for a sample size of 1000 rolls it doesn't have a lot
> of statistical power for a 20 sided die. Someone suggested that I use
> Bayesian inference, which I think is also called a "posterior probability
> density function," to analyze my data, but I'm having difficulty trying to
> understand how to do it.?****
>
> ****
>
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