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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