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 <email@example.com> 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.?**** > > **** > > Frequently asked questions (FAQ): http://mrmathman.com/faq<http://web.me.com/mrmathman/MrMathMan/FAQ.html> > > List Archives from 1994: http://mathforum.org/kb/forum.jspa?forumID=67 > ap-stat Resources: http://apstatsmonkey.com > > To search the list archives for previous posts go to > http://lyris.collegeboard.com/read/?forum=ap-stat > > To unsubscribe use this URL: http://lyris.collegeboard.com/read/my_forums/ > > To change your subscription address or other settings use this URL: > http://lyris.collegeboard.com/read/my_account/edit >