Re: Bayesian Model Selection versus Parameter Estimation

rsina wrote:
> Hi all,
>
> I am trying to check if a supposedly purely power-law distribution of data
> has in fact a break in it. Since this is a steep power-law, the data
> towards the end of the spectrum is very few. What I am trying to see is if
> the actual distribution is
>
> y(x)= A x^(-a) 
>
> or
>
> y(x)= A x^(-a)   x < x_*
> y(x)= A (x_*)^b x^(-a-b)  x > x_*
>
> obviously if b=0 (or x_* is infinite), the second model reduces to the first
> model.
>
> So my question is, is this a model selection problem, or a parameter
> estimation problem? Should I just start from the second model, and estimate
> the parameter b from the data? Or should I be doing a model selection test,
> and if the second model wins, then and only then try to estimate b and x_*?
> Also, if I treat this as a parameter estimation problem, does the fact x_*
> will be undetermined if b=0 (or close to 0) make the problem singular or
> unsolvable?
You could take either approach, but if your question is "Is there a 
break in the distibution?", then a model selection problem is perhaps 
the best approach fom that standpoint, but will be harder to fit.

I don't know how you're intending to fit the model, but if you're going 
to use WinBugs, then you should look at the changepoint example.  You 
can use an indicator variable to say whether there is a changepoint or 
not, although these models can get tricky.

If b=0, then the posterior for x_* is just equal to the prior, so it may 
not be a problem.  Just make sure you design your priors so that the 
parameters can't run away (e.g. use a uniform over the range of the data).

Good luck!

Bob

--
Bob O'Hara

Rolf Nevanlinna Institute
P.O. Box 4 (Yliopistonkatu 5)
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Finland
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