A bayesian statistics question

[bryce09]

I have a question im trying to work through, but im not too sure how to start it off.
the question is related to scientists studying populations of fish by tagging them. We are told that the scientists believe that they have tagged no more than 20% of the entire population, but that it is much more likely that the true proportion is about 5%. We dicide to take a random sample of the fish to determine the true proportion that have been tagged.

We are asked to give the appropriate beta proior distribution. Then go on to give the posterior distribution, mean, variance and the credible interval. I can do those last question no worries, it is just getting the prior beta distribution that i am having troube with.

Any help getting started as to how i can work out this distribution would be much appreciated.

 

[Dason]

What software are you using? Are you familiar with R? I have a few R functions that could probably help you with this.

 

[bryce09]

im just doing it by hand at this stage

 

[Dason]

By hand? I guess you could specify a mean and variance which you think will give you the criteria that you want and then backsolve for the parameters. But I think there are better ways to do it... but you really can't do them by hand since it requires matching quantiles of the beta which isn't really something you can do easily by hand.

 

[bryce09]

yea, its just for a small assignment. Backsolving to get the parameters is what i need to do which will be easy enough. The part im not sure on is what values to use for the mean and variance. We need to work it out from those values that we were given, but i wasnt sure how to do that.

 

[Dason]

It would probably be best if you could use actual quantiles in your specification (Say let the mean be .05 and let the .975 quantile be .2 - but without using a computer to solve the resulting equations this isn't really doable by hand). So if we're restricting ourselves then I'd say you should probably choose .05 for the mean and then use Chebyshev's inequality to choose a reasonable variance. Once you have those then you can backsolve for the actual parameters.

Actually choose a standard deviation that puts your "upper limit" at 3-4 standard deviations above the mean should probably be good enough. Chebyshev's inequality is quite conservative for most distributions.

 

[bryce09]

ok i'll try that, thanks for the help