Just to remind you, in many questions it would be impossible for you to do a complete calculation in 10 minutes. In such situations I want you to convince me that you know how to answer the question in lieu of actually doing the calculation.

The first bullet on the second side of the study sheet was the “tagged/untagged” fish survey. This is a “sampling without replacement” scenario, since we do not toss the fish back in the second part of the experiment (we don’t want to catch the same fish twice!) One student wondered if the likelihood depends on the order in which the tagged/untagged fish are caught. It doesn’t, because the denominator (in the example, for N=15) is always 15x14x13x12x…x6, no matter what the order, and if you look at the tagged (T) fish the numerator for those fish is always 10x9x8x7x6, and for the untagged fish it is always 5x4x3x2x1, regardless of the order. So, numerator and denominator are always the same.

For larger N, the denominator starts at N instead of 15, and the numerator for the untagged fish starts at (N-10) instead of 5. You can see that for N=15, we get exactly what’s in the picture below

For a prior, if you don’t have information about the size of the lake or the number of fish in it, it is probably safe to choose a power-law prior, since it’s more likely that the lake is small and has few fish, than that it is large and has many fish. A power-law prior is of the form shown in the following chart. Generally, the exponent a will be small, of order 1 or 2.

The likelihood is 0 for N<15 since we know that there are at least 15 fish in the lake. The following chart shows how one might set up the spreadsheet for power law priors with a=1 and a=2 (you have to pick one). A uniform prior is actually a power-law prior with a=0! All of these priors are defensible; I would expect you to give your reasons for picking the one you do.

The next bullet was couched in terms of a self-proclaimed psychic, but in class I noted that my intention was to distinguish between an exact null hypothesis of 0.85 for a fair die, and a possibly biased die that could have any (unknown) bias. The kind of die would be a 20-sided “dungeons and dragons” die with 3 sides marked one way (success) and 17 marked the other way (failure), for example.

Given states of nature Fair and Biased (F,B), the spreadsheet is very simple. It’s shown below. For the SON Fair, the likelihood P(D|F) is also easy to calculate and is shown at the bottom of the picture.

But for the biased die, there are 10 values of the bias, and these are also states of nature. We can calculate P(D|B) with a spreadsheet, where each possible bias is a SON. However, we have to be careful. We cannot be lazy and just put 1 for each prior, because we aren’t going to divide by the marginal at the end, which in most of our calculations cancels any arbitrary factor from the prior. We have to make sure that the prior adds up to 1. The spreadsheet for this (assuming a uniform prior, and you should defend this) assigns P(D|son,B)=0.1 to each of the values of bias we are considering. We then compute the likelihood (shown in the spreadsheet for several values of bias), the joint, and the marginal in the usual way. The marginal is what we need, since it is the value of P(D|B) that we want to put into the spreadsheet above. That allows us to finish that spreadsheet and get the posterior probabilites P(F|D), P(B|D) that we need to answer the question.

I hope to finish the review sheet on Monday. Have a nice weekend!

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