GENEALOGY-DNA-L Archives

Archiver > GENEALOGY-DNA > 2009-02 > 1235772937

From: (John Chandler)
Subject: Re: [DNA] Puzzle, resolved? (was: TMRCAs for groups of haplotypes)
Date: Fri, 27 Feb 2009 17:15:37 -0500

Ken wrote:
> The original problem was to infer generations, not a rate. But to use the
> above: something needs changing; if I am trying to measure or infer the rate
> it does not yet have one (other than any apriori P(m))? So I don't
> understand this expected value of the rate being 1/3 --- I'm trying to learn
> what the rate is by doing an experiment.

Yes, and we have done the experiment. I was speaking of the a posteriori
Bayesian expectation value of the rate. One trial, one failure to produce
a mutation. No prior knowledge.

> I'll alter the experiment to be that I look for a mutation over a time
> interval of t and see none; what can I learn from that? Then I repeat the
> experiment; what more do I learn, etc.?

That's not altering the experiment. We've fixed time t to be one unit.
In the simplest case, that's one generation, but it could be any unit.

> If I am trying to infer a mutation rate having not seen anything happen in
> time t, then seeing no mutations would not lead me to conclude mutation rate
> vanishes. Bayes theorem would suggest Prob(m -- given no muts in t) ~
> Prob(no muts in t -- given m) P(m) = exp(-mt) P(m)

As I mentioned originally, the a posteriori expectation of m is 1/3
mutation per unit time. However, that is not a sensible estimate of
the mutation rate, for the reasons I explained.

> As long as my a priori
> P(m) was broad I can conclude the probability distribution a posteriori is
> spread over the interval 0 to order 1/t in the exponentially decreasing
> manner. I do the experiment again and find no mutations, so my exp(-mt)
> changes to exp(-2mt) and aposteriori both experiments I have squeezed down
> the spread of reasonable inference of the mutation rate.

Again, I point out that the expectation value based on the second
experiment alone is still 1/3, same as in the first experiment. You
and James have both claimed that the expectation value is the most
meaningful estimate of the rate, but this situation shows otherwise.
You can repeat the experiment 100 times and get identical results each
time, so that there are 100 independent estimates of the mutation
rate, each giving the same value of 1/3 mutation per unit time.
However, if you then take all of the results together and apply the
same estimator to the combined test data, you do *not* get the same
result as in the individual experiments. Instead, the expectation
after 100 trials is 1/102. Clearly, the Bayesian expection value is
not a sensible estimator. Only the maximum likelihood value has
that property.

John Chandler