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Archiver > GENEALOGY-DNA > 2009-02 > 1235774704

From: "Ken Nordtvedt" <>
Subject: Re: [DNA] Puzzle, resolved? (was: TMRCAs for groups of haplotypes)
Date: Fri, 27 Feb 2009 15:45:04 -0700

Where'd this aposteriori expected value of 1/3 for mutation rate come from?

<m> = Integral { m dm Prob(no muts in unit t | m)
divided by Integral { dm Prob(no muts in unit t | m)

I don't think that comes out to 1/3 but rather 1
Then it comes out to 1/2 if experiment repeated.

----- Original Message -----
From: "John Chandler" <>
To: <>
Sent: Friday, February 27, 2009 3:15 PM
Subject: Re: [DNA] Puzzle, resolved? (was: TMRCAs for groups of haplotypes)

> 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
>
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