These were obtained by starting with this model

and then deriving and solving the resulting differential equation. This seems to be a pretty standard treatment.

But in Chapter 11 of Felsenstein, I found an alternative approach based on the Poisson distribution, which I think is quite wonderful.

To begin, he defines rates in terms of

*u*(= 3*λ), so the individual rates for changes from X to Y are

*u*/3. He says:

The easiest way to compute this is to slightly fictionalize the model. Instead of having a rateuof change to one of the three other bases, let us imagine that we instead have a rate 4/3*uof change to a base randomly drawn from all four possibilities. This will be exactly the same process, as there then works out to be a probability ofu/3 of change to each of the other three bases. We have also added a rateu/3 of change from a base to itself, which does not matter.

Now we use the Poisson distribution.

For a branch with time

*t*, the

probability of no events at all at a site, when the number expected to occur is 4/3*u*t, is the zero term of the Poisson distribution…the branch of timetconsists then of a vast number of tiny segments of timedt, each having the small probability of 4/3*u*dtof an event.

The probability of

*at least*one event is 1 minus the zero term:

The probability that it is a

*particular*event (say, A to C) is:

The probability of change to any one of the three nucleotides is then:

## No comments:

Post a Comment