Explore patterns of switches in HMM copying paths

[szhan]

Related to #18

[szhan]

I'm comparing the parent node ids of the paths obtained under different parameters:

• rho = 1e-08, mu = 1e-08
• rho = 1e-07, mu = 1e-07
• rho = 1e-06, mu = 1e-06

These histograms show the number of differences in parent node ids for each sample. These results show that we don't get the same results despite identical MMR.

[szhan]

The x-axis label is wrong above. It should be "Number of parent node id differences".

[szhan]

The MMR isn't simply equal to rho / mu. So, I've thinking about this wrong. Using the parameter combinations above, one shouldn't expect getting the same paths even though rho / mu is maintained.

[szhan]

h/t Duncan Palmer. rho and mu should be defined as follows.

import math

def compute_rho(mu, n, k):
    a = n * mu**k
    b = (1 - mu)**k + (n - 1) * mu**k
    rho = a / b
    return rho

def compute_k(mu, rho, n):
    a = math.log((rho/n) / (1 - rho + rho/n))
    b = math.log(mu / (1 - mu))
    k = a / b
    return k

[szhan]

I did another run setting mu to 1e-07 and 1e-08 and letting rho to be defined by the function above, setting k to 1 (which is MMR) and n to 1,400 (which is the size of the reference sample set used in the previous runs). Some paths are different still, but less so.

[szhan]

Also, see #21

[szhan]

Results of the experiments so far are consistent with numerical instability in the algorithm.

[szhan]

Next step is to compare sample paths obtained using tskit.lshmm and Duncan's lshmm.