nan uncertainty estimates

For certain types of overlapping distributions, the uncertainty estimate returned by BAR can benan

%matplotlib inline
import numpy as np
import pymbar
from matplotlib import pyplot as plt
for _ in range(10):
    fwd = np.random.normal(0, 10.0, size=50000)
    rev = np.random.normal(0, 175.0, size=50000)
    print(pymbar.BAR(fwd, rev))

Returns

(0.03461197183239051, nan)
(0.01789748600329588, nan)
(-0.041729050167845116, 0.006057615238226341)
(-0.0008426040270812507, nan)
/home/yutong/venv37/lib/python3.7/site-packages/numpy/core/_methods.py:48: RuntimeWarning: overflow encountered in reduce
  return umr_sum(a, axis, dtype, out, keepdims, initial, where)
(0.011851492025385824, nan)
(-0.011125827676117694, nan)
(-0.0011024556901020333, 0.006058115351639581)
(-0.14948909238990282, 0.006102804127491066)
(0.03077049043731961, 0.006088478465760845)
(-0.10509980343390524, nan)

The distributions look like:

Edit: G0 and G1 below is wrong, confused the distributions, there is probably no issue with the bias!

Futhermore, since the analytical partition functions are Z(sig)=1/(sig*sqrt(2pi)), so using G=-ln(Z) the analytical free energies are G0=3.221 and G1=6.083, The estimator also appears to be biased.

[mrshirts]

@jchodera - see the discussion above - the example was using U_1 and U_2, not U_1-U2 and U_2-U1, so the results were nonsensical because the data was not physical. Good question about what should be returned in those cases, though, and if they should be trapped (if possible? Might be hard to distinguish in many cases from numerically bad data).

[jchodera]

I’ve just suggested we also add a method to compute the number of effective samples: #427

I’m guessing these cases are seeing a collapse in the number of effective samples, meaning the estimates will be unreliable.

We should add some protection to make sure we don’t return nans, thought. And we may want to warn when there is a sample size collapse.