Problem with bounded variables when the value lies near the boundary

[lcontento]

When the value of a bounded variable lies near the boundary, the log-posterior becomes inf. For example,

import numpy as np
import pymc3 as pm

lb = -1.0
ub = 1e-7
x0 = np.nextafter(ub, lb)

with pm.Model() as model:
    pm.Uniform('x', testval=x0, lower=lb, upper=ub)

model.check_test_point()

prints

x_interval__   -inf
Name: Log-probability of test_point, dtype: float64

This is due to the formula used for linear interpolation in the backward transform of Interval
https://github.com/pymc-devs/pymc3/blob/b6a88f066e4d694a4cdafb1b7696772f045c9b07/pymc3/distributions/transforms.py#L273
which, when a and b are of different orders of magnitude, may return values outside of the closed interval [a, b].
To see this, run

x_interval = model.basic_RVs[0]
tau = x_interval.distribution.transform_used
tau.backward(tau.forward(x0)).eval() > ub

which gives True.

If such formula is substituted with e.g.

f = tt.nnet.sigmoid(x)
r = f *b + (1 - f) * a

(which is better behaved when the difference of magnitude is large), the problem disappears. Not being very familiar with PyMC3 I am not sure which linear interpolation formula would be the best choice.

[ColCarroll]

This looks like a fine solution to me -- would you be willing to write a patch?

[lcontento]

Sure. I will try to add a test for this problem too.