Mixture should not allow mixing of discrete and continuous distributions

[ricardoV94]

Right now the Mixture class allows to mix distributions of any kind, leading to improper logp evaluations where probability densities are mixed with probability masses.

norm = pm.Normal.dist(0, 1)
pois = pm.Poisson.dist(1)
mix = pm.Mixture.dist([0.5, 0.5], comp_dists=[norm, pois], shape=1)

# Observed integer could only have come from Poisson component, so logp should be:
np.log(0.5 * np.exp(pois.logp(0))).eval()  # -1.69

# But it's higher:
mix.logp(0).eval()  # -0.958

# Observed float could only have come from Normal component, so logp should be:
np.log(0.5 * np.exp(norm.logp(1/3))).eval() # -1.66

# But it's higher:
mix.logp(1/3).eval()  # -0.93

# This is not a problem when evaluating points where the domains do not overlap:
mix.logp(-1).eval()  # -2.11
np.log(0.5 * np.exp(norm.logp(-1))).eval()  # -2.11

Questions related to the mixture of continuous and discrete distributions crop up now and then on the Discourse:

• https://discourse.pymc.io/t/sampling-from-a-learned-mixture-of-zeros-and-lognormal/3671
• https://discourse.pymc.io/t/zero-inflated-student-t/6170
• https://discourse.pymc.io/t/zero-inflated-normal/6857
• https://discourse.pymc.io/t/mixture-of-continuous-and-discrete-logp/2392

The STAN forum has an informative discussion on this:

• https://discourse.mc-stan.org/t/zero-inflated-models-in-stan/19496

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It seems to me like this is an area where we could easily nudge users in the right direction, by raising an informative value error.

[ricardoV94]

#3582 raises the same concern (applied to the proposal distributions)

[brandonwillard]

This won't be worth addressing in v3, because v4 will dramatically change the implementation of mixtures, and mixed datatype issues like this can be addressed much more naturally and directly in Aesara.