Implement transforms for discrete variables

[ricardoV94]

I am marking this as a draft, as I might very well be shooting myself in the foot... but it seems like it should work?

import arviz as az
import numpy as np
import matplotlib.pyplot as plt
import pymc as pm

with pm.Model() as m:
    pm.DiscreteUniform("w", 0, 10, transform=None)
    pm.DiscreteUniform("wt", 0, 10)
    
    pm.Poisson("x", 5, transform=None)    
    pm.Poisson("xt", 5)
      
    pm.Geometric("y", 0.2, transform=None)
    pm.Geometric("yt", 0.2)
    
    pm.HyperGeometric("z", N=30, k=12, n=20, transform=None)
    pm.HyperGeometric("zt", N=30, k=12, n=20)
            
    trace = pm.sample(draws=5_000, chains=4)
    
az.summary(trace, var_names=sorted(map(str, m.free_RVs)))

One trace

     mean     sd  hdi_3%  hdi_97%  ...  mcse_sd  ess_bulk  ess_tail  r_hat
w   5.015  3.177     0.0     10.0  ...    0.033    4555.0    4930.0    1.0
wt  4.984  3.174     0.0     10.0  ...    0.016   20072.0   20000.0    1.0
x   5.021  2.226     1.0      9.0  ...    0.024    4303.0    5023.0    1.0
xt  4.936  2.197     1.0      9.0  ...    0.021    6805.0    5489.0    1.0
y   4.912  4.323     1.0     13.0  ...    0.094    1553.0    1585.0    1.0
yt  4.947  4.447     1.0     13.0  ...    0.069    1943.0    1816.0    1.0
z   8.030  1.286     6.0     10.0  ...    0.014    4265.0    4769.0    1.0
zt  7.987  1.299     6.0     10.0  ...    0.012    6036.0    5622.0    1.0

Another trace

     mean     sd  hdi_3%  hdi_97%  ...  mcse_sd  ess_bulk  ess_tail  r_hat
w   5.006  3.168     0.0     10.0  ...    0.033    4689.0    4554.0    1.0
wt  4.995  3.162     0.0     10.0  ...    0.016   19678.0   19990.0    1.0
x   5.008  2.235     1.0      9.0  ...    0.025    4397.0    4587.0    1.0
xt  4.970  2.222     1.0      9.0  ...    0.021    6671.0    5428.0    1.0
y   4.906  4.368     1.0     13.0  ...    0.104    1445.0    1307.0    1.0
yt  5.018  4.515     1.0     13.0  ...    0.084    1940.0    2067.0    1.0
z   8.002  1.265     6.0     10.0  ...    0.013    4551.0    4953.0    1.0
zt  8.020  1.286     6.0     10.0  ...    0.012    5338.0    5512.0    1.0

I purposely chose parameters I thought would benefit from transforms. In general it seems to improve slightly. The outlier clearly being the DiscreteUniform which now has a 100% acceptance rate (which will probably drive Metropolis tuning crazy :P)

Checklist

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• Link relevant issues (preferably in nice commit messages)
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Major / Breaking Changes

• Add interval transform for discrete variables.
  • As with transformed continuous variables, the respective value variables are now named f"{var}_dinterval" when manually evaluating the model logp/dlogp.

Bugfixes / New features

• ...

Docs / Maintenance

• ...

[codecov]

Codecov Report

Merging #6102 (da64bf4) into main (13e7c88) will decrease coverage by 31.84%.
The diff coverage is 64.28%.

Additional details and impacted files

@@             Coverage Diff             @@
##             main    #6102       +/-   ##
===========================================
- Coverage   92.02%   60.18%   -31.84%     
===========================================
  Files          95       95               
  Lines       16262    16328       +66     
===========================================
- Hits        14965     9827     -5138     
- Misses       1297     6501     +5204     

Impacted Files	Coverage Δ
pymc/distributions/distribution.py	63.59% <40.00%> (-33.48%)	⬇️
pymc/distributions/discrete.py	44.35% <58.82%> (-54.76%)	⬇️
pymc/distributions/transforms.py	68.98% <74.19%> (-30.39%)	⬇️

... and 66 files with indirect coverage changes

[ricardoV94]

Another use case showed up in https://discourse.pymc.io/t/variable-lag-and-slicing/12003 where proposing invalid values for discrete variables can cause the model to error out structurally: https://discourse.pymc.io/t/variable-lag-and-slicing/12003

[ricardoV94]

Reviving again. This is useful to avoid gotchas where categorical models have a "structural role" in the model, such as being used in indexing. In which case the samplers will just crash with IndexErrors and stuff like that. See https://discourse.pymc.io/t/abc-function-works-independently-but-fails-in-pm-simulator/12470/13

I don't see a good reason not to have transforms by default the same way we do for Continuous variables

[ricardoV94]

Another similar issue with discrete variables and indexing operations: #7066