Fix bug in Slice sampler and speedup test_step.py

[ricardoV94]

Closes #5815

I changed the logic when creating the initial interval according to MacKay (2005, page 375) [pdf].

Before

ql[i] = q[i] - nr.uniform(0, self.w[i])
qr[i] = q[i] + self.w[i]

After

r = nr.uniform()
ql[i] = q[i] - r * self.w[i]
qr[i] = q[i] + (1 - r) * self.w[i]

I think the previous was causing the proposals to be systematically biased towards higher values (right).

Also made sure that we set the boundaries of previous variables to the accepted values, which probably by mistake was only being done during tuning.

# Set qr and ql to the accepted points (they matter for subsequent iterations)
qr[i] = ql[i] = q[i]

If someone more familiar with these can confirm my changes to Slice make sense that would be great.

The sampler now converges in the multivariate example in #5815, and does an increasingly better job with more samples.

---

Also did some other minor changes, see individual commits for more details.

[codecov]

Codecov Report

Merging #5816 (d3cf283) into main (7b239bb) will increase coverage by 0.01%.
The diff coverage is 100.00%.

@@            Coverage Diff             @@
##             main    #5816      +/-   ##
==========================================
+ Coverage   89.36%   89.38%   +0.01%     
==========================================
  Files          74       74              
  Lines       13757    13769      +12     
==========================================
+ Hits        12294    12307      +13     
+ Misses       1463     1462       -1     

Impacted Files	Coverage Δ
pymc/distributions/logprob.py	97.65% <100.00%> (+0.19%)	⬆️
pymc/step_methods/compound.py	93.33% <100.00%> (+0.15%)	⬆️
pymc/step_methods/slicer.py	95.83% <100.00%> (+0.05%)	⬆️
pymc/step_methods/hmc/base_hmc.py	90.47% <0.00%> (+0.79%)	⬆️

[ricardoV94]

It seems like this is how the sampler used to behave until commit b988ba9 changed right_bound = left_bound + w -> right_bound = initial_point + w

[michaelosthege]

It seems like this is how the sampler used to behave until commit b988ba9 changed right_bound = left_bound + w -> right_bound = initial_point + w

Wow that means the Slice sampler was broken since version 3.2.

[ricardoV94]

Wow that means the Slice sampler was broken since version 3.2.

Should we backport the fix to V3?

[michaelosthege]

Should we backport the fix to V3?

I'm not sure if it's worth it, but an exception telling users to upgrade to v4 should be easy enough.

[ricardoV94]

It's the same amount of work to add an exception or change the 2 lines

[michaelosthege]

LGTM

[ricardoV94]

I should mention something in the Release Notes

[ricardoV94]

Added Release note