Fix mathematical notation in conditional_logprob docstrings

[amyoshino]

What is this PR about?
Closes #6798

This PR Fix mathematical notation in conditional_logprob docstrings

The changes are pointed out by @larryshamalama in #6798. The notation changes are:

• Given that $\Sigma$ is commonly used for covariance matrices, it is replaced by $\sigma$ which is a more common notation.
• $P(Y = y)$ is replaced by $P_{Y}(y)$ to avoid incorrect notation in the case of continuous-valued random variables.

Checklist

• Explain important implementation details 👆
• Make sure that the pre-commit linting/style checks pass.
• Link relevant issues (preferably in nice commit messages)
• Are the changes covered by tests and docstrings?
• Fill out the short summary sections 👇

Major / Breaking Changes

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📚 Documentation preview 📚: https://pymc--6821.org.readthedocs.build/en/6821/

[amyoshino]

@larryshamalama I have followed your suggestions in the issue #6798 but I am not sure if the way I represented the line L438 is correct: $\log p(Y = y \mid \Sigma^2)$ --> $\log p_{Y \mid s^2}(y \mid s^2)$. It looks odd to me because I am repeating the $s$ in the subscript and in the conditional side. Let me know if this is not the best way to represent it.

In the lines L448-L449 I shifted the $\Sigma$ with $s$ in your suggestion because I believe this is how the original version is described after replacing $\Sigma$ (subscript) with $s$ (value). Let me know if I am mistaken here as well.
https://www.pymc.io/projects/docs/en/stable/api/generated/pymc.logprob.conditional_logp.html#pymc.logprob.conditional_logp

[codecov]

Codecov Report

Merging #6821 (c44a5d2) into main (659f177) will increase coverage by 0.00%.
The diff coverage is n/a.

Additional details and impacted files

@@           Coverage Diff           @@
##             main    #6821   +/-   ##
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  Coverage   92.04%   92.04%           
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  Files          95       95           
  Lines       16261    16262    +1     
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+ Hits        14967    14968    +1     
  Misses       1294     1294           

Impacted Files	Coverage Δ
pymc/logprob/basic.py	94.28% <ø> (ø)

... and 1 file with indirect coverage changes

[larryshamalama]

Thanks @amyoshino for opening this PR, you're on a roll!

Apologies if the initial issue was confusing. I wrote a long text ahead, none of this is major but rather a clarification of what I previously meant. I can get quite picky with mathematical notation...

@larryshamalama I have followed your suggestions in the issue #6798 but I am not sure if the way I represented the line L438 is correct: log⁡p(Y=y∣Σ2) --> log⁡pY∣s2(y∣s2). It looks odd to me because I am repeating the s in the subscript and in the conditional side. Let me know if this is not the best way to represent it.

Your instinct is correct, we shouldn't have s as a subscript (random variable) and as an argument (value variable). In the hierarchical model definition, we just want random variables. In the mathematical function representing the log-density, arguments are the "value variables". I hope that this clarifies things.

We ideally want to adopt a notation that is more "standard" and easier to understand. That's mostly subjective... 1) I prefer using $\sigma^2$ to denote the variance. 2) Often, using both random variables as indices and arguments to denote what densities we are referring to can be too much... Here, I think that it's important, because we would like to delineate random variables from value variables in PyMC/PyTensor.

It seems that you use $\sigma^2$ as an argument and $s^2$ as the stochastic variable. (*) I personally would change the order, especially that we define sigma2_rv to be the PyTensor random variable and s_vv to be the valued variable. (*) We should probably even write s2_vv instead of s_vv. If you have any preferences, you can add your suggestion. Otherwise, I provided mine in this review.

In short, your instinct correctly addresses an error in mathematical notation, but the solution can be entirely subjective. Actionable items are presented by a (*) for clarity :)

[amyoshino]

@larryshamalama thank you for the comments and review! I think it makes way more sense to me now, my apologies for not making sense of your initial comments and changing the use of $\sigma$ and $s$. I believe it is now very clear to the readers.

I have commited your suggestions and added a last commit to cover the changes from s_vv to s2_vv. Sorry the number of commits as well, I started accepting one by one instead of accepting all in one batch, lesson learned for the next PR :)

I think everything is ok now, but let me know if I forgot to address anything or if you have any other suggestions or comments.

[larryshamalama]

@larryshamalama thank you for the comments and review! I think it makes way more sense to me now, my apologies for not making sense of your initial comments and changing the use of σ and s. I believe it is now very clear to the readers.

Ah please don't be sorry, I'm very happy to help :)

I have commited your suggestions and added a last commit to cover the changes from s_vv to s2_vv. Sorry the number of commits as well, I started accepting one by one instead of accepting all in one batch, lesson learned for the next PR :)

Don't worry about the number of commits! We often squash merge, but there is a certain "art" to writing nice commits. I'm still learning myself, @ricardoV94 writes great commits if you want to have some inspiration. Around the beginning of my open-source involvement, I was referred to this website, which I found somewhat helpful: https://wiki.openstack.org/wiki/GitCommitMessages#Structural_split_of_changes

[larryshamalama]

Looks good, thanks @amyoshino!

[amyoshino]

@larryshamalama Thank you for your guidance!! Thanks for also pointing out resources for me to improve, I found the linked document very informative, will carry it to the next PRs as best as I can.