Adding Multinomial and Nested Logit Models for Consumer Choice

[NathanielF]

Description

I'm adding two new model classes for discrete choice style models that I intend to be part of the consumer choice module.
As it stands i'm opening the PR as a draft for discussion around the implementation choices and API design I have for these models.

Related to this issue: #1653. There is a lot of potential in the discrete choice style models for Bayesian modelling in particular, the state of the art models in this domain involves a mixed logit parameterisation for which "vanilla" implementations are pretty straightforward using Bayesian hierarchical parameterisations.

Two New Models

Main things to flag: There are now two new model files in the consumer choice folder. The simple Multinomial Logit and the Nested Logit. As outlined in the issue i've restricted the nested logit to no more than two layers of nesting. I believe this will bring us to beyond parity with packages like mlogit in R and pylogit in python which allow for only 1 level deep nesting structures.

API Discussion

The API i'm suggesting for these models differs from the typical X,y inputs on the models in pymc marketing in general. Mostly this is because I feel the use of Wilkinson style notation here is important. For instance this is how you specify the Nested Logit Model currently:

We assume a wide-data input as well:

Causal Inference and Counterfactuals

The value that these models bring is their focus on causal inference. The entire history of discrete choice models stems effectively from the observation that multinomial logit models cannot support plausible counterfactuals around market interventions (due to IIA) and more sophisticated discrete choice models like the nested logit models are able to solve this. See for instance here how a pricing intervention on a multinomial logit results in proportional re-allocation of market share to the rest of the market.

We demonstrate this problem and solution by adding 2 new notebooks to the gallery.

In the second notebook for nested logit we show how the IIA is solved by this extra nesting structure:

Fixed Attributes and Alternative Specific Attributes

One thing i've done is to ensure that the models can identify parameters for the alternative specific attributes (e.g. price) and the individually fixed attributes e.g. (income). I've done my best to benchmark the parameter identification and recovery against R's mlogit package:

How to Proceed?

I have not done an extensive write up of the math behind these types of models and some of the functions need more documentation and tests. But I wanted to share what I have so far to generate discussion and maybe decide on how to proceed. One immediate improvement i could think of would be to remove duplication from the nested logit and multinomial logit model classes, making them instances of a more general "DISCRETE CHOICE" class where we could re-use e.g. the formula parsing functions. Additionally i'd like to benchmark the parameter identification with a second data set and example.

Longer term i think there is room for adding a vanilla mixed-logit example too.

Anyway, open to feedback. Adding a draft PR now to check which linting, and testing failures i have.

Related Issue

• Closes #
• Related to Adding Discrete Choice Models to Consumer Choice Module #1653

Checklist

• Checked that the pre-commit linting/style checks pass. Feel free to comment pre-commit.ci autofix to auto-fix.
• Included tests that prove the fix is effective or that the new feature works
• Added necessary documentation (docstrings and/or example notebooks) using numpydoc format.
• If you are a pro: each commit corresponds to a relevant logical change

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📚 Documentation preview 📚: https://pymc-marketing--1654.org.readthedocs.build/en/1654/

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❌ Patch coverage is 93.99293% with 34 lines in your changes missing coverage. Please review.
✅ Project coverage is 92.01%. Comparing base (900926b) to head (a755b57).
⚠️ Report is 98 commits behind head on main.

Files with missing lines	Patch %	Lines
pymc_marketing/customer_choice/nested_logit.py	93.71%	22 Missing ⚠️
pymc_marketing/customer_choice/mnl_logit.py	94.44%	12 Missing ⚠️

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  Lines        6819     7385     +566     
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[williambdean]

Why do some have suffix of _ and others do not?

[NathanielF]

I need to 0 out one of the alternatives alphas_ --> alpha and I was doing the same for betas_fixed_ for parameter identification.

e.g alphaz = pm.Deterministic('alphas', pt.set_subtensor(alphas_, -1, 0))....

[williambdean]

Can we pull out this into a function instead of method. A good separation might be:

1. parse_formula(formula) -> tuple[target, alt_covariates, fixed_covariates]
2. check_columns(covariates, df)
3. check_dependent_variable(target, ser)

[NathanielF]

Yeah, can do... probably not tonight though. Wrecked. Will try and pick it up during the week.

[williambdean]

no worries. take your time

[NathanielF]

I've pulled out the checks and made them staticmethods that are then invoked inside parse_formula. Not sure if that's exactly what you meant?

[NathanielF]

Ok folks. Making progress.

Added a bunch of type-hints and improved doc-strings to the nested logit class. Also added a light write up to both notebooks for multinomial and nested logit on the whys and hows.

Still missing a few doc-strings and will address @williambdean 's request soon too.

Thanks for the feedback @juanitorduz , @williambdean .

[juanitorduz]

great! (also, the notebooks are missing the watermarks and the end :) )

[NathanielF]

Added the watermarks too @juanitorduz :

[NathanielF]

Pretty happy with this now @juanitorduz , @williambdean . Think i addressed the concerns above.

[williambdean]

Line #3.        "gc ~ ic_gc + oc_gc | ",

Are the "|" required here?

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[NathanielF]

No, these should be handled here:

pymc-marketing/pymc_marketing/customer_choice/nested_logit.py

Line 224 in b13b67a

if "|" in covariates:

[williambdean]

Can we used dims in the various places here. I'm assuming that the 900 for P_central, P_room, and denom_top are same as obs (900)

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[NathanielF]

It's straightforward to add dims to the marginal P_central etc... it's not as straightforward to dynamically create the coords for every potential nest at each layer. Like it can be done, just a large headache. Trying not to overcomplicate this function if i can avoid it:

pymc-marketing/pymc_marketing/customer_choice/nested_logit.py

Line 353 in b13b67a

def _prepare_coords(df, alternatives, covariates, f_covariates, nests):

[williambdean]

Alright. Then let's leave out for now

[williambdean]

Really cool stuff. I liked those videos that were linked

[williambdean]

Is it possible to put X and F in a single xr.Dataset and pass as "X"

Or would that be unituitive?

[NathanielF]

I think that wouldn't be in the spirit of the model, the decomposition to fixed covariates and alternative specific covariates is pretty foundational to the literature and write up. Would avoid bundling them if i could.

[juanitorduz]

Thanks @NathanielF ! Once @williambdean gives the last ✅ we can merge and release to get feedback from the community! 🤘

[NathanielF]

Exciting @juanitorduz ! Real curious to see if these models are useful for the community. There is quite a few directions we could go with them.

[juanitorduz]

Actually, this PR looks great, and I am sure we can improve in future iterations. So let's merge and iterate :)

[NathanielF]

Awesome! Thanks @juanitorduz