Gamma-Gamma Spend Model: differences in likelihood vs distribution fit

1

Hi there, first off I’d just like to thank y’all for the awesome work you have put in both to PYMC and the extensive guides and documentation!

I am working to build a Gamma-Gamma model of customer spend. I can fit a model using the pm.Potential method similar to the PYMC Marketing method. But I’m curious why a model that is specified as a series of distributions and hyper-parameters doesn’t converge to the same results?

Here is my distribution model:

with pm.Model(coords=coord_gamma) as gamma_model_simple:
    q = pm.Uniform('q', lower = 1, upper = 10)
    y = pm.HalfNormal('y', sigma=3)
    v = pm.Gamma('v', alpha=q, beta=y)
    
    p = pm.HalfNormal('p', sigma=5)
    
    z = pm.Gamma('z', alpha=p*gamma_sample_df['num_orders'],
                 beta=v*gamma_sample_df['num_orders'], 
                 observed=gamma_sample_df['aov'])        
    
    gamma_model_simple_trace = pm.sample()

Where gamma_sample_df is my user-level observation data of average order value and the num_orders for each customer.

When I run this the model fits with no issues but the parameter estimates for p, q, and y are way off from the pm.Potential model:

with pm.Model() as model:
    p = pm.HalfNormal(name="p", sigma=10)
    q = pm.HalfNormal(name="q", sigma=10)
    v = pm.HalfNormal(name="v", sigma=10)

    def logp(x, m):
        return (
            pt.gammaln(p * x + q)
            - pt.gammaln(p * x)
            - pt.gammaln(q)
            + q * pt.log(v)
            + (p * x - 1) * pt.log(m)
            + (p * x) * pt.log(x)
            - (p * x + q) * pt.log(x * m + v)
        )

    likelihood = pm.Potential(name="likelihood", var=logp(x, m))

Unfortunately I can’t share any data, but I was hoping to get some general advice on why the approaches would differ? My ultimate goal is to fit a multilevel model and add customer-specific features to improve the estimates and I’m assuming that would be easier with the fully specific model.

Thanks in advance!

2

q and y are going to be pretty much the same as the prior as they only inform a single variable.

3

You may be interested in this dev notebook on the gamma-gamma. I also tried a direct model on summary stats which failed, in that case because it ends up with too many variables per observation: pymc-marketing/docs/source/notebooks/clv/dev/gamma_gamma_dev.ipynb at main · pymc-labs/pymc-marketing · GitHub

I could only model directly (without marginalizing via the potential) when using the individual observations instead of the summary stats.

I was trying something different then what you’re doing.

4

Perhaps @ColtAllen knows something regarding customer specific effects,

5

Hey @statmills ,

Your results differ because the first model estimates a spend distribution for an individual customer, whereas the pm.Potential model is for an entire population. The likelihood expression is derived with respect to the population expectation.

The Gamma-Gamma likelihood can be modified to support covariates for individual customers, which is something we want to see added to pymc-marketing in the coming months. Here’s the research note if you’re curious:

https://www.brucehardie.com/notes/045/

2 Likes
6

Thank you both @ColtAllen and @ricardoV94 for the quick responses. I think I understand the difference between the models so thanks for sharing the reference links. I think that I have enough to move forward with the likelihood model with a better understanding of how it works. Thanks!