I think this is minor, but I noticed the example doesn't use error bars for the customer numbers. The customer counts in the product/churn categories are counting statistics, so have an associated Poisson uncertainty. My guess is that considering the uncertainties when doing the likelihood parameter estimation won't fully obviate this issue, but I wonder how much it would help. I'm also not sure if real-world implementations commonly consider such uncertainties on their measured metrics.
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Hi - the reason there are no error bars for the number of customers in that 2x2 table is because there is no associated uncertainty. Those numbers are as you might find them in a dataset from summing with this particular problem. The problem we're looking at here is a binary churn/no churn problem, as opposed to one that looks at how many people churn out of a population.
That said, that absolute lack of uncertainty is itself the problem. The Maximum Likelihood approach to this sort of modelling implicitly assumes that the data you have is all there is to know about the problem you're working on and so can very easily overfit on a weird artefact like that. If you want to incorporate some kind of uncertainty in your estimates, then you either need to augment your dataset (i.e. include some random examples, roughly speaking), or estimate your model using the Bayesian approach which explicitly allows for uncertainty relating to the data itself.
Hope this clarifies!
> Those numbers are as you might find them in a dataset from summing with this particular problem. The problem we're looking at here is a binary churn/no churn problem, as opposed to one that looks at how many people churn out of a population.
But isn't that still a count per bin? Since it's a count it has a Poisson uncertainty.
> f you want to incorporate some kind of uncertainty in your estimates, then you either need to augment your dataset (i.e. include some random examples, roughly speaking), or estimate your model using the Bayesian approach which explicitly allows for uncertainty relating to the data itself.
Or use a likelihood that considers uncertainties. That also allows one to explicitly consider the uncertainties when maximizing the likelihood.
> But isn't that still a count per bin? Since it's a count it has a Poisson uncertainty.
Yes, as presented there. But in a binary classification setting, that's not how the data would be presented to the model. Instead you would have one row per customer with a churn/no churn label for that customer along with values for a number of independent variables you deem relevant. The reason I put it in that 2x2 table like that is just to make the problem more apparent. If you had potentially millions of customers (and therefore rows), the exact separation problem would not be as blatantly obvious to the naked eye as it is there - this is partly why I recommend using confusion matrices to check for whether this is happening.
> Or use a likelihood that considers uncertainties. That also allows one to explicitly consider the uncertainties when maximizing the likelihood.
Insofar as a set of parameters arrived at by means of MLE has associated standard errors, yes there is some uncertainty involved. If I understand you correctly, what you're talking about is modifying the likelihood to be flatter so that it can't get caught in this one localised whirlpool as easily. That's effectively regularisation - which I cover in the article. You could do it, but it's more or less a band aid. Really the data itself is the problem. Did you have a specific thing in mind when you were talking about likelihoods?
> But in a binary classification setting, that's not how the data would be presented to the model. Instead you would have one row per customer with a churn/no churn label for that customer along with values for a number of independent variables you deem relevant. The reason I put it in that 2x2 table like that is just to make the problem more apparent.
I see. Thanks for clarifying that.
> If I understand you correctly, what you're talking about is modifying the likelihood to be flatter so that it can't get caught in this one localised whirlpool as easily.
Effectively, yes. But by adding a term for the uncertainty on the measurements, not an uncertainty on the fit parameters (though those exist as well).
Well the Poisson uncertainty is only reasonable if the count is sampling a random process. But yea, they could try adding Poisson uncertainties and using a probabilistic emulator like a Gaussian process.