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I am training a classification model on a dataset of users on a website and each has 100 different measurements of their behaviour on the platform.

Most of these users are dormant but about 10% will reawken. We are interested in purchasing activity once awake.

We have separated the customers into two classes:

  1. Those who reawaken and purchase products 5+ times (positive class)
  2. Those who remain dormant and those who rewaken but purchase less than (or including) 4 times.

We also have a separate model that sorts customers into groups

  1. Will reawaken.
  2. Remain dormant

My question is:

If we output the probability of being in the positive class for both of these classifiers, what will the product of these two probabilities mean semantically?

Will it be P(5+ and reawaken), or will it be something else?

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Will it be P(5+ and reawaken), or will it be something else?

The events are not independent, so one cannot assume that p(A and B) = p(A) * p(B).

Let's denote the events as follows:

  • A = reawakens (not A = remains dormant)
  • B = purchases at least 5 times

The event "reawakens and purchases more than 5 times" is "A and B". In general we have:

p(A and B) = p(A/B) p(B)
p(A and B) = p(A/B) (p(A and B) + p(not A and B))

But it's impossible for a customer to stay dormant and buy anything, so:

p(not A and B) = 0 

Which gives us:

    p(A and B) = p(A/B) p(A and B)
    p(A/B) = 1

Then we obtain:

p(A and B) = p(B)

Note that this makes sense intuitively: the probability of buying at least 5 times is the same as awakening and buying at least 5 times.

Currently the labels of the two models overlap so I don't think you can infer much from combining their outputs. A way to make it usable would be to make the first model consider only awakening customers in order to avoid the overlap. But it might be more useful to train a single joint model to get a clear picture of your data. In general such a model would classify between 4 categories:

  • not A and not B
  • not A and B
  • A and not B
  • A and B

But since p(not A and B) = 0 there are actually only 3 labels corresponding to:

  • stays dormant
  • awakens and buys less than 5 times
  • awakens and buys at least 5 times
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