Two part model - using the output of first as feature to second? I am working on a two-part model to predict a continuous target variable $Y$, given a feature set $F$. $Y$ is product of two continuous variables - $A$ and $B$, where $A \epsilon [0, 1]$ and $B \epsilon (0, \infty)$.
The use-case is to predict the revenue from an advertisement click. We fit a two-part model (sequential decision model ie Probit + OLS) to first predict the probability of a purchase after an ad click and then given a purchase happens estimate the revenue from that purchase.
Now, what we observe in our unique use-case is that a higher probability of conversion (output of first model) is a strong indicator of lower revenue. As such, if we include the output of the first model as a feature in the second OLS model it improves the fit (ie R^2) of the second model.
Also, what we realized is that since the conversion probability is a feature for the revenue OLS model, the predicted revenue values from the model are already dampened (ie further multiplying it with the output of the first model ie conversion probabilities as is traditionally done for two-part model actually degrades the quality of the final prediction)
We observe this empirically but is there any literature around such scenarios where we have "dissonant effects" in a two-part model ie the output of the first step is a predictor for the second step?

You have a model R = f(CP, F1, ..., Fn) + eR, where

*

*R is the revenue

*CP is the conversion probability

*F1, ..., Fn are further features

*f is the function describing the model

*eR is some noise.

Furthermore, you have a second model for the conversion probability CP: CP = g(G1, ..., Gm) + eCP, where

*

*CP is the conversion probability

*G1, ..., Gm are the features you use to predict CP

*g is the function describing the model for CP

*eCP is some noise.

Now, everything depends on how you choose the functions f and g, what learnable parameters and hyperparameters you give them, and what you presume for the distribution of the noises. This describes a simple Bayesian network (BN), and there are several learning algorithms for BNs varying in complexity, e.g. the EM algorithm.
And there will be no a priori degradation of the quality of the final prediction.
