Regression for only positive values prediction

Question

I already referred these posts here. Please don't mark it as duplicate. This post is continuation of my other post here. So, am not providing the full context here. You can refer the link here

a) Is there any other regression model that can take raw input (without any data standardization/transformation using log,sqrt etc) and give us positive predictions only? or any other relevant regression model for customer revenue prediction problem?

Answer 1

YES

Generalized linear models with appropriate “link” functions can force predictions to be positive. For instance, logistic and probit regression squeeze into $(0,1)$ the values given by the linear prediction. Other link functions can allow for different allowed prediction values, such as a $\log$ link forcing values to be positive (but any positive number is possible).

Notably, this is different from transforming the outcome variable. Remember that you are (probably) estimating an expected value. For a nonlinear link function $g$:

$$ \mathbb E\left[g\left( Y\vert X=x\right)\right] \ne g\left( \mathbb E\left[ Y\vert X=x \right] \right) $$

Therefore, using a link function is not equivalent to transforming the outcome.

This idea can be extended to models such as generalized additive models and neural networks. If $R$ is the regression function that has unconstrained outputs (so $R(X)=X\beta$ for a generalized linear model), you can take $ g(R(X)) $ to assure yourself of positive values.