How can we model the class which maximizes the event probability?

If you want to predict the best class of input variable, why don't you take the input variable itself as the output? And say the probability or a boolean of whether the event is happening or not as a input. I guess with enough training data, it may predict right. But that might not be an ideal solution, since you already specify the model the best class for which the event happens, so your approach is a better one.

As you are looking for a conditional distribution of a variable given another one, graphical models come to mind.

In your method, you would build a model which maps the input features (ad type, customer descriptors) to the output (purchase made = 0/1). You can use a model such as a logistic regression, or a decision tree (which can model the interaction effects between the ad type and the customer descriptors.) This seems to be reasonable and is one of the approaches commonly used for ad selection.

A few other options based on customer similarity:

• Near-neighbour based approach: Based on the customer descriptors, identify similar neighbours of the current customer. Compute the probability of conversion for each ad type for all the neighbours put together, and pick the ad type which has the highest probability of conversion.

• Cluster based approach: Form clusters of customers based on the customer descriptors. Map the current customer into one of the clusters, and pick the ad type which maximizes conversion probability for that cluster.

With respect to designing a system of ad selection using such a model, it is also important for the system to explore new regions of the space, so some small randomization of predictions will help here.