Suppose we have $n$ random variables $X_n$ - let's say these are measures of customer engagement - and we sample these $m$ times through a set of designed trials. The resulting $m$ data points define the 'behavior' for each customer. We end up with an $m$ by $n$ matrix of samples. Now, each one of these random variables is also parametrized $X_n = X_n(a, b, ...,z)$ where $a, b, ..., z$ can be both categorical and numerical variables. Let's say $a, b, ..., z$ are information about the customer (age etc.) that can be collected beforehand and that does not change throughout our sampling trials.
The task is to come up with a model that clusters customers with similar behavior and then to use this model to classify new customers into one of these clusters. However, for new customers, sampled data/behavior information is not available -- only the parameter values $a, b, ..., z$. There is most certainly a relationship between the $a, b, ..., z$ parameters and the behavioral samples for each customer, and we would like to somehow calibrate and exploit this in the model.
At this point, I'm struggling to clearly understand what the essence of the problem is. It's an unsupervised learning task where we have samples of many random variables that are themselves parameterized by some categorical or numerical values. We would like to come up with a model or representation that makes it easy to classify a new random variable if we can only observe its parameters.
I'm not sure what the most obvious approach is to try here (I only have basic background in ML) but I'm hoping that this is actually a fairly standard problem in literature that I don't know about.
Any help or suggestions would be helpful!