Scenario:
Imagine having $2$ distinct set of features S and U and $2$ distinct sets of labels Ys and Yu.
Each element of the set S is a k-dimensional vector. Each element of the set U is a l-dimensional vector.
We can conduct Bayes inference on each of this sets independently so that we obtain $p(y|s)$ and $p(y|u)$, where $s$ and $u$ are a $k$- and $l$-dimensional vector $s$ respectably.
Example:
Think about breast cancer detection from images and breast cancer detection based on blood samples. The set of patients that are described by S and U is empty or of very small cardinality.
Question 1:
Assuming $s_x$ and $u_x$ describe an observation in features of the S and U sets respectably. Is there a way to construct $p(y_x|s_x,u_x)$ ?
Question 2:
Assuming there are $n$ observations ($n$ vector pairs $(s_i, u_i)$) that are known to describe the of the same process (think of the same patient) and $n \ll \min(|S|,|U|)$.
What is is the best way co construct the posterior distribution?