# Learning problem when we have data from distributions $(p_i)$ when we care about (known) distribution $p^*$?

Suppose we have a dataset $$D$$ or multiple datasets $$(D_i)$$, with distributions $$p_i:X\to \mathbb R$$. Suppose there is another distribution $$p^*$$. All distributions are known, including $$p^*$$, but the $$p_i$$ are different from $$p^*$$, with different support.

Suppose also there is a function $$f:X\to Y$$, and we are interested in the distribution of $$f(X)$$ according to $$p^*$$, or some statistics of this distribution, e.g. the expectation of $$f(X)$$.

• Does this problem, or a similar or more general version of the problem, have a name, so I can look it up?

• Are there algorithms for making estimates that are appropriate for this (what are their names)?

• How does Bayesian statistics deal with this?

ps. I'm not necessarily asking for an explanation, just for the terminology/general direction so I can look it up.

EDIT: To clarify, we know the distribution on X but we don’t know f. E.g. suppose X is a set of variables, $$X=(X_1,...,X_n)$$, where for a given individual $$x_1$$ is their height, $$x_2$$ is their age, etc. Now suppose $$f(x)$$ measures the life expectancy. Suppose that we have datasets from people in various European countries, the US, etc, but we don’t have data about Japan. In this case $$p^∗$$ is the distribution of personal characteristics in japan, and the pi from Germany, the US, etc. But we can only measure the $$f(X)$$ from the datasets in Europe and the US.

Then we want to estimate the life expectancy of someone from Japan. However, we cannot just blindly estimate a function f on the data from Europe and the US, and confidently generalize to Japan, because (1) the distribution of $$X$$ is different for Japan (different support even), and different areas of the domain of $$X$$ may have different effects on life expectancy (e.g. old age+exercise may be bad for life expectancy even if young age + exercise is good)

• How does p* relate to the problem? Nov 10, 2019 at 10:44
• Could you probably give a more specific example of this problem? Why do you need the datasets $D_i$ and distributions $p_i$ for? If you do know the distribution, obtaining expectation is straightforward. Nov 12, 2019 at 10:42
• Ok, please let me know, if this is your question: $p_i$, the distribution of individual characteristics $(x_1,..,x_n)$ in country $i$ is observed from the data $X_i$ that we have for this country. Using this information we can construct a statistical model for the distribution of $\hat f_i(x)$ (mean and other moments) for the function $f(x)$. For Japan we do not have data $X_j$ and therefore do not know the distribution $p_j$ and can't build a model $\hat f_j(x)$ from the data. What are the ways to get $\hat f_j (x)$ based only on other countries' data $X_i$ without dataset $X_j$ for Japan? Nov 12, 2019 at 11:11
• @user56834 It seems that I get you. Why don't you put your neat toy example into the question statement? Nov 12, 2019 at 11:16
• Your comments contain some critical details that are missing from the question. In particular, that $f$ is unknown, and that the datasets contain sampled values for both $X$ and $Y$. Could you edit the question to include these details? Including your life expectancy example would also help clarify the question. Nov 12, 2019 at 16:56