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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 $f(X)$.

I would like to learn more about this type of problem. What keywords/references should I look for?

  • 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)?

  • Is there learning theory about this (that I can look up)?

  • 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)

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  • $\begingroup$ How does p* relate to the problem? $\endgroup$ Nov 10, 2019 at 10:44
  • $\begingroup$ 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. $\endgroup$
    – Konstantin
    Nov 12, 2019 at 10:42
  • $\begingroup$ 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? $\endgroup$
    – Konstantin
    Nov 12, 2019 at 11:11
  • $\begingroup$ @user56834 It seems that I get you. Why don't you put your neat toy example into the question statement? $\endgroup$
    – Konstantin
    Nov 12, 2019 at 11:16
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    $\begingroup$ 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. $\endgroup$
    – user20160
    Nov 12, 2019 at 16:56

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I might have misunderstood your question, but this seems to me like a job for multilvel/hierarchical models, possibly with poststratification.

Models like this are often used for opinion polling. Let's say you want to know what is the public support for gun control in Alabama, but you have only few pols, or none in Alabama, but you have polls from Texas, Mississipi, Georgia etc. Then you can infer support for each demographic group (e.g. black, white, educated, rich) in Alabama based on those other polls, and use posstratification to get the overall state level support, taking into an account difference in demographic characteristics.

There are whole books dealing with these problems, the main one is Data Analysis Using Regression and Multilevel/Hierarchical Models by Gelman and Hill, that deals with frequentists and bayesian approaches to problems like this (not sure if it includes poststratification).

So the name is hierarchical/multilevel models which can then by followed by poststratification. This is sometimes also called random effects or mixed effects models. This would also lead you to learning theory or algorithms for these problems. In general, these are just regressions of regressions thus the algorithms and theory is pretty standard. Bayesian statistics deals with this by putting a prior over the individual coefficients that is then estimated from the data (i.e. emprical bayes)

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