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I want to estimate the mean of some quantity over a two-dimensional space, say the average height of some bounded geographic area.

it's convenient to sample like so

  • choose random latitudes,
  • choose random longitudes
  • form the Cartesian product of these lats and longs
  • query at those points

This sample is not IID, but can we still say anything about the convergence of the sample mean to the population mean.

(related: Interpretation of cartesian product of the support of marginal distribution)

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Cool question! The samples aren't iid, but you can partition them into subsets that are each internally iid. If you've chosen $x_1, ... x_n$ and $y_1, ..., y_n$ all independent, then for any fixed $j$, you can select a subsequence of iid samples by only using samples of the form $x_i, y_{i+j}$ (where the $y$ index is modulo $n$). The average over each subsequence should converge under the usual assumptions (e.g. no half-Cauchy-distributed mountain summits). The grand mean can then be written as an average of those terms, one per $j$, and each of them converges individually, which guarantees convergence overall (by Slutsky's Theorem).

EDIT: For the geographical area example, your $x$ and $y$ samples should be uniform. In other words, to approximate $\frac{\int h(x,y)dxdy}{\int dx dy}$, you rewrite as $\int h(x,y)f(x)g(x)dxdy$ where $f,g$ are the PDF's you're sampling from. Those PDF's should be constant. Otherwise you're doing importance sampling, and you need to account for the difference between the weights of the average you want (uniform/constant) and the weights of the average you're computing ($f(x)g(y)$ where $f$ and $g$ are the actual pdf's you're using).

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  • $\begingroup$ wow! that's such a great approach! $\endgroup$
    – fgregg
    May 7, 2020 at 15:49
  • $\begingroup$ Thanks! I haven't done anything that clever since college! Just out of pure curiosity, what are you doing? And what makes this scheme more convenient than vanilla Monte Carlo? $\endgroup$ May 8, 2020 at 13:07
  • $\begingroup$ I edited to clarify about uniform sampling. It just means you should take simple random samples of each database. $\endgroup$ May 10, 2020 at 22:20
  • $\begingroup$ returning to this, i don't quite see how the subsequences are iid. imagine that we sample three latitudes and three longitudes, so we'll have three subsequences, the first would be: [(x_0, y_0), (x_1, y_1), (x_2, y_2)] given the first element in the sequence (x_0, y_0), we know that Pr(x = x_0) = Pr(y = y_0) = 0 for all other x and y in the sequence. how can that be iid? $\endgroup$
    – fgregg
    Jun 30, 2021 at 14:51
  • $\begingroup$ (the above only comment only holds if we are sampling x and y without replacement). $\endgroup$
    – fgregg
    Jun 30, 2021 at 14:55

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