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We have an $n$-variate distribution $X\in\{0,1\}^n | \sum_i^n X_i = k$. Or, in other words, we are guaranteed that only $k$ variables will be $1$ in any given trial. Therefore, there are only $n \choose k$ possible outcomes. We will call $P(X_i = 1) = p_i$, $P(X_i=1 ~\land~X_j=1) = p_{ij}$, and so on.

We are given $m$ samples from this distribution. With the samples, we can estimate the true distribution using an empirical estimator (or MLE, maximum likelihood estimator). We call the estimated distribution $\hat{X}$. What is the smallest $m$ can be for which we can make sure that $\hat{X}$ is close to $X$? Specifically, how small can $m$ be such that $P(\norm{\hat{X} - X}_1 \geq \epsilon) \leq \delta$ for some $\epsilon$ and $\delta$? How small can $m$ be such that $P(\norm{\hat{X}-X}_2 \geq \phi) \leq \gamma$ for some $\phi$ and $\gamma$?

I'm interested in the case where $n$ is known and there is no restriction on $k$.

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  • $\begingroup$ Are you looking for universal values of $m$ or for formulae that depend on the $p_i,$ $p_{ij},$ etc.? $\endgroup$ – whuber Jun 13 '19 at 22:03
  • $\begingroup$ I'm looking for values of $m$ specific to this type of distribution. I assumed $p_i, p_{ij},etc.$ would be relevant in the analysis but not appear in the final result. The final result should be in terms of $n, k, \epsilon, \delta$ (or $\phi, \gamma$ for the L2 distance). Side note: I feel like $m$ shouldn't depend on $n$ but that's a guess...I'm pretty stuck on how to proceed. $\endgroup$ – jax.adan Jun 13 '19 at 22:22
  • $\begingroup$ $m$ will depend strongly on $n$ and $k,$ so don't drop them from consideration. Because you are looking for a universal bound, the $p_i$ etc. are wholly irrelevant. I suspect, without having checked, that the worst situation (in both metrics) is the highest-entropy one, where all of the subsets are equally likely. $\endgroup$ – whuber Jun 13 '19 at 22:27
  • $\begingroup$ I'm trying to find the expectation values of $d_{l1}(\hat{X}, X)$ and $d_{l2}(\hat{X}, X)$. I think that should be good enough -- at least to claim $\epsilon$ accuracy after a certain number of samples. $\endgroup$ – jax.adan Jun 14 '19 at 15:43
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  • $\ell_2$ won't be "interesting." You can learn any discrete distribution to $\ell_2$ distance $\phi$ with probability $1-\gamma$ with $O( \log(1/\gamma)/\phi^2 )$ samples, with no dependence on the domain size. And this will be tight even with your constraint (pretty much any estimation, even on a single bit, will require that many samples).

  • Now, in total variation/$\ell_1$. You basically have, as you say, an arbitrary distribution on $M = \binom{n}{k}$ elements. Learning this to total variation distance $\varepsilon$, w.p. $1-\delta$, can be done with $$ O\left( \frac{M + \log(1/\delta)}{\varepsilon^2} \right) $$ samples, and this is tight (up to constants). So the answer is $$ \Theta\!\left( \frac{\binom{n}{k} + \log(1/\delta)}{\varepsilon^2} \right) $$ Note: the upper bound is not going to be efficient in $n$ (but is efficient in the number of samples, and achieved by the empirical estimator) if $k$ is not a constant, as then you have a sample complexity super-polynomial in $n$ (e.g., $k=n/2$ leads to roughly a $2^{n}/\sqrt{n}$ dependence).

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  • $\begingroup$ Yea, I was hoping there was something better than reducing to the univariate case. Does this change if we are only interested in learning p_i's and p_{ij}'s? $\endgroup$ – jax.adan Jun 14 '19 at 18:56
  • $\begingroup$ @jax.adan that would be much cheaper (something like $\log(n/\delta)/\varepsilon^2$ to simultaneously approximate these ~$n^2$ values up to an additive $\varepsilon$), but that won't give you any meaningful guarantee about approximating the distribution itself $\endgroup$ – Clement C. Jun 14 '19 at 19:16
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    $\begingroup$ Got it. Can you give me more info on how you got $log(n/ \delta) / \epsilon^2$? And what distance metric is being used in that case? That any $p_i$ or $p_{ij}$ is within $\epsilon$ from the approximation? $\endgroup$ – jax.adan Jun 14 '19 at 19:19
  • $\begingroup$ Just take enough samples to estimate all these parameters (basically, they are Bernoulli coin flips to an additive $\varepsilon$, with failure probability $\delta'=\delta/n^2$ (using the same samples for all) then take a union bound over all $n(n+1)/2$ possible failures. You only need $O(\log(1/\delta')/\varepsilon^2)$ samples for that. $\endgroup$ – Clement C. Jun 14 '19 at 19:28
  • $\begingroup$ @jax.adan there is no natural metric there, though you can phrase it as $\ell_\infty$ for the vector of $n(n+1)/2$ parameters considered (marginal and pairwise means). $\endgroup$ – Clement C. Jun 14 '19 at 19:40

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