# Hypothesis testing for not identically distributed random variables conditioned on the outcome of a subset

I encountered the following problem (I give more details of the problem at the end of the post) and I am trying to figure out the best way of performing a null hypothesis testing. I looked for similar questions (like this) but it does not fit exactly my problem.

I have a random vector $$X = (X_1,...,X_N)$$ of $$N$$ random binary variables, not necessarily independent and non identically distributed. These $$N$$ variables are divided into two subsets: $$A$$ with $$N_A$$ random variables and $$B$$ with $$N_B$$ variables ($$N_A + N_B = N$$), so I can also write $$X = (X_A,X_B)$$. I know the marginal distribution of each of the binary variables, as well as the first and second moments of the random vector.

Now I consider another random vector $$Y = (Y_1,...,Y_N) = (Y_A, Y_B)$$, from which I can only sample in two steps: first sample $$Y_A$$ (obtaining some string $$(a_1,...,a_{N_A})$$), and then sample $$(Y_B | Y_A = (a_1,...,a_{N_A}))$$. The null hypothesis is that $$Y$$ follows the same distribution as $$X$$.

The problem that arises here is that the set of possible outcomes for $$A$$ is too large, which means that the probability to obtain the same $$a= (a_1, ..., a_{N_A})$$ is negligible. Thus, the distribution of the random variables in subset $$B$$ changes in each iteration. Since I cannot repeat the sampling under identical conditions I cannot use the usual central limit theorem to approximate the experimental mean by a Gaussian and perform typical Gaussian hypothesis tests.

You can imagine this as having $$N$$ biased coins, each bias being different, and the coins may not be independent. First I throw $$N_A$$ of the coins, which conditions the possible outcomes of the second set $$B$$.

How can I test my null hypothesis under these restrictions?

More details of the problem: I am dealing with a problem in quantum mechanics, having a state of $$N$$ spins that might be entangled (thus non independent variables). The data corresponds to measuring part of the system first (subsystem $$A$$), thus collapsing the whole state and conditioning the possible outcomes of the rest of the system (subsystem $$B$$). Because the set of possible outcomes for subsystem $$A$$ is very large and because when I measure I destroy the state, sampling two times subsystem $$A$$ and obtain the same result is highly unlikely.

Thank you very much in advance! Any idea or suggestion is highly appreciated!

• What is your null hypothesis here? It's not clear to me. Mar 4 '20 at 10:49
• @MattF. I have edited my question and hope it is more clear. I did not write down the distribution corresponding to the null hypothesis because I want to leave it in a general formulation, as (see end of the post) it arises from the physical problem I have. It is a very general binary distribution: each binary string (a_1,...,a_N) has a certain probability, but it is not given by any well known distribution. I can compute expectation values and correlations theoretically and I would like to see if certain data, that has been sampled in this particular 2 steps-way, matches my model.
– ECR
Mar 4 '20 at 12:33
• I still don't find this clear. It would help to have a simple example, maybe with $A=2, B=1$, and maybe using sequentially chosen decks of cards rather than "non-independent coins". For now, I can't tell how this sequential sampling is different from normal sampling, or whether the sample of $Y$ is being compared with a distribution, or with a sample from $X$. Mar 4 '20 at 15:14

(ii) Without taking into account how entanglement works, you cannot devise a reasonable test here. I don't actually know how entanglement works, but suspect that, because of the concentration of measure phenomenon, the conditional distribution of $$Y_B$$ given $$Y_A$$ may depend (at least approximately, with high probability) on just a low-dimensional functional of $$Y_A$$. If so, you can get a desired dimensionality reduction, which would allow a reasonable test.