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In the software i'm contributing to, there's a procedure to take samples from a certain 2D distribution (if you're curious, it's for sampling a BSDF, used in rendering) and a procedure to return the corresponding pdf (over the hemisphere in terms of solid angle measure).

In order to test the implementation of the sample() and pdf() functions, i want to test if the binned sample counts agree with the bin probabilities $P_i$ i get from the pdf function (by integrating pdf over each bin). Basically i want to compare the two distributions in a significance test way and based on the result, let the unit test for the software pass or fail.

In the literature and similar software projects i mainly see people using Chi-squared tests, which can work to a degree, but produces problems with bins with zero counts or in general when the bin counts vary over several orders of magnitude.

Therefore i started thinking how to tackle this problem from a Bayesian standpoint and came up with the following:

  • I take n samples and bin them into N bins, interpreting them as counts drawn from a multinomial distribution.
  • I construct the posterior for the bin probabilities using a Jeffrey's prior, giving me a Dirichlet distribution: $p(\theta|\text{bincounts})=\text{Dirichlet}(\alpha_1+\frac{1}{2},\cdots,\alpha_N+\frac{1}{2}$)

Now my first idea was to compute a credible interval (volume in this case) from the Dirichlet distribution and check if the bin probabilities that i computed from pdf() are inside this region to decide on the test result.

But since i have on the order of a thousand bins, directly computing this region for a given significance level is not feasible.

So instead i had the idea to invert the idea and basically compute the significance level (or credibility) given our bin probabilities and Dirichlet distribution by doing the following:

  • calculate $\text{threshold}=p(\text{pdf}(P_1,\cdots,P_N)|\text{bincounts})$ by evaluating our posterior with the bin probabilities we got from pdf().
  • integrate the indicator function (showing which regions have lower posterior density than our threshold)$$ I(\theta) = \left\{ \begin{array}{lr} 1 :& 0 \lt p(\theta) < \text{threshold} \\ 0 :& \text{otherwise} \end{array} \right. $$ over the whole support of the Dirichlet posterior density.
  • to make this numerically tractable, estimate the integral by using importance sampling over the posterior density itself

This should then result in a number between 0 and 1 showing how much posterior 'volume' has lower density than our threshold giving us a kind of credibility that we can then use to decide if the test fails.

I implemented this in Mathematica and there my algorithm looks like this:

DirichletCredibility[observedcounts_, probs_, n_] := Module[
  {dist, mcsamples, pdf, threshold},
  dist = DirichletDistribution[observedcounts + 1/2];
  threshold = PDF[dist, Most[probs]];
  mcsamples = RandomVariate[dist, n];
  N@Mean[Composition[Boole[# <= threshold] &, PDF[dist]] /@ mcsamples]
]

I hope i explained well enough what i'm trying to achieve, if not i'm happy to clarify and give more details.

Now what happens when i use this is twofold: - if i test DirichletCredibility with bin counts drawn from a multinomial distribution and bin probabilities which don't match the multinomial probabilities the resulting credibility goes to zero with a high enough number of total samples, which is great! - if on the other hand i test DirichletCredibility with perfectly matching multinomial bin probabilities the result doesn't approach one, but instead tends to a uniform distribution (by repeating the credibility computation with different samples from the multinomial distribution).

My questions are: - Do you think my methodology can work in principle or does it have major flaws that i should address or consider? - If this is not feasible how would you tackle the problem in a Bayesian way to get a useful significance test for comparing sampled bin counts with reference probabilities?

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