PTED: Permutation Test using the Energy Distance

Think of it like a multi-dimensional KS-test! It is used for two sample testing and posterior coverage tests. In some cases it is even more sensitive than the KS-test, but likely not all cases.

Install

To install PTED, run the following:

pip install pted

Usage

PTED (pronounced "ted") takes in x and y two datasets and determines if they come from the same underlying distribution. For information about each argument, just use help(pted.pted) or help(pted.pted_coverage_test).

The returned value is a p-value, an estimate of the probability of a more extreme instance occurring. Under the null hypothesis, a p-value is drawn from a random uniform distribution (range 0 to 1). If the null hypothesis is false, one would expect to see very low p-values and so one can set a limit such as p=0.01 below which we reject the null hypothesis. In this case 1/100th of the time even when the null hypothesis is true, we will reject the null.

Example: Two-Sample-Test

from pted import pted
import numpy as np

x = np.random.normal(size = (500, 10)) # (n_samples_x, n_dimensions)
y = np.random.normal(size = (400, 10)) # (n_samples_y, n_dimensions)

p_value = pted(x, y)
print(f"p-value: {p_value:.3f}") # expect uniform random from 0-1

Example: Coverage Test

from pted import pted_coverage_test
import numpy as np

g = np.random.normal(size = (100, 10)) # ground truth (n_simulations, n_dimensions)
s = np.random.normal(size = (200, 100, 10)) # posterior samples (n_samples, n_simulations, n_dimensions)

p_value = pted_coverage_test(g, s)
print(f"p-value: {p_value:.3f}") # expect uniform random from 0-1

How it works

Two sample test

PTED uses the energy distance of the two samples x and y, this is computed as:

$$d = \frac{2}{n_xn_y}\sum_{i,j}||x_i - y_j|| - \frac{1}{n_x^2}\sum_{i,j}||x_i - x_j|| - \frac{1}{n_y^2}\sum_{i,j}||y_i - y_j||$$

The energy distance measures distances between pairs of points. It becomes more positive if the x and y samples tend to be further from each other than from themselves. We demonstrate this in the figure below, where the x samples are drawn from a (thick) circle, while the y samples are drawn from a (thick) line.

In the left figure, we show the two distributions, which by eye are clearly not drawn from the same distribution (circle and line). In the center figure we show the individual distance measurements as histograms. To compute the energy distance, we would sum all the elements in these histograms rather than binning them. You can also see a schematic of the distance matrix, which represents every pair of samples and is colour coded the same as the histograms. In the right figure we show the energy distance as a vertical line, the grey distribution is explained below.

The next element of PTED is the permutation test. For this we combine the x and y samples into a single collection z. We then randomly shuffle (permute) the z collection and break it back into x and y, now with samples randomly swapped between the two distributions (though they are the same size as before). If we compute the energy distance again, we will get very different results. This time we are sure that the null hypothesis is true, x and y have been drawn from the same distribution (z), and so the energy distance will be quite low. If we do this many times and track the permuted energy distances we get a distribution, this is the grey distribution in the right figure. Below we show an example of what this looks like.

Here we see the x and y samples have been scrambled in the left figure. In the center figure we see the components of the energy distance matrix are much more consistent because x and y now follow the same distribution (a mixture of the original circle and line distribution). In the right figure we now see that the vertical line is situated well within the grey distribution. Indeed the grey distribution is just a histogram of many re-runs of this procedure. We compute a p-value by taking the fraction of the energy distances that are greater than the current one.

Coverage test

In the coverage test we have some number of simulations nsim where there is a true value g and some posterior samples s. For each simulation separately we use PTED to compute a p-value, essentially asking the question "was g drawn from the distribution that generated s?". Individually, these tests are not especially informative, however their p-values must have been drawn from U(0,1) under the null-hypothesis. Thus we just need a way to combine their statistical power. It turns out that for some p ~ U(0,1) value, we have that - 2 ln(p) is chi2 distributed with dof = 2. This means that we can sum the chi2 values for the PTED test on each simulation and compare with a chi2 distribution with dof = 2 * nsim. We use a density based two tailed p-value test on this chi2 distribution meaning that if your posterior is underconfident or overconfident, you will get a small p-value that can be used to reject the null.

Interpreting the results

Two sample test

This is a null hypothesis test, thus we are specifically asking the question: "if x and y were drawn from the same distribution, how likely am I to have observed an energy distance as extreme as this?" This is fundamentally different from the question "how likely is it that x and y were drawn from the same distribution?" Which is really what we would like to ask, but I am unaware of how we would do that in a meaningful way. It is also important to note that we are specifically looking at extreme energy distances, so we are not even really talking about the probability densities directly. If there was a transformation between x and y that the energy distance was insensitive to, then the two distributions could potentially be arbitrarily different without PTED identifying it. For example, since the default energy distance is computed with the Euclidean distance, a single dimension in which the values are orders of magnitude larger than the others could make it so that all other dimensions are ignored and could be very different. For this reason we suggest using the metric mahalanobis if this is a potential issue in your data.

Coverage Test

For the coverage test we apply the PTED two sample test to each simulation separately. We then combine the resulting p-values using chi squared where the resulting degrees of freedom is 2 times the number of simulations. Because of this, we can detect underconfidence or overconfidence. Specifically we detect the average over/under confidence, it is possible to be overconfident in some parts of the posterior and underconfident in others. Underconfidence is when the posterior distribution is too large, it covers the ground truth by spreading too thin and not fully exploiting the information in the prior/likelihood of the posterior sampling process. Sometimes this is acceptable/expected, for example when using Approximate Bayesian Computation one expects the posterior to be at least slightly underconfident. Overconfidence is when the posterior is too narrow and so the ground truth appears as an outlier from its perspective. This can occur in two main ways, one is by having a too narrow posterior. This could occur if the measurement uncertainty estimates were too low or there were sources of error not accounted for in the model. Another way is if your posterior is biased, you may have an appropriately broad posterior, but it is in the wrong part of your parameter space. PTED has no way to distinguish these modes of overconfidence, however just knowing under/over-confidence can be powerful. As such, by default the PTED coverage test will warn users as to which kind of failure mode they are in if the warn_confidence parameter is not None (default is 1e-3).

GPU Compatibility

PTED works on both CPU and GPU. All that is needed is to pass the x and y as PyTorch Tensors on the appropriate device.

Citation

If you use PTED in your work, please include a citation to the zenodo record and also see below for references of the underlying method.

Reference

I didn't invent this test, I just think its neat. Here is a paper on the subject:

@article{szekely2004testing,
  title={Testing for equal distributions in high dimension},
  author={Sz{\'e}kely, G{\'a}bor J and Rizzo, Maria L and others},
  journal={InterStat},
  volume={5},
  number={16.10},
  pages={1249--1272},
  year={2004},
  publisher={Citeseer}
}

Permutation tests are a whole class of tests, with much literature. Here are some starting points:

@book{good2013permutation,
  title={Permutation tests: a practical guide to resampling methods for testing hypotheses},
  author={Good, Phillip},
  year={2013},
  publisher={Springer Science \& Business Media}
}

@book{rizzo2019statistical,
  title={Statistical computing with R},
  author={Rizzo, Maria L},
  year={2019},
  publisher={Chapman and Hall/CRC}
}

There is also the wikipedia page, and the more general scipy implementation, and other python implementations