PTED: Permutation Test using the Energy Distance

PTED (pronounced "ted") takes in x and y two datasets and determines if they were sampled from the same underlying distribution. It produces a p-value under the null hypothesis that they are sampled from the same distribution. The samples may be multi-dimensional, and the p-value is "exact" meaning it has a correctly calibrated type I error rate regardless of the data distribution.

NOTE: PTED is basically just the Maximum Mean Discrepancy (MMD) except using the Euclidean distance as the kernel. It took me a while to realize this, and in the meantime I made a nice python package. So I consider it a win for everyone.

Install

To install PTED, run the following:

pip install pted

If you want to run PTED on GPUs using PyTorch, then also install torch:

pip install torch

If you want to use JAX arrays as inputs, then also install jax:

pip install jax

The two functions are pted.pted and pted.pted_coverage_test. For information about each argument, just use help(pted.pted) or help(pted.pted_coverage_test).

What does PTED do?

You can think of it like a multi-dimensional KS-test! Although it works entirely differently from the KS-test, this gives you some idea of how useful it is! 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.

PTED is useful for:

• "were these two samples drawn from the same distribution?" this works even with noise, so long as the noise distribution is also the same for each sample
• Evaluate the coverage of a posterior sampling procedure, and check over/under-confidence
• Check for MCMC chain convergence. Split the chain in half or take two chains, that's two samples that PTED can work with (PTED assumes samples are independent, make sure to thin your chain accordingly!)
• Evaluate the performance of a generative ML model. PTED is powerful here as it can detect overfitting to the training sample (ensure two_tailed = True to check this).
• Evaluate if a simulator generates true "data-like" samples
• PTED (or just the energy distance) can be a distance metric for Approximate Bayesian Computing posteriors
• Check for drift in a time series, comparing samples before/after some cutoff time
• The energy distance is differentiable, so it can be used to train machine learning algorithms

And much more!

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

Note, you can also provide a filename via a parameter: sbc_histogram = "sbc_hist.pdf" and this will generate an SBC histogram from the test¹.

You can also generate a Probability Integral Transform (PIT) plot via pit_plot = "pit.pdf" for pted_coverage_test. The PIT plot shows the empirical CDF of the p-values against the expected uniform CDF (1:1 diagonal), along with a shaded band determined by pit_confidence (95% by default). The band is simultaneous: under the null the whole curve stays inside it with probability pit_confidence, so any portion of the curve that pokes out is evidence of non-uniform p-values at significance 1 - pit_confidence, indicating a potentially miscalibrated posterior.

The band is built from the exact null distribution of the p-values. The ECDF can only move where the p-values can actually land, so rather than constraining order statistics the band constrains the counts there. Working with counts rather than order statistics is what makes this usable here. Permutation p-values are discrete — after enumeration they are exactly uniform on a grid of L = len(permute) + 1 points — and ties are precisely what a binomial count expects. pted_coverage_test works out the lattice size and passes it in for you. Ultimately, this means that the fact we only use a finite number of simulations to compute the coverage test does not affect the validity of the bands in the PIT plot; any point that sticks out of the grey shaded area indicates possible miscalibration at the given significance level (default 95%).

How it works

Two sample test

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

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

The within-group sums run over distinct pairs, skipping the i = j terms which are zero by construction. That makes d an unbiased estimator of the population energy distance, so under the null it scatters around zero and can come out slightly negative (and strongly negative if x and y are more alike than chance allows, e.g. if they share samples). Only its rank among the permuted values matters. For equal sample sizes this ranks the permutations identically to the 1/n^2 normalisation of Székely & Rizzo, so the p-value is unchanged; for unequal sizes the two differ slightly.

The energy distance measures distances between pairs of points². 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. The procedure goes like this, first you sample from your prior: g ~ Prior(G). Then you sample from your likelihood: x ~ Likelihood(X | g). Then you sample from your posterior: s ~ Posterior(S | x), you will want many samples s. You repeat this procedure nsim times. The g and s samples are what you need for the test.

Internally, 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 possibly not especially informative (unless the sampler is really bad), however their p-values must have been drawn from U(0,1) under the null-hypothesis³. Thus we just need a way to combine their statistical power. It turns out that for some p ~ U(0,1), 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 simple doubling procedure (2 * min(p_right, p_left)) to get the p-value meaning that if your posterior is underconfident or overconfident, you will get a small p-value that can be used to reject the null.

Example: Containment Test

pted asks "are these the same distribution?". pted_containment_test asks the weaker, directional question "does y cover x?" — a sample drawn from a tighter distribution than y passes, while one that spreads beyond it, sits off to one side, or throws a few points clear of it fails.

from pted import pted_containment_test
import numpy as np

y = np.random.normal(size = (500, 10))
inside  = np.random.normal(size = (100, 10)) * 0.5
outside = np.random.normal(size = (100, 10)) * 1.6

print(pted_containment_test(inside,  y))  # large: contained
print(pted_containment_test(outside, y))  # small: not contained

Unlike every other test in the package this one is not symmetric — swapping the arguments asks the other question and will usually give a different answer. That asymmetry is the point, and it is why the energy distance alone cannot answer it: the energy distance is just as large when x is tighter than y as when it is broader, so a one-tailed energy test rejects 99.8% of the time on a perfectly contained sample.

Instead, every x point gets a depth — its mean distance to the points labelled y — so peripheral points score high. This is nearly identical to pted_coverage_test, and we use the same -2 sum log p formula to combine the p-values for each x. However, instead of interpreting the result as a $\chi^2$ distribution, we use a secondary permutation test to calibrate it.

The test is exact where x and y share a distribution and conservative inside the null, which is the goal for a containment test. It measures depth rather than literal support: a tight cluster of x sitting in a low-density pocket well inside y counts as contained. So use a heavy dose of caution when interpreting the results.

Read the plot as well as the p-value

-2 sum log p is a sum whose per-point floor is zero against a null mean of two, so a bulk of x sitting deep inside y banks slack that can hide a handful of points y cannot reach at all. In testing, 5% of x placed five sigma outside the prior predictive went undetected, while every other failure mode — data broader than y, data offset into its tail — was caught cleanly. So pass pit_plot:

p = pted_containment_test(data, prior_predictive, pit_plot = "containment.pdf")

The plot shows the empirical CDF of the per-point depth p-values — one step per point of x — on a logarithmic p axis, because the whole diagnostic lives at the left edge. It carries two reference marks and no diagonal.

The upper bound is the one-sided simultaneous ceiling for n1 p-values that really are uniform: the same as the ordinary PIT plot, with the lower edge dropped. Only the upper edge means anything here. Any part of the CDF below this line indicates the x values are likely contained at the threshold level (95% by default).

The vertical line marks threshold p-value (0.05 by default). Parts of the CDF to the right of this line are embedded in y at the threshold level and so likely are contained. Points to its left are peripheral relative to y and so this may suggest the values are not contained. However, when many values of x are being tested some leakage to low p-values are expected, which is why the uniform-upper-bound line is also plotted.

Read them together rather than as a decision rule. A curve with points to the left of the vertical line and above the uniform-upper-bound is a fair indication that x is not contained in y. Though results either way are not definitive, as is the nature of null hypothesis testing, and even moreso here.

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 or other 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).

Necessary but not Sufficient

PTED is a null hypothesis test. This means we assume the null hypothesis is true and compute a probability for how likely we are to have a pair of datasets with a certain energy distance. If PTED gives a very low p-value then it is probably safe to reject that null hypothesis (at the significance given by the p-value). However, if the p-value is high and you cannot reject the null, then that does not mean the two samples were drawn from the same distribution! Merely that PTED could not find any significant discrepancies. The samples could have been drawn from the same distribution, or PTED could be insensitive to the deviation, or maybe the test needs more samples. In some sense PTED (like all null hypothesis tests) is "necessary but not sufficient" in that failing the test is bad news for the null, but passing the test is possibly inconclusive⁴. Use your judgement, and contact me or some smarter stat-oriented person if you are unsure about the results you are getting!

Arguments

Two Sample Test

def pted(
    x: Union[np.ndarray, "Tensor", "jax.Array"],
    y: Union[np.ndarray, "Tensor", "jax.Array"],
    permutations: int = 1000,
    return_all: bool = False,
    n_landmarks: Optional[int] = None,
    two_tailed: bool = True,
    prog_bar: bool = False,
    batch_size: Optional[int] = None,
    rng=None,
) -> Union[float, tuple[float, np.ndarray, float]]:

• x (Union[np.ndarray, Tensor, jax.Array]): first set of samples. Shape (N, *D)
• y (Union[np.ndarray, Tensor, jax.Array]): second set of samples. Shape (M, *D)
• permutations (int): number of permutations to run. This determines how accurately the p-value is computed.
• return_all (bool): if True, return the test statistic and the permuted statistics with the p-value. If False, just return the p-value. bool (default: False)
• n_landmarks (Optional[int]): if not None, estimate the energy distance from a rectangular distance matrix instead of the full pairwise matrix. n_landmarks points are drawn from the pooled sample as "landmarks", and only the distance from every sample to each landmark is computed, so the cost drops from O(n^2 d) to O(n m d) for m = n_landmarks. A value covering the whole pooled sample, or None, runs the exact full-matrix computation.
• two_tailed (bool): if True, compute a two-tailed p-value. This is useful if you want to reject the null hypothesis when x and y are either too similar or too different. If False, only checks for dissimilarity but is more sensitive. Default is True.
• prog_bar (bool): if True, show a progress bar to track the progress of permutation tests. Default is False.
• batch_size (Optional[int]): number of permutations evaluated per matrix product. Larger values are faster (especially on GPU) at O(batch_size * n) extra memory. None picks the largest size that keeps the batch-scaling tensors within 1 GiB and bounds one batch's matrix product, so a run cannot allocate unboundedly whatever the sample shape.
• rng: seed, np.random.Generator, or None to draw from the global numpy state, so np.random.seed still controls reproducibility.

Coverage test

def pted_coverage_test(
    g: Union[np.ndarray, "Tensor", "jax.Array"],
    s: Union[np.ndarray, "Tensor", "jax.Array"],
    permutations: int = 1000,
    warn_confidence: Optional[float] = 1e-3,
    return_all: bool = False,
    n_landmarks: Optional[int] = None,
    sbc_histogram: Optional[str] = None,
    sbc_bins: Optional[int] = None,
    pit_plot: Optional[str] = None,
    pit_confidence: float = 0.95,
    prog_bar: bool = False,
    batch_size: Optional[int] = None,
    rng=None,
) -> Union[float, tuple[np.ndarray, np.ndarray, float]]:

• g (Union[np.ndarray, Tensor, jax.Array]): Ground truth samples. Shape (n_sims, *D)

• s (Union[np.ndarray, Tensor, jax.Array]): Posterior samples. Shape (n_samples, n_sims, *D)

• permutations (int): number of permutations to run. This determines how accurately the p-value is computed.

• return_all (bool): if True, return the test statistic and the permuted statistics with the p-value. If False, just return the p-value. bool (default: False)

• n_landmarks (Optional[int]): if not None, estimate the energy distance from a rectangular distance matrix instead of the full pairwise matrix. n_landmarks points are drawn from the pooled sample as "landmarks", and only the distance from every sample to each landmark is computed, so the cost drops from O(n^2 d) to O(n m d) for m = n_landmarks. A value covering the whole pooled sample, or None, runs the exact full-matrix computation.

  Because the ground truth is a single point, the per-simulation test runs in the singleton regime, where the permutation subgroup reaches only n - m distinct label assignments. Keep n_landmarks well below the number of posterior samples.

• sbc_histogram (Optional[str]): If given, the path/filename to save a Simulation-Based-Calibration histogram.

• sbc_bins (Optional[int]): If given, force the histogram to have the provided number of bins. Otherwise, select an appropriate size: ~sqrt(N).

• pit_plot (Optional[str]): If given, the path/filename to save a Probability Integral Transform (PIT) plot of the per-simulation p-values against the expected uniform distribution, with a shaded KS confidence band.

• pit_confidence (float): Confidence level for the PIT plot's simultaneous band. Default is 0.95 (95%). Only used when pit_plot is not None.

• prog_bar (bool): if True, show a progress bar to track the progress of simulations. Default is False.

• batch_size (Optional[int]): number of permutations evaluated per matrix product. Larger values are faster (especially on GPU) at O(batch_size * n) extra memory. None picks the largest size that keeps the batch-scaling tensors within 1 GiB and bounds one batch's matrix product, so a run cannot allocate unboundedly whatever the sample shape.

• rng: seed, np.random.Generator, or None to draw from the global numpy state, so np.random.seed still controls reproducibility.

GPU Compatibility

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

Example with PyTorch:

from pted import pted
import numpy as np
import torch

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

Example with JAX:

from pted import pted
import numpy as np
import jax.numpy as jnp

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

Memory and Compute limitations

If a GPU isn't enough to get PTED running fast enough for you, or if you are running into memory limitations, there are still options! We can use an approximation of the energy distance; the test stays exact, it just becomes less sensitive than it would otherwise be. Instead of building the full n x n pairwise distance matrix (n = n_samp_x + n_samp_y), PTED builds a smaller rectangular n x m matrix of distances from every sample to m "landmark" points drawn from the pooled sample — the Nyström-style subsampling familiar from kernel methods. Set n_landmarks = m to choose how many.

Building the full matrix costs O(n^2 d) and each permutation O(n^2); the rectangular matrix costs O(n m d) and each permutation O(n m). So for a fixed number of landmarks PTED grows linearly with dataset size, much like other large scale (machine learning oriented) two sample tests. Permutations are evaluated in batches as a single matrix product rather than one at a time, and everything heavy stays on whichever backend your arrays live on.

Example:

from pted import pted
import numpy as np

x = np.random.normal(size = (5000, 10)) # (n_samples_x, n_dimensions)
y = np.random.normal(size = (4000, 10)) # (n_samples_y, n_dimensions)

p_value = pted(x, y, n_landmarks = 200)
print(f"p-value: {p_value:.3f}") # expect uniform random from 0-1

Enumerating small permutation groups

If the permutation subgroup turns out to have no more than permutations members, PTED evaluates all of it rather than sampling. Sampling a small group keeps redrawing the observed labelling, and each of those ties inflates the p-value under the >= convention: with a group of 151 and 199 draws, P(p <= 0.02) came out at 0.0169 instead of 0.0200. Walking the group instead puts it at 0.0195, leaving only the unavoidable granularity of a 151-point lattice — and it costs less compute, since 151 evaluations beat 199. This mostly matters for pted_coverage_test, where the ground truth is a single point and the group is only n - m large.

One consequence: when this kicks in, return_all gives back reference_size - 1 permuted statistics rather than permutations of them. The p-value is then exact rather than sampled.

Why using landmarks is still exact

The subtlety is that the landmarks cannot simply be re-split between the two groups after each shuffle. The landmark set L is chosen using the group labels (so that both groups are represented), which means the labels are no longer uniformly distributed once you condition on L. PTED handles this by confining permutations to the subgroup that fixes L: labels are shuffled within the landmark positions and within their complement, never between. L then stays put under every permutation and the observed labelling is exchangeable with the permuted ones. That is what makes the p-value exact. (L is chosen from sample positions and group labels only, never from the data values — picking landmarks by maximin, k-means or leverage would break the argument. "Landmark" here means only "a point everything is measured against", never "a point chosen for its importance".)

What you give up is sensitivity: the null spread grows like sqrt(n / m), so the smallest detectable energy distance scales as (n m)^-0.5 rather than n^-1. The detection threshold degrades as one over the square root of the compute.

You also give up some p-value resolution, and this is most significant when one group holds a single point — which is exactly the per-simulation test inside pted_coverage_test. The lone point is kept out of L — it has no within-group pairs, so landmarking it would buy nothing — and its label then roams over the n - m positions outside, which is the whole reachable set. The smallest attainable p-value is about 1 / (n - m) however many permutations you draw, so PTED raises a PermutationResolutionWarning when that set is too small to resolve the p-value you asked for.

This is the sharpest case of a general rule: landmarks are for m << n. At m = n / 2 the rectangular matrix saves only a factor of two over the exact test, which does not pay for the sensitivity and resolution it costs. Reach for them when the full matrix will not fit or will not finish, not to shave a constant factor.

Does restricting the permutations still measure the energy distance?

Yes — the restriction is on how the statistic is calibrated, not on what it measures. Every block mean still averages genuine distances between genuine members of the two groups: the cross term pairs all n_x samples against the large-group landmarks and all n_y samples against the small-group landmarks, so no sample is demoted to a mere marker. Each block mean is an unbiased estimate of exactly the population quantity the full test estimates, which makes the whole statistic an unbiased estimator of the population energy distance — an incomplete U-statistic in the sense of Janson (1984), averaging over a subset of the pairs rather than computing a different function of them.

Freezing the per-group landmark counts is a restricted-randomisation device, the same idea as conditioning on the margins in Fisher's exact test: it removes a nuisance source of variability from the null rather than changing the estimand. Empirically it is power-neutral. Compare it against the obvious alternative — draw L uniformly by position, so that L is independent of the labels and the full permutation group is legal — and both reject at the same rate (n = 200, m = 40, 500 trials: 0.16 vs 0.16 at a 0.25σ shift, 0.31 vs 0.35 at 0.40σ). The power lost relative to the full test comes from subsampling, not from the subgroup.

What the subgroup buys is the freedom to choose L by design. Drawing L blind to the labels leaves the smaller group with no landmarks at all — and its within-group term unestimable — distressingly often once the samples are unbalanced: with n_x = 3, n_y = 200 and m = 40, 52% of permutations have no x-landmarks. Choosing L so that it covers the small group fixes that, and the subgroup is what keeps the test exact when you do.

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

As for the posterior coverage testing, this is also an established technique. See the references below for the nitty gritty details and to search further look for "Simulation-Based Calibration".

@article{Cook2006,
     title = {Validation of Software for Bayesian Models Using Posterior Quantiles},
    author = {Samantha R. Cook and Andrew Gelman and Donald B. Rubin},
   journal = {Journal of Computational and Graphical Statistics},
      year = {2006}
 publisher = {[American Statistical Association, Taylor & Francis, Ltd., Institute of Mathematical Statistics, Interface Foundation of America]},
       URL = {http://www.jstor.org/stable/27594203},
   urldate = {2026-01-09},
    number = {3},
    volume = {15},
     pages = {675--692},
      ISSN = {10618600},
}

@ARTICLE{Talts2018,
       author = {{Talts}, Sean and {Betancourt}, Michael and {Simpson}, Daniel and {Vehtari}, Aki and {Gelman}, Andrew},
        title = "{Validating Bayesian Inference Algorithms with Simulation-Based Calibration}",
      journal = {arXiv e-prints},
     keywords = {Statistics - Methodology},
         year = 2018,
        month = apr,
          eid = {arXiv:1804.06788},
        pages = {arXiv:1804.06788},
          doi = {10.48550/arXiv.1804.06788},
archivePrefix = {arXiv},
       eprint = {1804.06788},
 primaryClass = {stat.ME},
}

If you think those are neat, then you'll probably also like this paper, which uses HDP regions and a KS-test. It has the same feel as PTED but works differently, so the two are complimentary.

@article{Harrison2015,
  author = {Harrison, Diana and Sutton, David and Carvalho, Pedro and Hobson, Michael},
   title = {Validation of Bayesian posterior distributions using a multidimensional Kolmogorov–Smirnov test},
 journal = {Monthly Notices of the Royal Astronomical Society},
  volume = {451},
  number = {3},
   pages = {2610-2624},
    year = {2015},
   month = {06},
    issn = {0035-8711},
     doi = {10.1093/mnras/stv1110},
     url = {https://doi.org/10.1093/mnras/stv1110},
  eprint = {https://academic.oup.com/mnras/article-pdf/451/3/2610/4011597/stv1110.pdf},
}

Footnotes

1. See the Simulation-Based Calibration paper by Talts et al. 2018 for what "SBC" is. ↩

2. Yes, I see it now. That formula is the same as for MMD. Just think of it as a special case of MMD that uses the Euclidean distance kernel. In fact this way it is often more sensitive than the RBF kernel typically used for MMD. ↩

3. Since PTED works by a permutation test, we only get the p-value from a discrete uniform distribution. By default we use 1000 permutations, if you are running an especially sensitive test you may need more permutations, but for most purposes this is sufficient. ↩

4. actual "necessary but not sufficient" conditions are a different thing than null hypothesis tests, but they have a similar intuitive meaning. ↩