CoLT: The conditional localization test for assessing the accuracy of neural posterior estimates
Authors: Tianyu Chen
Vansh Bansal
James G. Scott
Venue: NeurIPS 2025 spotlight ()
Keywords: Statistical Hypothesis Testing; Neural Posterior Estimation; Likelihood-Free Inference; Bayesian Methods; Simulation-Based Inference
Primary area: probabilistic_methods
TL;DR: We propose a new method to test equality between the true and estimated posterior distributions, establishing necessary and sufficient conditions for distributional equivalence , with both theoretical guarantees and practical scalability.

Abstract: We consider the problem of validating whether a neural posterior estimate $q(\theta \mid x)$ is an accurate approximation to the true, unknown true posterior $p(\theta \mid x)$. Existing methods for evaluating the quality of an NPE estimate are largely derived from classifier-based tests or divergence measures, but these suffer from several practical drawbacks. As an alternative, we introduce the *Conditional Localization Test* (**CoLT**), a principled method designed to detect discrepancies between $p(\theta \mid x)$ and $q(\theta \mid x)$ across the full range of conditioning inputs. Rather than relying on exhaustive comparisons or density estimation at every $x$, CoLT learns a localization function that adaptively selects points $\theta_l(x)$ where the neural posterior $q$ deviates most strongly from the true posterior $p$ for that $x$. This approach is particularly advantageous in typical simulation-based inference settings, where only a single draw $\theta \sim p(\theta \mid x)$ from the true posterior is observed for each conditioning input, but where the neural posterior $q(\theta \mid x)$ can be sampled an arbitrary number of times. Our theoretical results establish necessary and sufficient conditions for assessing distributional equality across all $x$, offering both rigorous guarantees and practical scalability. Empirically, we demonstrate that CoLT not only performs better than existing methods at comparing $p$ and $q$, but also pinpoints regions of significant divergence, providing actionable insights for model refinement. These properties position CoLT as a state-of-the-art solution for validating neural posterior estimates.

CoLT: The conditional localization test for assessing the accuracy of neural posterior estimates
Tianyu Chen', Vansh Bansal"
James G. Scott
Department of Statstics and Data Sciences
Department of Statistics and Data Sciences
UT Austin
McCombs School of Business
(tianyuchen, vansh @utexas. edu
UT Austin
j james wwwt wwottmmccommms utexas. edu
Abstract
We consider the problem of validating whether a neural posterior estimate q(O x)
is accurate approximation the true, unknown true posterior p(0 x)). Existing methods for evaluating the quality of an NPE estimate are largely derived from bassifferrbbsed tests or divergence measures, but these suffer from several practical drawbacks. As an alternative, we introduce the Conditional Localization Test (CoLT), principled method designed to detect discrepancies between p(0 x) and ( 2)) across the full range of conditioning inputs. Rather than relying on exhaustive comparisons or density estimation at every æ. CoLT learns localization function that adaptively selects points (z)) where the neural posterior deviates most strongly from the true posterior p for that x. This approach is particularly advantageous in typical simulation- based inference settings, where only single draw 8 ~ p(0 x) from the true posterior is observed for each conditioning input, but where the neural posterior g Iz) can be sampled an arbitrary number of times. Our theoretical results establish necessary and sufficient conditions for assessing distributional equality across all æ, offering both rigorous guarantees and practical scalability. Empirically, we demonstrate that CoL. not only performs better than existing methods at comparing p and q, but also pinpoints regions of significant divergence, providing actionable insights for model refinement. These properties position CoLT as state-of-the-arn solution for validating neural posterior estimates.
1 Introduction
This paper proposes new method for determining whether two conditional distributions p(0 x)) and q(O x) are equal, at least close, across all conditioning inputs. One of the most important applications of this idea arises in validating conditional generative models for neural posterior estimation, NPE, which is rapidly growing
of simulation-basee inference. Here 0 represents the parameter of scientific model with prior p(0), while x ~ p(x represents data assumed to have arisen from that model. In NPE, we simulate data pairs (8,z) drawn from the joint distribution (x, 8) ~ p(z, 0) = p( p(0)p p(( 8). A conditional generative model such as a variational autoencoder [1], normalizing flow [2], diffusion model [3, 4, 6]. or flow-matching estimator [7]-is then trained on these (a., 0) pairs to approximate p(0 x)) with a learned distribution g(O x)) The problem of neural posterior validation is assess whether the learned g is good approximation to the true p.
This setting poses challenges not present in simpler problem of testing for equality of unconditional distributions, with For one thing, we must verify that q() z) approximates p(0 2) not merely for a single given æ, but consistently for all x, without having to explicitly consider all possible
equal contribution
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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points. Moreover, most practical problems present a severe asymmetry in the available number of samples from p and q. In NPE, for example, we observe just single "real" sample p(0 x) for each x, yet we can generate an arbitrary number of "synthetic" samples 0 ~ q( x) by repeatedly querying our NPE model. Any successful method for assessing distributional equivalence of p and must account for this imbalance.
Existing methods. Several methods have been proposed to assess the accuracy of a neural posterior estimate. But each has shortcomings. One popular method called Simulation- Based Calibration (SBC) [8] uses simple rank-based statistic for each margin of q(O z). but this provides only necessary (not sufficient) condition for posterior validity. Moreover, since rank statistics are computed separately for each margin, the statistical power of SBC suffers badly from multiple-testing issues in high-dimensional settings. TARP [9] provides a condition that is both necessary and sufficient for the neural posterior estimate to be valid. However, TARP's practical effectiveness depends heavily the choice of inon-trainabl probability distribution to generate "reference" points that are needed to perform the diagnostic, and the method can perform poorly under suboptimal choice of this distribution. Finally, the classifer two-sample test, or C2ST [10], involves training a classifier to distinguish whether given 0 sample originates from the true posterior or the estimated one. It then uses the classifier output to construct an asymptotically normal test statistic under the null hypothesis that p = q. But as many others have observed, the C2ST hinges on the classifier's is ability to effectively learn a global decision boundary over X' and θ simultancously. In practice, the classifier may struggle to do so, due to insufficient training data, limited model capacity, or the inherent complexity of the task. Moreover, to perform well, the C2ST usually needs a class-balanced sample, which entails multiple draws of f from the true posterior at a given x. This is often impractical, as in many settings we only have access to single @ ~ p(0) x)) at a given Z.
Our contributions. Our paper addresses these shortcomings with a principled and efficient ap- proach, called the Conditional Localization Test (CoLT), for detecting discrepancies between p and g. CoLT is based on the principle of measure theoretic distinguishability: intuitively, if two conditional densities p(0 I) and q(0 æ)) are unequal, they must exhibit nonzero difference in mass over some specific ball of positive radius. The basic idea of CoLT is to find that ball-that is, to train a localization function 0₁ X + θ that adaptively selects the point (z) where, for given æ, p and q are maximally different in the mass they assign to a neighborhood of (z). Intuitively, a neural network that learns smooth mapping (z: should be well suited for this task: if two conditioning inputs x and 0 are close, we might reasonably expect that any differences between and would manifest similarly (i.e. in nearby regions of 0 space) for both and ã. This smoothness allows the network to generalize local differences across nearby regions in x space, making the search for discrepancies both efficient and robust.
Of course, the principle of measure- measure-theoretic distinguishability is well established, and so one might fairly ask: why has it not been widely exploited in machine learning tool for comparing conditional distributions? This is likely for two reasons, one geometric and one computational, both of which CoLT successfully addresses.
First, directly comparing mass over high-dimensional Euclidean balls can be ineffective for testing, as the Euclidean metric may not align with the geometry of how p and are most readily distinguishable. To address this, we use trainable embedding function é that maps points from the parameter space θ into latent Euclidean space, where distances can better reflect the concentration of probability mass. We then assess mass equivalence over Euclidean balls this latent space, i.e. over balls B& (0, R)- = O0 E O (0') - x(0)]12 < R]. We show how the necessary machinery from real analysis can be rigorously adapted to this setting, with modest requirements on Φ.
Second, even when assessing equivalence over non Eucl-eddd metric balls, naively training a local- ization function (z) would seem to require repeatedly sampling from both p and q at some x, comparing their local (Monte Carlo) integrals over all possible balls. This is intractable for all but the smallest problems. Luckily, we show that training 0 can be done far more efficiently. The essential idea involves using a single observed draw from p(0) | æ) to anchor our comparison of whether the conditional mass of y .r) aligns with p(0 xr), in expectation over x. This single draw, combined with the localization function 0,, can be used to carefully construct one-dimensional ball probubility rank statistic that is uniformly distributed if and only if p and agree on all local neighborhoods around (z)). We rigorously construct this rank statistic, and we show how it leads to practical
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14a-
(a) True posterior.
(b) a = 1.5
(c) a 0004
(d) Power curve across
Figure Results on the toy tree-shaped example. a increases (larger perturbation), the distribution becomes blurrier and deviates from the true manifold shown in Panel A. CoLT with learned metric embedding maintains strong statistical power even modest perturbations (Panel B), whereas the (Panels C/D).
C2ST, SBC, and TARP all perform poorly even for much larger ones like
optimization algorithm for (z). Moreover, the rank statistic naturally induces a valid integral probability metric (IPM), offering continuous measure of the distance between the two distributions. This is especially valuable in NPE settings: by moving beyond binary assessment, CoLT allows user of NPE methods to quantify improvements across training runs, benchmark multiple posterior approximators, or make targeted improvements to model architecture based on where specifically the neural posterior q is performing poorly.
Finally, our empirical results demonstrate that CoLT consistently outperforms current state-of-the -oe------ttt-tttt methods across wide range of benchmark problems. The evidence shows that CoLT is able to consistently identify subtle discrepancies that classifier-based approaches routinely miss, providing strong empirical support for our theoretical analysis.
toy example. To provide an initial demonstration of CoLT's effectiveness, we begin with a toy example. Panel A of Figure shows p(0 2)) as living structured manifold, with branches A (bottom left) and (top right) representing distinct regions of probability mass, as introduced in [11]. We sample a conditioning input as ~ (0, 1), with the true conditional distribution defined as:
Branch if 2 0,
p(0 x) ~
Branch if or < 0.
Our goal here is to assess whether method can reliably detect even small perturbations of This example, while simple, effectively targets a common fallure mode of generative models: producing samples that lie near, but not exactly on, the true manifold of the posterior.
To benchmark CoLT's performance, we constructed "perturbed" posterior samples 0 ~ q(0) x) by adding a small amount of isotropic Gaussian noise to "correct" samples: that is, 0 + e, where draw from p(0 æ) and each component of has standard deviation 0.01 We then varied a, which controls the degree mismatch between p and g. and we tested the power of CoLT versus TARP, SBC, and the C2ST for each a. The nominal Type-I error rate was set 0.05 for all methods. To ensure a fair comparison, we trained the C2ST classifier and CoLT localization function with similar model capacities (number of layers and size of each layer); see Appendix C for details.
When a = 1.5 (Panel B), the samples from fall very slightly off the correct manifold. CoLT can reliably detect this difference (power 0.877), while C2ST failed entirely (power 0.000). At larger value power of 0.065. Panel D shows that, while performing bit better than C2ST, neither TARP nor SBC are competitive with CoLT at any a. These results highlight our method's performance advantage in scenarios where the posterior lives structured manifold, and the discrepancy between p and is reasonably small. We also emphasize that CoLT doesn't 't merely detect the difference; as our theory shows, it can also quantify the difference via integral probability metric.
= 4.0 (Panel C), CoLT achieves perfect power (1.000), whereas C2ST only reaches
2 Theoretical Results
In this section, we present our main theoretical results; all proofs are given in the Appendix. Through- out, we denote the Lebesgue measure by ml-) and use d0 represent Lebesgue integration. also use the shorthand notation q(O x)) p(0 x)), or simply y. to denote that q(0 x) p(0 x))
for almost every (0,x) θ x X. Throughout, we assume that p and q are absolutely continuous with respect to Lebesgue measure for all z.
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2.1 The conditional localization principle
CoLT relies on what we might call the localization principle: to check whether p and q are different, search for the point f1 and the local neighborhood around 8, where the mass discrepancy between p and q is as large as possible. If the largest such discrepancy O, the two distributions are equal.
Taken at face value, however, the localization principle seems deeply impractical. First, if we wish to conclude that p(0) x) q(0) x)) for all 2;, it seems that we would need to apply the principle pointwise over a grid of z-- v-values. Second, for each x; we would need to search for the point 0 that maximizes the discrepancy in local mass between p and q. if one exists for that z. Finally, we would need to draw many samples from both p(0 x) and q(0 x) to obtain reliable Monte Carlo integrals. The sheer number of evaluations needed many z-values, many candidate O1-locations per æ, and many Monte Carlo samples per (0), 2) feir--eenders this naïve approach not just intractable, but nestedly intractable.
Luckily, we can do much better than the naive approach. In fact, our subsequent results can be thought of as peeling back these layers of intractability one time.
We begin with a key definition. Specifically, we consider balls of the form B3 R) (l)(0") - (0)]l I R}. where Φ is an embedding function. By defining neighborhoods through dæ. we can shape our regions of comparison to better reflect meaningful differences in probability mass. The following imposes a mild, but useful, geometrie regularity condition on the metric ø
{ E θ
Definition 1 (Doubling Condition). Let O be a set equipped with a map é and let m be a measure on O. For each 0 E θ C CRP and R > 0, define the ø-ball
B2 R) = {0' E e 1|x(0)) 6(0)]l2 S R).
We say that o satisfies the doubling condition with respect to if there exists constant C >> 0 such that for all 0 E O and all R > 0,
m (0, 2R) S C Cm
(I)
Intuitively, this condition ensures that the metric balls B& do not distort the underlying geometry of RDD too severely, for instance by creating regions of infinite density or measure.
A straightforward sufficient condition for this is that the embedding & be bi-Lipschitz of any order, which guarantees the above global doubling condition. However, strict bi-Lipschitz map is not necessary. As more flexible and practical alternative, can define (()) kg(O,-) -) as a deep- kernel embedding [12], which uses any Lipschitz encoder & DRD RT to extract features. The corresponding kernel-based distance, given by
li 0", yeilds a local doubling condition, defined in Appendix A. This weaker, local condition sufficient for our following localization result to hold. We provide a detailed proof and discussion for both the bi- bi-Lipschitz and deep-kernel cases in Appendix A.
kg (0, 0) + (0'
With this definition in place, we can state our first result about the equality of conditional distributions. This result replaces the stringent requirement of verifying an equality-of-mant condition for each with a weaker condition that involves averaging over x. We formalize this idea in terms of a localization function (x)): : X θ, which identifies the most informative localization point based on æ. Intuitively, 0z(zz) serves as witness to any potential discrepancy between p(0 x)) and q(0 zz).
Theorem (Conditional localization). Let p(0 x) and q(0 x)) be defined as before, and define the difference function (0) = p(0 x) q(0 x). Let dig + + -R+ be the distance function, in induced by the embedding map 6, satisfying the doubling condition with respect to Lebesgue measure. further Let B& (Bi(z)) R) denote the é-ball of radius R centered at (z)). Assume further that p(z) >> i density on X which is strictly positive almost everywhere.
If. for every measurable function O2 X + θ and every R > O, we have
1x p(x)) V B. d_ do dr 0,
then (0) for almost every (x, 0) in X O
full proof is provided in Appendix B. The sketch is as follows: The theorem's 's assumption the
average discrepancy over all z is zero-is challenging because positive and negative discrepancies

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could cancel. However, a crucial feature of the theorem is that the center of the metric ball, fr(x;), is allowed to depend on z via localization map. The proof uses a measurable selection argument to construct this adversarial localization function (3)) that, for each æ; intentionally centers the ball Bg in the region of maximum discrepancy. Applying the theorem's hypothesis to this "worst-case" selector forces this maximum discrepancy to be zero for almost every which implies the discrepancy is zero for all balls. From this, the Lebesgue Differentiation Theorem the doubling condition-alle allows to conclude that if the average difference over all shrinking balls zero, the pointwise difference p(0|z) ((0)a) must itself be zero almost everywhere. Moreover, p 9 implies that the supremum of
applies due to
J p(xx) VESTeNON (p(0 (x) - q(0) x)) d0 dr
over all measurable choices of (.)+) and all R must be O. This gives us natural target for optimization over the choice of the localization function (z)).
2.2 The ball probability rank statistic: a practical condition for mass equivalence
Theorem eliminates the need for an exhaustive search over æ. But its direct application still appears to require many draws from both p(0 r) and q(0 2)) to verify the equality of mass over metric balls. Testing this condition via Monte Carlo would typically involve repeatedly sampling from both distributions at the same and comparing their local integrals. This remains computationally demanding even in principle. Moreover, the typical setup where this methodology might be applied, the situation is asymmetric: p(x) and p(0 x)) correspond to a real unknown distribution that generated the training data, meaning that for any observed x, we often have access to only a single corresponding draw from p(0 z²) By contrast, q(0 zz) represents (conditional) generative model that we can query arbitrarily many times for a given æ. A practical formulation must leverage this structure by treating the single "true" 0 draw an anchor and evaluating whether the conditional mass of q aligns with in expectation over x.
Our next result establishes precisely this adaptation, ensuring that the comparison suggested by Theorem can done feasibly. The basic idea as follows: we can draw a random sample (0" 2) x),, compute the localization point (z(),, and let the radius be implicitly determined as R(00)= d& (0r(z). 0*). As the number of samples gets large, this turns out to be equivalent to checking all radii in Theorem 1. We now formalize this equivalence below, temporarily dropping the dependence on the conditioning input r to lighten the notation.
Theorem 2. Let p and q be defined as above. Fix reference point 0: E O, and define the metric ball
B, (0 E e (B 0) S r).
For any 0" e define the ball probability rank under
Ug(6*) = Powa do (f,, 8) S
Then, the condition that p and q assign the same probability all balls centered at Or, i.e.
v(B,) = (( for all radii 0 < sup d(O',0)) e'ce
is equivalent to the statement that, when 6* ~ the random variable (,,(0) ) is uniformly distributed on []. That is, checking whether, for all choices of 0t, U,( Unif(0, 1) under 0" ~ p is both necessary and sufficient for
Intuitively, if and differ, then there must exist some point 6 and some radius R for which the two distributions assign different mass to the ball ((,, R). This mismatch causes the distribution of (f") ( to deviate from uniformity when 0" ~ Conversely, if U2 (0*) ~ Unif(0, 1) under 6* for every choice 0,, then p and g must agree on the mass of such balls, and hence be identical. Thus taken together, Theorems 1 and collapse a daunting, high-dimensional equality -f- requirement into a one-dimensional uniformity condition that can serve as the basis for tractable optimization problem.
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2.3 From local-mass uniformity to an IPM
Theorem shows comparing the ball call-probability rank statistic Ug.r a uniform distribution gives us a test for whether = The next result shows that, once we optimize over every allowable localization map 8, every embedding Φ, and every ball radius. the same uniformity test yields an integral probability metric (IPM) that we call the averaged conditionally localized distance (ACID). Concretely, let
B = (B, ()) 8: X +0, 0: R> 0]
the class of indicator functions of metric balls whose centers depend on æ. The corresponding IPM is
ACLD(p. 4) = sup
Ex Bz=p(z)
[B-p(o)=) Eo-
JEB
Our next theorem connects this distance to the ball probability rank statistic from Theorem 2.
Theorem (Ball-probability IPM). Let p(O x) and q(O x)) be absolutely continuous conditional densities on common parameter space θ for E X. and suppose p(x) >> 0 a.e. X. For x-dependent) center (z(z) E and for the metric (o(0,)) 0*) = lox(0) - 6(0")) )l2 induced by an embedding @ θ + 1 satisfying the doubling condition, define
U (0)) = {Po-q(o-=ddd(11))) dd(((z))))) S ((0)(()),0)}
Now let d(p, q) be the worst-case Kolmogorov distance of U (g*) averaged over a, from the Uniform distribution:
d(p.9) = sup (0.
Em-p(a) ( (0') sa -a
Then
d(p,g) 9) = ACLD(P,
The theorem establishes that the largest possible deviation from uniformity that one can provoke in Uq.e, by freely choosing the localization function, embedding, and ball radius, is numerically identical an IPM built from indicator balls. Hence training the localization network (x) maximise the distance between Uz.z and U(0, 1) is equivalent to computing ACLD(p, q). If the optimizer fails to increase this distance beyond sampling noise, we have empirical evidence that q(O 2) has passed the full mass- eusivaleuvee test implied by Theorem 1. Conversely, if Uq.e is not uniformly distributed,
then its empirical KS distance to (0, i) gives both a p-value based on the classical KS test, and estimates a distance between p and
3 The CoLT Algorithm
The key insight from Theorem 2 is that searching for embedding & and localization function (x)) that maximally distort the ball probability rank statistic Ug away from uniformity is equivalent to detecting regions where q fails to match p. This forms the basis of our optimization procedure. We represent both the metric embedding ó and the localization function as neural networks, (((2;; 1)) with learnable parameters vb. Our strategy is roughly follows:
Generate rank statistic: Draw a minibatch of "anchor" points (0) x;) E from p00, æ). the true joint distribution. By construction, each (0; xx)) has conditional distribution p(0 | (x)). For each anchor point i, sample M synthetic draws { from q(0 xz), and compute the empirical ball probability rank statistic:
1 Σ (d((()()))))) < do (d((()))))))))) M
Ú.(b,,)
Measure non-uniformity: As a loss, we use negative divergence from uniform distribu- tion, L(b, 6) -D(Ū, Φ), Uniform). We discuss the choice of divergence below.

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Optimize: Gradient descent is applied to the loss function. If p q. optimization will stall, as no choice of (o, 8, (z; v)) will yield substantial deviation from uniformity. Otherwise, the optimizer finds localization map that exposes the failure of
This approach is detailed in Algorithm (training phase) and Algorithm 2 (testing phase). We first apply Algorithm to train the embedding network ø and localization network 0,, aiming to maximize the discrepancy between the empirical U, values and the uniform distribution. Then with the trained networks and a test of {(0,, 3.)) we compute
statistic and corresponding p-value using the one-sample Kolmogorov-Smine -iiiroo (KS) test in Algorithm 2.
Algorithm 1 Conditional Localization Test (CoLT): Training Phase
1:
procedure COLT( x.)] p(0 x)p(x),, sampling distribution q(0 x))
2:
Generate K samples {0., ~ 90 x.) for E [N], [K]
3: Define d0 0²) [6(()) ((0')l2
4: Initialize o, (z, (6) as neural networks
5: while not converged do
6: for i I...... do
7:
8 - (2;;
8:
U, = ΣØ= ((dp(0,,, 01) (0,
9:
end for
L( Φ) (U,, Uniform)
IMaximize divergence
11:
Perform gradient update on v4 end while
13: Return 0z. Φ
14: end procedure
Algorithm 2 Conditional Localization Test (CoLT): Testing Phase
1: procedure (o):
p(0 r)p(x), sampling distribution q(0 =), 0g Φ)
2: Generate K samples (0:,] } ~ ( x;) for E [M], E [K]
3: Define (0, 0') (0) (00'
4: for = 1,..., N do
5: O: (z22))
6: U: = R Eラ 1(ds (0:000000 0t) ds.0..)))
7: end for
8: 1.P KS test( ... UN} Uniform)
l/test statistic & p-value
9: Return t,p
10: end procedure
We make three remarks about this algorithm. First, because U3 involves an indicator function. gradients cannot propagate directly; we therefore use the Straigh-Thhrouge Estimator (STE) trick [14] to enable gradient-based optimization. Second, we represent the distance embedding network Φ as a neural network due to its flexibility and capacity to approximate wide range of transformations. Moreover, neural networks are typically Hipschtz-continnon ensures that the doubling condition (Definition 1) is satisfied; see Appendix A. Alternatively, a fixed, non-trainable form & can be specified, and our theoretical guarantees will still hold, but power may be reduced. For example, setting & the identity reduces d(.,-) to the & distance.
under mild conditions [15], which
Third, in Algorithm 1, various divergence measures can be used quantify the discrepancy be- tween the empirical distribution of rank statistics U; and the uniform distribution. While the Kol- mogorov- Smirnov (KS) distance is natural choice motivated directly by our theory, is not ideal for gradient-based optimization, which would need to propagate gradients through sorting and max operations. To address this, we instead use Sinkhorn divergence [16], an entropy- reguuaaieee version of Wasserstein distance that retains geometric sensitivity while offering a smooth objective. Em- pirically, we find that Sinkhorn divergence leads to stable optimization and good performance. We emphasize that Sinkhorn divergence is used only during the training phase to learn the localization and embedding maps. test time, we use the KS statistic, as suggested by our theory, to compute p-values based on the empirical rank distribution.
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(a) Bivariate normal
(b) Perturbed normal
(c) Latent normal
(d) Perturbed latent normal
Figure 2: (A) Bivariate Gaussian with correlation 0.9. (B) Example perturbation of (A) yield heavier tails. (C) Latent Gaussian with transformation. (D) Example perturbation of (C) with heavier tails in the latent space, In our benchmarks, CoLT and similar methods are tasked with distinguishing the ground-truth distributions (A and C) from perturbed variations (B and D, respectively). These are both large perturbations (large a) and should be easy detect; smaller a yields subtler perturbations. Details of the perturbation schemes are provided in Appendix C.2.
4 Experiments
Benchmark tasks. To evaluate CoLT against established NPT methods, we use suite of benchmark tasks introduced by [17]² (details in Appendix C.2). Each benchmark defines reference posterior p(0) iz), then introduces a family of perturbed alternatives q(0 x; a),, where the scalar parameter a 2 O controls the severity of deviation. As increases, so does the discrepancy between and allowing us to generate smooth performance curves that quantify the sensitivity of each NPT method.
We evaluate CoLT on two such benchmark familics. The first is based on multivariate Gaussian posteriors with data-dependent mean and covariance. Specifically, we sample N(1 Im) and define
p(0 x) = (N Ex), Px = Wiæ, Σg - IW x -
where W1 E Rsxm and W2 E Rx fixed matrices constructed from i..d. Gaussians, and E a Toeplitz matrix with entries Σ, = - using = 0.9. The alternative q(O x; is then constructed by applying structured perturbations either to μ4 or Σ,,, detailed in Appendix C.2. This setup allows us to simulate NPE errors such as mean shifts, covariance inflation, or distortions of multimodal structure. See Figure Panels A-B.
The second family of benchmarks introduces geometric complexity by drawing latent Gaussian samples according to the same recipe as above, and then applying a nonlinear transformation, 0 = f(0) = A h( (O), where / ~ N(u,, Σz) h is coordinate-wise sine nonlinearity, and A E Rdxd B E Rdx* are fixed matrices. This creates a posterior distribution p(0 | I) concentrated on a smooth, curved manifold of intrinsic dimension s in Rd. To generate q, perturbations are applied in the latent Gaussian space (i.e. before transformation). See Figure 2, Panels C-D.
Baselines and settings. For our method, evaluate two variants: CoLT Full, where both the embedding network o and the localization network 02 are jointly optimized; and CoLT ID, where ø is the identity and only the localization network 9 is trained. We assess both Type I error (at 0) and statistical power (for a > 0) across all methods. Both versions of CoLT are compared against three established approaches: C2ST [10], SBC [8], and TARP [9]. To enable fair and meaningful comparisons, we adapt each baseline to produce a p-value, as follows. For C2ST, we sample one 0 from q(0 .r) for each to create balanced training and test datasets, using the asymptotically normal test statistic described in [10]. For SBC, we conduct the KS test between the rank statistics and the uniform distribution for each dimension independently, followed by Bonferroni correction to control for multiple testing. For TARP, we select random reference points and the TARP test statistic to perform KS test against the uniform distribution.
In both benchmark families, we vary the input, parameter, and latent dimensions (m, s, d) and report power a function of a. We sample 100 pairs x(),, from the true joint distribution p(0 x x))(()), along with 500 samples from q( z) for each corresponding x during training. After training, we evaluate method's 's power by sampling 200 additional batches with the same sampling budget. For
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(b) Selected power curves on the manifold benchmark family.
Figure 3: Statistical power curves (high is better) for four perturbation types under both benchmark
families: (a) Gaussian posterior with data-dependent mean and covariance, and (b) its nonlinear transformation onto curved manifold. Each panel refers to a specific perturbation type, with the horizontal axis (a+) describing the severity of the perturbation. Selected settings for (s, d, m) are shown here, with results on wider variety of settings shown in the Appendix C.2.
all methods, we set nominal Type-l error rate of 5%. We repeat experiments with three random seeds and report averages. Further implementation details and design choices are in the Appendix C.
Results. Figure 3 summarizes the performance of the various testing methods across both bench- mark families and four specific perturbation types: covariance scaling, anisotropic covariance distortion, heavy-tailed deviations via i-distributions, and the introduction of additional modes. In the simpler Gaussian benchmark (top row), both variants of CoLT (Full and ID) match or exceed the performance of C2ST while consistently outperforming SBC and TARP. CoLT ID-which measures mass over fixed Euclidean malls-performm well in cases like covariance scaling and additional modes, where the geometry of the discrepancy aligns well with the ambient space. C2ST also performs reasonably in these non-curved settings, particularly for tail adjustment and additional modes.
In contrast, the manifold benchmark (bottom row) reveals a clear advantage for CoLT Full, which learns a flexible embedding function localize discrepancies. As with the toy example in Figure 1. this learned geometry appears essential in detecting errors, especially under tail adjustments and anisotropic distortions. CoLT ID, which lacks this geometric adaptability, performs notably worse than CoLT Full in these settings, although it still generally meets exceeds the performance of other methods. These results highlight an important inductive bias: while fixed Euclidean balls suffice for flat posteriors, Iearned embeddings are crucial for detecting structured mismatch curved or low-dimensional manifolds. Taken together, the results confirm that CoLT is competitive across a range of settings and is especially effective when for posteriors with complex geometry.
Additional experiments appear in Appendix C.3, which demonstrates our method's applica- tion to diffusion based generative posteriors, and in Appendix D, which includes expanded re- sults across more perturbation types and dimensional configurations. We provide the code at https: /ttp:://tttttt com woomm//ianyyoooigg//ooaa
Discussion and limitations. Our theoretical and empirical results establish CoLT as a principled and practical approach for detecting local discrepancies between conditional distributions, with state- of-the-art performance compared to existing methods. But CoLT does have limitations. The method relies on learning both localization function 0 (z) and an embedding ((0), introducing inductive

---

bias through architectural and optimization choices. If either component is underparametterized or poorly trained, CoLT may fail to detect real discrepancies. Its sensitivity also depends on the quality of the rank statistic, which can degrade with limited samples. And while CoL.T yields continuous IPy-style metric, interpreting this scalar, especially in high dimensions, can be challenging, as the underlying IPM function class is non-standard and implicitly defined by the learned components.
The benchmarking framework also has its limitations. Although designed to reflect realistic failure modes in NPE, the benchmarks are inherently synthetic and simplified. Perturbations are applied in controlled, parametric ways that may not capture the full complexity of real- woolld approximation errors. Moreover, the true posterior is always known, enabling rigorous evaluation but diverging from practical settings where ground truth is inaccessible. Despite these caveats, the suite provides clear, extensible testbed, probing number of common failure modes of NPE methods.
References
[1] Diederik P Kingma, Max Welling, al. Auto-encoding variational bayes, 2013.
[2] George Papamakarios, Eric Nalisnick, Danilo Jimenez. Rezende, Shakir Mohamed, and Balaji Lakshminarayanan. Normalizing flows for probabilistic modeling and inference. Journal of Machine Learning Research, 22(57):1-64, 2021.
[3] Jonathan Ho, Ajay Jain, and Pieter Abbeel. Denoising diffusion probabilistic models. Advances in neural information processing systems, 33:6840-6851, 2020.
[4] Tomas Geffner, George Papamakarios, and Andriy Mnih. Compositional score modeling for simulation-based inference. In International Conference on Machine Learning, pages 11098-11116. PMLR, 2023.
[5] Tianyu Chen, Vansh Bansal, and James G Scott. Conditional diffusions for amortized neural posterior estimation. In International Conference on Artificial Intelligence and Statistics, pages 2377-2385. PMLR, 2025.
[6] Manuel Gloeckler, Michael Deistler, Christian Weilbach, Frank Wood, and Jakob H Macke. All-in-one simulation-based inference. arXiv preprint urXiv:2404.06336 2024.
[7] Jonas Wildberger, Maximilian Dax, Simon Buchholz, Stephen Green, Jakob H Macke, and Bernhard Schölkopf. Flow matching for scalable simulation-based inference. Advances in Neural Information Processing Systems, 36:16837-166666 2023.
[8] Sean Talts, Michael Betancourt, Daniel Simpson, Aki Vehtari, and Andrew Gelman. Val- idating bayesian inference algorithms with simulation- based calibration. arXiv preprint arXiv: 2018.
[9] Pablo Lemos, Adam Coogan, Yashar Hezaveh, and Laurence Perreault-Levasseus Sampling- based accuracy testing of posterior estimators for general inference. In International Conference on Machine Learning, pages 19256-19229 PMLR, 2023.
[10] David Lopez- Paz and Maxime Oquab. Revisiting classifier two-sample tests. arXiv preprint 161.005610.06555, 2016.
[11] Tero Karras, Miika Aittala, Tuomas Kynkäänniemii, Jaakko Lehtinen, Timo Aila, and Samuli Laine. Guiding a diffusion model with a bad version of itself. Advances Neural Information Processing Systems, 37:52996-53021 2024.
[12] Feng Liu, Wenkai Xu, Jie Lu, Guangquan Zhang, Arthur Gretton, and Danica J. Sutherland. Learning deep kernels for non-parametric two-sample tests, 2021. URL https: /ttpss/////xxii /arxiv. org/abs/2002 09116.
[13] Juha Heinonen, Pekka Koskela, Nageswari Shanmugalingam, and Jeremy T. Tyson. Sobolev Spaces on Metric Measure Spaces: An Approach Based on Upper Gradients. New Mathematical Monographs. Cambridge University Press, 2015.
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[14] Yoshua Bengio, Nicholas Léonard. and Aaron Courville. Estimating or propagating gradients through stochastic neurons for conditional computation. arXiv preprint arXiv: 2013.
[15] Henry Gouk, Eibe Frank, Bernhard Pfahringer, and Michael J Cree. Regularisation of neural networks by enforcing lipschitz continuity. Machine Learning, 110:393-416, 2021.
[16] Marco Cuturi. Sinkhom distances: Lightspeed computation of optimal transport. Advances in neural information processing systems, 26, 2013.
[17] Tianyu Chen, Vansh Bansal, and James Scott. NPTBench: A benchmark suite for neural posterior testing. Technical report, UT Austin, 2024. URL https://githul: com/ TianyuCodings/ Naainngg/NPTI TBench.
[18] Elias M. Stein and Guido Weiss. Introduction to Fourier Analysis on Euclidean Spaces, volume 32 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 1971. ISBN 978-0-691-08077
[19] Tero Karras, Miika Aittala, Timo Aila, and Samuli Laine. Elucidating the design space of diffusion- generative models. Advances in neural information processing systems, 35: 26565-26577, 2022.
[20] Trevor Hastie and Werner Stuetzle. Principal curves. Journal of the American statistical association, 540406):502-16,, 1989.

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A Notes on Ó
Bi-Lipschitz condition The key requirement on é is that it satisfies the doubling condition with respect to Lebesgue measure. One sufficient condition for this to hold that e be a bi-Lipschitz function, where there exist constants Ci, C2 such that, for all 0,, O2.
ci 02l12 z (lo(01) (122) s C2l101 02l12
C2 D
If this condition holds, then the doubling condition holds with doubling constant C To see this, observe that for such a o, the metric balls satisfy
) 0.
B O2 RR C B R) C BEuc C2R)..
Then we have
m (Buu (0,, CiR)) CP ml (Or R)) and m m mBBaci (O,, C2 m (Brac R)).
Hence, from it follows that
Braci C) S Bb(0, S (0 C2R),
CP m (BEudu (0, R)) I m (B,(0, R)) I C2 (BBuuc R)).
which shows that m (0, R)) scales like R to a constant factor.
Deep kernel-based distances [12]:
We also propose using the following the deep kernel-based distance
do(0,0') = |lx00) - 20) = v (0, (0,
(0 0') 2kg (0,
(2)
where the infinite dimensional embedding @(0) = (0,0,-) E H4 is defined through the following deep kernel:
[ = - k (E(()), k2,,
for E 10, RBF Gaussian kernels k (: and (2((,-) given by ki (x. = exp
1 2
and any Lipschitz embedding function & RD + R". The kernel kg(..)) is known to be characteristic kernel [12], and hence defines valid distance metric. The following lemma shows that the kernel-based distance defined in (2) satisfies local-doublig condition, i.c., for any fixed radius M > O, there exists a constant C'ss > such that for all E θ and all radii r E (0, M). we have ml (0.2r)) < C.M.
m(Bu(8rr)
Lemma Let E IR." Pg-- R. be a Lipschitz continuous function with constant L2. The distance metric de defined in (2) satisfies the following:
1. (Global Upper Bound) There exists a constant C1 >> 0 such that for all 0 0' E ERD.
d')(0,0) < C2 ||0 0'|].
2. (Local Lower Bound) For any M > 0, there exists a constant Cs > 0 such that for all 8, 8' ER. with ||e 8'll S M,
ds(0,0)) 0') 2 CI.M 110 8'11
Hence, dg locally doubling defined above.
We defer the proof to Appendix B.5. The result of above lemma is critical as satisfying the local- doubling condition makes (O, dg. m) a Vitali measure space [13, Theorem 3.4.3), which suffices for the Lebesgue Differentiation Theorem to hold and makes our Theorem applicable.
19

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B Proofs
B.1 A preliminary lemma
To prove Theorem we first need the following lemma, which adapts standard measure- - results to the case of non-Euclidean metric based on an embedding function that satisfies the doubling condition in Definition 1.
Lemma 2. Let p(0 x) and q(O x)) be defined as above, and let & be an embedding function that induces a metric ds on θ defined as
ds(01,03) =
Further, assume that satisfies the doubling condition (1) with respect Lebesgue measure.
Suppose that for almost every 0₁ E G we have
J ()
p(0
2)) d0
q(0 x)) do
((
for all metric balls B(8, R), defined as
B(O,, Rt) 10 do (t) R}.
Then p(0 (x) q(0) x)) almost everywhere (0).
Proof. Define the difference function
(0) = p(0 x)) g(0 x).
The goal is show that (0) almost everywhere in θ using the given integral condition.
Because e is assumed to satisfy the doubling condition with respect to Lebesgue measure, we have for some C >> 0 that
m( (0, 2R)) < Cm(B,(O, R))
Now since (0) is locally integrable (as a difference probability densities), we apply the Lebesgue Differentiation Theorem for doubling measures [18], which implies:
lim R m m(Bo(O. R)) S (0') de' = ( for a.e. 8.
However, by assumption, we know that for all and all sufficiently small R.
f B, d0') d8' 0.
Since m(B(0, R)) > 0. dividing by the Lebesgue measure of the ball and taking the limit yields:
(0) = for almost every
Since (0) 0 a.e., follows that p(0 x)) <(0 .x) almost everywhere in O.
B.2 Proof of Theorem 1
Let B(0, R) be ball with radius R. Assume the doubling property: 3C 1 such that
m(B(O, 2R)) < C' Cm(B(O, R)) AOE R > 0,
and that ball boundaries have m-measure zero.
Let X be open subset of _ with Borel measure μ. Suppose
p: X + (0, oo) is a continuous probability density;
(x. + p(0 x) and (x, 0) + q(0 xr) are continuous in x and belong to ())(() for
every
20

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for every measurable function 8, : X + O and every R > 0,
Jxx p(x)) pl Unoncon (p(0 x) - 10 x)) do do 0.
((O())))))
Then show that p(0 xe) q(0) x)) for m-a.e (x, 0).
Proof. The proof proceeds in two main steps. First, we fix R >> 0 and use measurable selection argument to show that the inner integral is zero for almost every æ. Second, we apply the Lehesgue Differentiatinn Theorem to show that the integrand itself must be zero almost everywhere.
Step 1: Show ball integrals are zero for Z. Let (0) p(0 x)) q(0 2). For fixed
R > 0, define the ball integral
FR(X,0) 0) == JB(O. (0) do'.
The function FR( 0) is Carathéodory function: that is, it is measurable in for each fixed (by Fubini's theorem, as dz is continuous in x:) and continuous 0 for each fixed æ (by the Dominated Convergence Theorem, dı E Live and m0B R)) 0).
Define the pointwise suprema:
(x) = sup (1, 8), 8EB
S (I) == sup - FR 0)),
Gn(Z) sup
Fn (z. 0)1 (x),
Since ④ C RD is separable, let D = { be fixed countable dense subset. Because 0 + FR 8) is continuous and D is dense, the suprema over θ are equal to the suprema over D. Specifically, let (x) == sup+21 F2(x, 0z) and S == supb>1(-FA) (F((F 0x)). These are measurable functions, since they countable suprema of measurable functions, and we have (x) $x (x) and S (z) $ (x) for all
Now, we construct measurable e- e-maximizing selectors. Fix n E N. Define:
k (x) == min k 2 1: FR(E, 0n) 2 (x) - =
(z) = min k 221: 1: -FR( 0x) 2 (z) 1
These minima well-defined and finite. By the definition of the supremum, for any e 1/n > the set of indices l: satisfying the condition is guaranteed to be non-empty. By the well-ordering principle, non-empty subset of N has minimum. The maps x A# (x) measurable, as the sets (a: ket (I) }}} are formed by measurable comparisons. Thus, the selectors 0 (x) 0 (xt and 8% (x) = 0x (m)) are well defined and measurable.
By construction, these selectors satisfy:
Fn(2, (x)) 2 (+) and Fn(Z,07e.. (x)) (x) for all x.
Applying the theorem's hypothesis with O: =Oh gives:
0 = - p(x) Fr Fr( (z) dx 2 / p(x) (5 - - dx 1 (())S de S
/ p(x) dx.
Similarly, applying the hypothesis with 0, ft,n gives:
0 = (lx)FFt (2. (x)) dx < p(x) (-5 + 2)) dr = | p())5 dr - p(z) dr.
21

---

Since Gn(Z)= I (kp(2) Sp(x) = (x) + S (2), we have
J Glo)Gn( 2 [ p()) dr + / p(x) (2) dr
2 / p(x) dr
n x
This must hold for all n, and so we must have fx ((x)) dx O. Since p(x) and (R(x) O, we must therefore have GR(z) for p-a.e, v. This means that for p-a.e, I, we have FR( 0) = 0 for all / E θ.
Step 2: Apply the Lebesgue Differentiation Theorem. From Step 1, we know there is a set
Xn C X with (X)) such that for any Xo. JB(e,R) d, (0) dl' 0 for all R 0 and all
0 E
The Lebesgue Differentiation Theorem for doubling spaces states that for any function f E (0, (0,
S f(0") do' for m-..e O. ((0,R)
f(0) lim R- m mB(O, R))
For any fixed x E Xo, the function 0 -> dz (0) is in oo (9) by hypothesis. Applying the theorem gives:
[
0
(0) = lim
BBOO.)) d0') d0'
lim
0
R-+0 m B(0,, R))
R-0 ml (B(0,, R))
for m-a.e 0 E O.
Since this holds for every x E Xo where μ(X' \ Xx) = 0 (as is density for μ0), we have (0) 0 for μ @ m-a.e (). This means p(0 z) = q(0 z) for μ & m-a.e (z, 8).
Remarks. We conclude with remarks.
1. Necessity of quantifying over all measurable selectors. The assumption that the inner integral vanishes for every measurable selector 0; X' - θ is crucial. the condition held only for constant maps 0 (z) Oo. it would assert only that the x-average of the localized integrals vanishes:
S (mle)
( dx( dB
dr 0.
This would allow for cancellations across and would not imply that the inner integral vanishes pointwise in æ. The logic of the proof requires the freedom to vary the center of the ball adaptively with æ to prevent these cancellations.
2. Continuity in can be relaxed. The theorem and proof remain valid if the maps x p(0 x) and x + q(0 x) are merely measurable rather than continuous, provided that the resulting function F2 (ar, 0) = SE(., dz (0') d0' is Carathéodory function (measurable in z, continuous in 0). This condition holds under weaker assumptions than continuity, such as joint measurability of (a, 0) + d, (0). The continuity-in-r assumption is straightforward condition that guarantees this property, which all that is needed for the measurable selection argument to succeed. Moreover, the measurability of the set N {(x, E Xo, E ( where O is the set of non-Lebesgue points for the function (g) = d(x 0). follows from the joint measurability of (!).
B.3 Proof of Theorem 2
We first need the following lemma.
Lemma 3. Let (O,dg) dg) be a metric space, and let 0 E O be fixe. Define the function
R(0) (0, 0), 0 E O.
22

---

Now let be probability measure on Q. For any 0" E G define the ball probability rank of 0* under as
Ug(0")= = Paiag (R(O) I R(0")) Porq (0:,0) 0) & ds(O,,0)) 0)
Then, if we also have that 0* ~ q, then the random variable (0())) is distributed as Uniform(O,, 1), i..,
Po-~a (V, S u) =u, Vu E [0, 1].
Proof. Define FR as the cumulative distribution function (CDF) of the random variable (O) d(Or, 0), where 0 ~ i.c.
Fn(r) = Pong (R(0) sr)
By definition of (0"), have
(²((0) Pa~g (R(0) < R(0')) FR(R(()))
But by assumption, we have 8* q. Accordingly, R(0*) itself a random variable drawn from the distribution whose CDF is FR. it follows from the probability integral transform that for any localization point 02.
Po-~g (0*) S u) u.
(3)
Thus, Ug(07) Uniform(O, 1), completing the proof.
The key observation from Lemma is that the probability mass assigned by to the ball of this radius, centered at 0,, follows a uniform distribution when 0* p. Thus, if equation (3) (which states that U(8") ~ Uniform(0. 1)) holds for all possible choices of f1, then the conditional distributions p(0 z) and q x) must be identical. Intuitively, this because the process of drawing 0* and measuring probability mass within its corresponding ball implicitly tests equality of mass across all possible radii structured way. If the distributions p and q were different, there would exist some localization point Og where the resulting uniformity condition fails, revealing discrepancy in their induced probability measures.
With this lemma in place, we can now prove Theorem 2.
Proof. (=) Suppose that p(B,) = q..)) for all r 0. Consider the cumulative distribution function (CDF) of the distance variable R((0)) d(0;, 0*), when 0* ~
(r(r) = ~e (R(0') < r) =p(B,).
Similarly, under ~ g, the corresponding CDF
Fr(r) Pee-~ (R(00) r) ((B,).
By assumption, these two CDFs are identical, i.c., F2(r) Fr(r) for all Now, by the definition of (0*),
(0") = Ponyg (R(0) < R(0))) = Fg(R(O))).
Since Fg- = Fp. we obtain
U₂(0") ()()))))
From Lemma 3, we know that (R(O)))) ~ Uniform(0, 1) when 0" ~ p. which implies that (O*) ~ Uniform(0,) 1) under 0r p. Thus, the distributions of (0"))) under p and q must be identical.
(+=) Now suppose that U,(0*) (m'')) Then, for any E 00, 1].
PU, u) Po w (U,(0") S u).
Rewriting in terms of the CDFs, this implies
Po-~p F, (R())) < u) = Po- (R(0") S u).
By the probability integral transform, since (R(())) ~ Uniform (0, 1) under both p and g. follows that (R(()))) F, (R(( in distribution. This means that F, Fg. implying p(B,) 4( (B)) Vr.
Thus, the probability assigned to each metric ball is identical under and
23

---

B.4 Proof of Theorem 3
From Lemma 3, we have that (8 2) follows uniform distribution when rr g(8") | x),, thus satisfying
IP
U S al
F (x) e~~ag(0-1a)
|0g(0" |=) S a).
(-))
Next, define the radius (x) as follows
(o) (a) == inf (r:
P B (s(((()) - (~q((02)
Since q(0 I) is absolutely continuous with respect to the Lebesgue measure, the mapping a
( (a) is a bijection. Additionally, by definition of U, | <)), it follows that
(0r U,(0" ax) S a) = (0( (1), (o)())(a)).
Combining these observations, we obtain:
d(p,g) sup
E Ex>p(xx
IP (Ug(0') x) S s 0r 0"~a(00 (a) ('~0(011x)
(0'(0 1x) S ai]
01()),,,a
sup 8: (-),b,a
E Ex)p(x)
IP
[Bo(O((r), (o((o)
PP
(z),
0 0"~p(0 (x))
1~~ (an)
= ACLD (p.9) (B)( RI]
sup (-),
E Ex)p(x)
P (~~p(0-12))
[B(()((z), R)]
P 0~-0(0-11)
B.5 Proof of Lemma
Proof. Let u |000|
[(00)) 20
2 and
Since 0, 0' E ERP,u 2 0 and 2 0. The squared
distance is given by
10,00 - 2 21 [(1 e)e +0)
Proof of the Global Upper Bound We can rewrite the expression for the squared distance as:
L0 0') 2(1 10ll-eh e)e (u-o) ZEe " 2(1 E (a(t))) 2ee "(1 - e)
2(1
For any z 2 0, the standard inequality e < holds. Furthermore, since E (0, e
and - c 2 0, the second term is non-positive. We can therefore bound the expression:
L, 0') < 2(1 e (utoo) < 2(u + v).
Substituting the definitions of and yields:
[10
1 (l)(0) (10))12)
110 -0-00 0'11
+ l1b(0) - ((0')12
(0, 2
20²
20
o²
of
By the Lipschitz assumption on d, we have [])(0) - -(00') s L2 110
0'1,, which implies:
1 (0, 0') < gz (10- 0'1 13 (000-111))) 8'1²) (
Lg
100 0'12.
+
oo
Taking the square root provides the global upper bound with the constant C2
4g or
Proof of the Local Lower Bound We rewrite the term 1 - kgg 0'):
(1-e-") + (1 e)e (1 "). w 2 0, the second term is non-negative. This allows us to Since - > 0, -m > 0, and bound the expression below:
1- k=,=(0,0') 1 - [(1 -
+ e)e
1 - 00, > -k
24

---

This provides lower bound on the squared distance:
10 0'11²
(0,00) = k0 2 2(1 - ")= 1--xx] exp
203
To establish a linear relationship between dg 8*) and |0 8'11, we analyze the function g(2) 1-e"" for 0. By L''Hópital's rule, li g(2) 1. The function g(z) ) is continuous and strictly positive on any compact interval [0, / By the Extreme Value Theorem, must attain a minimum holds.
value Gmin on this interval. Therefore, for all (0, mma]] the inequality e Z Cmin
Let restrict domain to bounded set where |0 8' S M for some constant M 0. This 00-001
implics that u above:
Zmax On this domain, we can apply the linear inequality derived
2x
2
|10 0'
(0,000) 2 2(1 -)= 2Cmin t
20min
Cmin
2 |0 - 0'12.
20
0
Taking the square root gives the local lower bound (0,00)) 2 CI.M |0 0'l with the constant
C1,Mr Venn which depends on M through Cmin-
02
D completing the proof.
This yields the local-doubling condition with constant CM
CL.M
C Experiments Details
C.1 Toy Example
We construct synthetic ground-trute data distribution I² by defining Gaussian mixture model (GMM) whose components procedurally placed according recursive branching process. We use the code from paper [11] whose generation process can be found in https: /ttp::///gttii /githui..b... com/ NVlabs / edm2 bblpb//b/t//oocccc/c hobb//maii main/ main/toy.... tettttoe__m_ppppeee py. To maintain the completeness of our paper, we include the generation process here.
Gaussian Mixture Representation
The base distribution is modeled weighted sum of multivariate Gaussian components:
p(z) Σ rk N( (z pa: Eg o²1),
where:
Φe E R4 are normalized mixture weights,
Pg R are the component means,
Dg 22x2 are the component covariance matrices,
E R. where set 1e 2 for p(0 z) and (1 + a) for q(0 z).
Each component is assigned a weight and covariance that decays with tree depth, producing finer-scale detail deeper recursion levels.
Recursive Tree-Structured Composition
The mixture components are positioned according to a recursive tree-like structure:
Two primary classes (A and B.) are generated, each initialized the root origin and with distinct initial angles (e.g., /4 and 5 /4).
each recursion level (up to depth 7), branch is extended in a given direction, and eight Gaussian components are placed uniformly along the branch.
25

---

Table Six types of perturbations used assess the sensitivity of CoLT.
Explanation
MI(1 + Hoops Fig)
Mean Shift: Introduces a sy systemaric bias by shifting the mean.
Nip.. (1 -oj8a)
Covariance Scaling: Uniformly influtes variance.
M.).
N(ya. E, aA)
Anisotropic Covariance Perturbation: Adds variability along the minimum- variance eigemvectoe of Yg: Yale-
(olke, Ex)
Adjustment I-Distribution: Introduces degrees freedom 1/(a + c). uppronching Guassian as a
tails, with
WW.. Eo) a.( (.... N()..E))
Additional Modes: introdaces spurious multimodaiity.
11 aWWi,. taM
Mode Collapse: g loses multi modal stractur..
Each component's mean is computed by interpolating along the current direction vector, and the covariance anisotropically scaled align with the branch's 's orientation.
Each branch spawns two child branches recursively, with angles perturbed stochastically to simulate natural variability.
Component Covariance Structure
The covariance of each Gaussian component is constructed as:
= (dd" + (I dd?) ttick²) diag(size)?,
where d is the normalized direction of the branch, thick controls orthogonal spread, and size scales with recursion depth.
This construction ensures that components are elongated along the branch direction and narrow orthogonal to it, producing tree-like density patterns.
C.2 Experiments with perturbed Gaussians
We give further details the experiments in Section 4. For these benchmarks, construct ground-truth conditional distribution by first simulating latent Gaussian variables with redependent means and variances:
8 (H Ex),
Hs = Wix, Σ, = W ø EE
where 0 e R., W1 E R and W2 E x are fixed weight matrices with standard normal entries. The matrix E E xx a fixed Toeplitz correlation matrix, with entries Eig = lirl and corr = 0.9 to simulate strong structured correlations.
In all cases we sample conditioning inputs x Lm),,
C.3 Diffusion Training and Sampling Procedure
Diffusion Model To further evaluate our method in the context of generative posterior estimation, construct benchmark based diffusion model. We begin by sampling x Unif 1.5x, 4.5x] and defining 0 = (ar c2s(2). x sin)))), which induces highly nonlinear and non-Gaussian posterior structure. We then train a diffusion model approximate this distribution. During evaluation, generate samples via reverse diffusion and treat the output after 20 reverse steps as a notional ground- truth posterior (Figure 4a)-not because it is the the best available approximation produced by the model. Outputs from earlier steps (20 a) serve as degraded approximations to this endpoint. This yields a generative-model- perturbation scheme parameterized by (Figures 4b and 4c).
target distribution, but because it represents
monotonic
Beyond simple rejection, our method provides fine- fine-graine grained quantitative insight: statistic t from Algorithm to measure how close each approximate posterior (from fewer reverse steps) is to the reference posterior (20-step output). This allows us quantify posterior degradation as a function of reverse diffusion progress. As shown Figure 4d, the test statistic increases monotonically with a, reflecting growing divergence from the posterior.
the test
As described in Section 4, we sample ~ Unif(1.55 4.5m] and define 0 (ar cos(s), x sin())). This produces a nontrivial two-dimensional manifold for posterior inference.
26

---

O
NTM
(a) Step = 20 (a = 0)
(b) Step 16 (a: 4)
(c) Step = 15 (a =
(d) Test statistic
Figure Posterior estimation using a diffusion model. The output at step 20 is treated as the ground-truth posterior. Reducing the number of reverse steps results in increasingly degraded
approximations. Our method reflects this degradation with a monotonically increasing test statistic, indicating sensitivity to model quality.
We generate 32,768 samples train diffusion model using 50,000 training epochs and learning rate of 1 x 10-2 The diffusion model is trained and sampled following the implementation provided EDM [19].
For posterior approximation, we set the total number reverse diffusion steps to 20, treating the output step 20 as the ground-truth posterior. To construct approximate posteriors varying levels of fidelity, we also record intermediate samples from reverse steps 15 through 19. These intermediate outputs serve posterior estimates for evaluation against the step-20 reference.
C.4 Sampling and Training Hyperparameters
In this section, we provide detailed configurations for the data generation process and training setup used throughout our experiments (see Section 4). We evaluate our method across various combinations of (dim(r),dim(()): (3,3), (10, 10), (50, 10), and (100, 100). We denote N as the number of sample pairs {:), drawn from the true joint distribution p(0 =))(zz), and K as the number samples drawn from the estimated posterior {(0 x) for each conditioning value x.
In the CoLT Full setting, we utilize distance embedding network Φ with input dimension equal to d0) and output dimension set to dim Although alternative output dimensions for may potentially improve performance, we fix the output dimension to dim(O) to avoid additional hyperparameter tuning and ensure fair comparison across settings.
All neural networks in our method (including o O2 and C2ST classifier) are implemented as 3-layer multilayer perceptrons (MLPs) with 256 hidden units per layer.
The table below summarizes the range of perturbation levels a tested for each experiment type, along with the sampling and training hyperparameters for both CoLT and C2ST.
Pecturbation
Alphas
NV
A
#Eval
CoL. Epochs
ColT LR
CIST Epochs
CIST LR
(0.0, 0.00, 0.1, 0.20,
100
1000
Ic
Covariance
Ariactmpic
(0.0, 1.5, 3.0)
Kuroois Adjusment via Disuibttin
(0.0, 0.2, 0.3. 0.6)
100
1c-5
Additinal
(0.0, 0.1. 0.15, 0.2,
0.4)
5--5
Mode Collupse
(0.0, 0.1, 0.3, 0.3, U.4, 0.5, 0.6)
100
500
2001
1000
1e- 5
1000
1e-5
Bind Pricr
100
500
200
1000
1e-5
1000
1c-5
The (Toy Esample)
(0.0, 0.5, 1.0, 1.5. 2.0. 2.5. 3. 5.5, 4.0)
1000
100
200
9000
In-5
5000
1e-5
Diftaiio
1000
200
200
1000
1c
1000
1c
Table 2: Experimental configurations for each type of posterior perturbation. Columns specify the perturbation type, tested a values, sample counts, evaluation batch size, and training hyperparameters for CoLT and C2ST methods.
C.5 Curvature Transformation and Calculation
In Section 4, we introduce the concept of curvature in the parameter space. Specifically, we construct a transformation network increase the curvature of The network consists of fully connected layer ('torch. nn. Linear') with input dimension equal to the dimension of 0, hidden layer of
27

---

size 128, followed by an element-wist wise sine activation, and another linear layer mapping from 128 to the original dimension of 0. The weights the linear layers are initialized using PyTorch's PaneeyyTTrreee random initialization.
default
To compute the eurvature, apply the principal curve algorithm from Hastie and Stuetzle [20]. The resulting principal curves are shown in Figures 2a and 2c.
We observe that before applying the curvature-inducing transformation, the principal curve closely resembles straight line, with total absolute curvature of approximately 62, as expected for a highly correlated Gaussian distribution. After applying the transformation, the resulting parameter space exhibits significantly increased curvature, with a total absolute curvature of around 400.
This transformation provides an effective mechanism for inducing curvature in 0 space, allowing us to study the performance of methods under non- con-Fuclidean geometries.
D Additional Experimental Results
D.1
Tree Task
We present additional visualizations for the toy tree-structured posterior under various levels of perturbation As increases, the sampled points become increasingly dispersed and less con- centrated around the underlying structure. The shaded region indicates the true posterior manifold corresponding to 0.
(a)
(b) 0.5
(c) 1.0
(d) 1.5
(c) 2.0
2.5
(g) 3.0
(h) 3.5
(i
4.0
Figure 5: Tree samples across varying values.
The statistical power for different methods, including CoLT Full and CoLT ID, is shown in Figure 6.
0.0
C2ST
Coll ID
0.2
TARP
0.0
Figure 6: Statistical power for tree-structured tasks across all evaluated methods, including CoLT variants.
28

---

D.2 Perturbation
In this section, we present additional results across varying dimensions of (z. 0) for different tasks, evaluated under multiple perturbation magnitudes in the non-curvauure setting. The results demon- strate the robustness of CoLT and baseline methods under wide range of perturbations. Specifically, we show:
Mean shifts: Figure 8
Covariance scaling: Figure 9
Anisotropic covariance perturbation: Figure 10
Kurtosis adjustment via t-distribution: Figure 11
Additional modes: Figure 12
Mode collapse: Figure 13
D.3
Blind Prior
In addition to the above perturbation strategies, we also include a Blind Prior setting, where the estimated posterior ignores the input z entirely: g(8 xx) = p(0), i.e., the posterior estimate is simply the prior distribution of f. This scenario serves as important pathological case, as it has been shown cause both SBC and TARP (with random reference points) fail-these methods are unable to detect the distributional discrepancy between q() z) and the true posterior p(0 x). By contrast, we demonstrate that our proposed method remains sensitive and effective even in this setting.
Blind Prior In Table 3, we present results under the Blind Prior setting, where the estimated posterior ignores the conditioning input and is set as q(O x) = p(0). This case is particularly challenging. both TARP and SBC fail detect the resulting distributional discrepancy.
In contrast, our proposed BaathodsColl.. across all dimensional settings. Notably, while C2ST is effective in low dimensions, its power deteriorates significantly as the dimensionality increases. Our methods maintain high power even in high-dimensional regimes, demonstrating their robustness and effectiveness in detecting subtle posterior mismatches.
and CoLT successuly detect this violation
Method
x 3,0
10,
50,
100, 100
CoLT ID
1.000 + 0.000
1,000 0.000
1,000 + 0.000
1.000 = 0.000
CoLT
0.975
0.778 0.222
0.307
0.452 0.260
C2ST
1.000 2 0.000
1.000 2 0.000
0.847 2 0.038
0.122 0.107
TARP
0.053 + 0.004
0.047 0.012
0.035 0.009
0.068 0.004
Table 3: Statistical power (mean + stderr) under the Blind Prior setting for each method, evaluated
across increasing dimensions. Only CoLT variants consistently maintain high power as dimensionality increases.
In addition to reporting the statistical power of each method in Table 3, we provide their corresponding Type error rates Table 4. Since the p-value threshold is set to 0.05, all methods successfully control the Type error within the expected range, indicating that none falsely reject the null hypothesis under the correctly specified posterior.
Method
3,
# 10,8 10
# 50,
100, 100
C2ST
0.0767 1 0.0044
0.0433 = 0.0067
0.0200 + 0.0050
0.0233 4 0.0073
CoLT
0.0517
Col.T ID
0.0567 + 0.0093
0.0550 0.0076
0.0467
0.0044
0.0433 + 0.0017
SBC
0.0400 0.0104
0.0350
0.0350 0.0000
0.0350 0.0087
TARP
0.0367 0.0093
0.0383 0.0017
0.0517 0.0109
0.0433 1 0.0017
Table 4: Type error (mean + stderr) under the Blind Prior setting for each method, evaluated across
increasing dimensions.
29

---

D.4 Diffusion Sampling Results
In this section, we provide additional visualizations related to the diffusion- posterior approxi- mation. Figure illustrates the underlying data manifold used train the diffusion model, well as the sampling results from reverse steps 15 through 20. These samples allow us to visualize the quality of intermediate outputs as the reverse process progresses.
We also include power and Type error curve that quantifies how the performance our method changes with respect the number of reverse steps. As expected, the statistical power increases as the number of reverse steps approaches 20, while Type error remains well-controlled.
32
(a) Training Data
(b) Step 20
(c) Step 19
(d) Step 18
(e) Step 17
(f) Step 16
(g) Step 15
(h) Power and Type I Error
Figure 7: Visualization the diffusion-bars based posterior sampling process. (a) shows the data manifold used for training. (b)-() show sampled distributions various reverse diffusion steps (15 20), where step 20 is treated as the ground truth. (h) plots the statistical power and Type error as function of the step gap from the final posterior.
Ablation Studies
In this section, present additional ablation experiments to validate the design choices and examine the robustness of our proposed method. These studies explore the impact of different architectural and algorithmic components, providing comprehensive understanding of the method's performance across the design space.
Model Capacity. As long the localization and embedding networks possess sufficient represen- tational capacity, we expect them to achieve comparable performance. As detailed in Section C, we employ 3-layer MLPs with 256 hidden units and observe consistent results across datasets of varying dimensionality and distributional complexity. This indicates that, given adequate model capacity, performance remains stable and robust.
$.2
0.5
044
06-
DVT
0.5
044
100
02-
02-
225
820
Figure 8: Mean Shift
30

---

0.5
0.3
046
OLT
0.4
100
0.2
3.5
Figure 9: Covariance Scaling
Coealed, Devi
DeriswoO), De100w00
DevOIMIEE OHASTEIR
1.3
02-
0:0
0.27
0.7
06-
0.55
GLT
0.55
0.3
SEC
5.53
0.22
0.11
0.0
2.6
28
Figure 10: Anisotropic Covariance Perturbation
Ole30-0, Duni0-3
D1mo1-16. 10
Ciniae-8,, Dleati-10
Clvo1-103, Dee
1.5
1.2
0.2
0.8-
0.3
0.
3.2
0.-
OM
fall
Iver
08
01
05
00
0.5
0.6
28
0.
04
Figure 11: Kurtosis Adjustment via t-Distribution
DevoleICs Owee
0.
Cod
0.4
04
04
W00
0.
0.2
Figure 12: Additional Modes
Dinb: 10. Deeay-
Dieci 18 Dir0tt 10
Clmos 101--118... cledd 162
1.2
1.2
1.0
0.3
035
0.5
SEC
02
00
33
01
02
13
24
65
66
00
01
62
83
05
18
00
03
1.
00
03
04
Figure 13: Mode Collapse
31

---

To further assess the effect of network depth, we conducted an ablation study under mean-shift perturbations with a 0.2 and a 0.3. The results. presented in Table show that reducing the number of layers slightly weakens statistical power, while deeper networks yield marginal gains. Nevertheless, the overall performance remains within similar range, suggesting diminishing returns beyond moderate depth. Although full architecture search is beyond our current scope, we expect further improvements with more refined architectural design.
Hidden Layer= =
Hidden Layer =
Hidden Layer 4
a 0.2 (CoLT-Full)
0.61
0.63
0.91
a 0.3 (Col.T-Full)
0.88
0.89
1.00
Table 5: Effect of MLP depth on the CoLT-Full performance under mean-shift perturbation with different a. Increasing the number of hidden layers provides slight improvements, while maintaining overall consistency.
Divergence Functions. We conduct additional experiments using various divergence objectives and find that training remains stable across all variants, with the loss consistently decreasing over time. For instance, shown in Table 6, in the kurtosis adjustment task under t-distribution perturbations with a = 0.2 and = 0.3, the Sinkhorn divergence achieves the best performance. We attribute this its smoother loss landscape, which facilitates optimization [16].
Sinkhorn
MMD
Wasserstein
KS
0.2
0.13
0.08
0.09
0.10
0.3
0.27
0.14
0.20
0.18
Table 6: Statistical power under different divergence objectives for the kurtosis adjustment task with t-distribution perturbations. The Sinkhorn divergence consistently outperforms others, likely due to its smoother and more optimization- friennll loss landscape.
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NeurIPS Paper Checklist
1. Claims
Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
Answer: [Yes]
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Reviews:
[{"review_id": "tx7u11GqMa", "reviewer": "Reviewer_ugqN", "summary": "The paper presents *the conditional localisation test* (CoLT), a method for assessing the closeness of two conditional distributions. This is a common scenario in simulation-based inference, where one wishes to compare an approximate conditional posterior, $q(\\theta | x)$, to the true conditional posterior, $p(\\theta | x)$. The paper is based on the insight that if two distributions are unequal, they must differ in mass within a ball of some specific radius, and proposes to use a trained localisation function to find such a neighbourhood. The authors prove that finding this neighbourhood is possible (also empirically) through three theorems stating 1) that it is sufficient to compare the distributions' mass for an average over $x$ instead of for all $x$; 2) that the uniformity of a proposed ball probability rank statistic is necessary and sufficient to conclude if the distributions are identical; and 3) that finding the largest possible deviation from uniformity of this statistic provides an estimate of the distance between the distributions. \n\nThe authors provide algorithms for training and evaluating CoLT, which they apply to a benchmark suite of synthetic experiments, demonstrating that it works for a variety of conditional posteriors with varying complexity.", "questions": "1. Do you have some intuition (or practical experience) for how the capacity of the localisation and embedding networks should scale with the dimensionality and complexity of the posterior? \n2. Is there a practical limit to the dimensionality of the problem that CoLT is useful for? As an extreme example, could CoLT be used to assess the performance of an implicit distribution over neural network parameters (i.e., millions or billions of parameters)?\n3. You mention that $\\phi$ must be Lipschitz, but I cannot find such a statement for $\\theta_l(x)$ except that it should be smooth. Apologies if I missed it, but what are the requirements for the localisation function?\n4. You mention that the localisation map can be used to expose where $q$ fails to match $p$, but I’m not sure how one would find these discrepancies in practice. Can you give an example?\n5. Your theoretical results seem to suggest that CoLT provides a measure of the global discrepancy between distributions, but is this also the case in practice? When we optimise the localisation function, isn’t CoLT then restricted to assessing the local neighbourhood of that point?", "limitations": "Yes.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 4, "strengths_and_weaknesses": "**Strengths**\n1. The paper presents a novel and original take on an important problem, which should be of wide interest to the NeurIPS community.\n2. The authors are able to rigorously develop a practical algorithm for a seemingly intractable problem.\n3. The paper is very well-written. I enjoyed reading it, in particular the authors' intuitive explanations of the theorems and the overall method.\n4. The method seems likely to become a new benchmark for neural posterior estimation.\n\n**Weaknesses**\n1. While the experiments are interesting, they are also quite simple -- which the authors acknowledge too. I understand that it is difficult to define parametric, yet realistic posteriors for such benchmarks, but I am missing some more intuition for how to use the method in practice, for instance:\n\t1. How does the performance depend on the capacities of the localisation and embedding functions?\n\t2. Is there a practical limit to the dimensionality or complexity of the distributions where CoLT is effective?\n\t3. How is the stability of the training affected by different choices of the localisation and embedding functions and divergence measure?\n2. It is unclear how computationally intensive the method is and how it compares to other methods.", "was_revised": true, "final_justification": "The authors have addressed my questions and concerns, and given the paper's novel contribution and potential significance, I recommend acceptance.", "rebuttal": "We thank the reviewer for the positive feedback and for recognizing the novelty, originality, and significance of our work. We address the reviewer’s concerns below.\n\n> W1.1-How does the performance depend on the capacities of the localisation and embedding functions?\n\n1. As long as the localization and embedding networks have **sufficient capacity, we expect them to yield similar performance**. As discussed in **Lines 853–854**, we use 3-layer MLPs with 256 hidden units, and observe consistent performance across various data dimensionalities and distribution complexities. This suggests that, given adequate model capacity, performance remains robust.\n\n2. We also conducted **an additional ablation study** using the mean shift perturbation with $\\alpha=0.2$ and $0.3$ to examine the **impact of network depth**. Reducing the number of layers slightly decreases statistical power, while increasing layers slightly improves it. However, the overall performance remains in a similar range. While we did not perform a full architecture search, we believe further performance gains are possible with more carefully designed networks.\n\n|                         | **Hidden Layer = 2** | **Hidden Layer = 3** | **Hidden Layer = 4** |\n|-------------------------|----------------------|----------------------|----------------------|\n| $\\alpha=0.2$ (CoLT-Full) | 0.61                 | 0.63                 | 0.91                 |\n| $\\alpha=0.3$ (CoLT-Full) | 0.88                 | 0.89                 | 1.00                 |\n\n\n> W1.2-Is there a practical limit to the dimensionality or complexity of the distributions where CoLT is effective?\n\n1.  Our experiments span a broad range of simulation-based inference settings, **from low-dimensional cases (e.g., $\\dim(x)=3$, $\\dim(\\theta)=3$) to high-dimensional ones (e.g., $\\dim(x)=100$, $\\dim(\\theta)=100$)**. We also evaluate CoLT on distributions of **varying complexity, including unimodal (e.g., Gaussian) and multimodal (e.g., Gaussian mixtures) cases**. As shown in Figure 3, Table 3, Table 4, and Figures 8-13, CoLT consistently performs well across these scenarios, and **we do not observe any inheren limitations tied to dimensionality or distributional complexity**.\n\n2. In practice, **scaling to much higher-dimensional data (e.g., image domains) introduces challenges in optimization and training dynamics**, which may require **careful design choices such as data preprocessing, learning rate tuning, and appropriate architecture selection**. We agree that extending our method to these settings is a valuable direction for future work. Rather than viewing this as a limitation, we see it as a **domain-specific adaptation of our framework to new area**.\n\n> W1.3 – How is the stability of the training affected by different choices of the localisation and embedding functions and divergence measure?\n\n1. Based on our experiments, **we do not observe any training instability resulting from different choices of localization or embedding functions**. These components are parameterized using a fixed neural network architecture (3-layer MLPs with 256 hidden units, as described in Lines 853–854), and **all experiments exhibit stable training behavior**. Furthermore, in our ablation study for W1.1, **varying the number of MLP layers does not lead to instability**. Our method is robust to architectural variations, and we perform only minimal hyperparameter tuning, as described in Section~C.4.\n\n2. Similarly, **we do not observe training instability with different divergence measures. While the choice affects the final performance**, we conduct **additional experiments with different divergences** and find that training remains stable across all variants, with the training loss consistently decreasing. For example, in the kurtosis adjustment task using t-distribution perturbations with $\\alpha = 0.2$ and $0.3$, the Sinkhorn divergence yields the best performance. As discussed in Lines 252–261, we attribute this to its smoother loss landscape, which is known to benefit optimization~[1].\n\n\n[1] Marco Cuturi. Sinkhorn distances: Lightspeed computation of optimal transport. Advances in\n368 neural information processing systems, 26, 2013.\n\n|        | **Sinkhorn** | **MMD** | **Wasserstein** | **KS** |\n|:----------------:|:-------------:|:--------:|:----------------:|:-------:|\n| $\\alpha=0.2$     | 0.13         | 0.08    | 0.09            | 0.10   |\n| $\\alpha=0.3$     | 0.27         | 0.14    | 0.20            | 0.18   |\nTab. Statistical power with different divergence loss.\n\n> W2.It is unclear how computationally intensive the method is and how it compares to other methods.\n\nWe provide a **conceptual comparison of computational costs in terms of FLOPs** (floating point operations). Let $n$ be the number of samples from the true posterior, and for each $x$, let $m$ be the number of samples from the estimated posterior. Let $H$ denote the FLOPs required for a single forward pass of the neural network, and $E$ the number of training epochs. Ignoring dimensionality:\n\n- SBC and TARP require $\\mathcal{O}(nm)$ FLOPs,\n\n- C2ST requires $\\mathcal{O}(6nEH + 2nH)$ FLOPs,\n\n- CoLT requires $\\mathcal{O}(6nmEH + 2nmH)$ FLOPs.\n\nWhile CoLT incurs higher training cost, the **neural networks we use are lightweight, and the actual wall-clock training time is short**—especially when compared to the computational cost of training the approximate posterior $q(\\theta|x)$, which is typically parameterized by a generative model and requires significantly more computation resources.\n\n> Q1. ... how the capacity of the localisation and embedding networks should scale with the dimensionality and complexity of the posterior?\n\n1. We evaluate CoLT across **a broad range of posterior dimensionalities (from 3 to 100) and complexities (unimodal to multimodal), and find that a 3-layer MLP with 256 hidden units is generally sufficient for reliable performance**. While increasing network capacity can improve power, we believe that in most statistical testing settings—typically involving moderate-dimensional posteriors—a 3-layer architecture is adequate. **For higher-dimensional domains such as image data, increased model capacity is necessary. In such cases, more expressive architectures like ResNet-18 may be appropriate replacements.**\n\n2. Please also refer to our responses to W1.1 and W1.2.\n\n> Q2. Is there a practical limit to the dimensionality ... could CoLT be used to assess the performance of an implicit distribution over neural network parameters (i.e., millions or billions of parameters)?\n\n1. When applying CoLT to more complex distributions—such as an implicit distribution over neural network parameters where both $\\theta$ and $x$ may lie in extremely high-dimensional spaces—**the primary consideration becomes selecting a network architecture with sufficient capacity. In such cases, carefully designing the training dynamics and tuning the optimization hyperparameters become crucial for stable and effective training.**\n\n2. While since our method ultimately relies on a ball-rank statistic (a one-dimensional summary), the localization and embedding **networks used in CoLT can be significantly lighter than the networks required to parameterize $q(\\theta \\mid x)$.**\n\n3. Please also refer to our response to Q1, where we discuss the scalability and stability of our approach in more detail.\n\n> Q3. You mention that $\\phi$ must be Lipschitz, but I cannot find such a statement for $\\theta_l(x)$ except that it should be smooth....\n\nWe do not need to make any assumptions about the regularity (e.g., smoothness) of $\\theta_\\ell(x)$. It is only required to be a Borel-measurable function over the space $\\mathcal{X}$.\n\n> Q4. You mention that the localisation map can be used to expose where $q(\\theta \\mid x)$ fails to match $p(\\theta \\mid x)$, but I’m not sure how one would find these discrepancies in practice. Can you give an example?\n\n1. Certainly. Suppose we are given a true posterior $p(\\theta \\mid x)$ and an approximate posterior $q(\\theta \\mid x)$, and we wish to identify where they differ most for a given $x$. After training, **we can feed $x$ into the localization network to obtain a point $\\theta_{\\ell}(x)$ in the parameter space**. This point corresponds to an anchor where the discrepancy between $p(\\theta \\mid x)$ and $q(\\theta \\mid x)$ is most pronounced.\n\n2. Once this anchor is obtained, **we define a neighborhood (e.g., a ball) around it in the embedding space induced by the learned distance embedding $\\phi$**. Given an input $x$, the localization and embedding networks together allow us to identify a local region where $p(\\theta \\mid x)$ and $q(\\theta \\mid x)$ differ the most, enabling more targeted and interpretable comparison.\n\n> Q5.Your theoretical results seem to suggest that CoLT provides a measure of the global discrepancy between distributions, but is this also the case in practice? When we optimise the localisation function, isn’t CoLT then restricted to assessing the local neighbourhood of that point?\n\nThis gets to the heart of how CoLT operates. **While the optimization does indeed focus on a specific location, the resulting value is a valid measure of discrepancy**. To clarify, it's helpful to contrast two types of discepancy measures: ones like KL divergence, which compute a discrepancy based on an expectation (an average) over the entire sample space; and IPMs based on a supremum, such as the Kolmogorov-Smirnov (KS) distance. **Although the KS calculation hinges on finding a specific worst-case location, the resulting distance D is a well-established global metric for the discrepancy between distributions P and Q**.  CoLT operates on this same principle. The optimization of the localization function is precisely the search for this point of maximal local discrepancy. By finding the location and scale where the two measures differ most, CoLT defines a global discrepancy score based on this worst-case local separation."}, {"review_id": "cu8hnHiLE6", "reviewer": "Reviewer_k1j4", "summary": "The authors propose a method for validating amortized variational posteriors; that is, those that define $q(\\theta \\mid x)$ for arbitrary $x$. The questions assessed is whether $p(\\theta \\mid x) = q(\\theta \\mid x)$ for all $x \\in \\mathcal{X}$. Although this would seem computationally intractable, the authors obtain tractability by i) assessing in expectation over $x$, ii)  using embeddings and localization points $\\theta_l$ over which to assess the discrepancy between the two. The resulting algorithm involves training an embedding and localization point mapping on $x$ to *maximize* the discrepancy, and a KS test relative to a uniform distribution to make the final assessment.", "questions": "- One downside of the method is that small perturbations of $p$ are often OK; maybe even desirable (cf. line 103). For example, in the toy example, most practitioners would probably be thrilled if their variational posterior $q$ looked like panel (b) with target being panel (a).  \n- In fact, the NPE target (forward KL divergence), is known to yield overdispersed posteriors, and this has even been argued to be desirable for downstream applications such as importance sampling. This should be explained or at least touched on, maybe with reference to some of [1][2][3][4][5].\n- It’s not particularly novel to consider an expectation over $p(x)$ instead of matching every single $x$, which is impossible (as noted by the authors). Indeed, NPE targets the expected forward KL divergence in the first place. Some type of reference to this fact should be given, preferably with citations (see below).\n- Does $\\Theta$ being compact suffice to satisfy the doubling condition?  If so, maybe this could be added to line 150.\n- Theorem 1 seems pretty straightforward to me; I would recommend this be renamed a Proposition.\n- The paper is light on citations; I would recommend citing some of the works below\n\n[1] McNamara et al. Globally Convergent Variational Inference. \n\n[2] Papamakarios and Murray. Fast $\\epsilon$-free Inference of Simulation Models with Bayesian Conditional Density Estimation\n\n[3] Le et al. Revisiting Reweighted Wake-Sleep for Models with Stochastic Control Flow\n\n[4] Bornschein and Bengio. Reweighted Wake-Sleep\n\n[5] Ambrogioni et al. Forward Amortized Variational Inference.", "limitations": "Yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n- The proposed method fills a gap in a literature by answering a pressing question in NPE that is as yet relatively unaddressed.\n- The proposed method is motivated well theoretically; the uniform distribution of the ball probability rank and formulation of the IPM metric elegantly tie the result together.\n- The exposition is clear and well-motivated intuitively. \n\n\nWeaknesses:\n- Any statistical “test” that involves as much machinery as training a neural network is a bit dissatisfying nominally in that it can not be easily applied  “out of the box”\n- The motivation of the method is questionable in some areas: I would probably disagree with the assertion that  $p(\\theta \\mid x)$ and $q(\\theta \\mid x)$ need to match across all conditioning inputs $x \\in \\mathcal{X}$ (line 22). If the data space $\\mathcal{X}$ is, for example, $\\{0,1\\}^{784}$, the space in which binarized MNIST digits would reside, we would only need these to match over a (practically) much smaller space. This is later implicitly accounted for by considering an expectation over $p(x)$ defined on $\\mathcal{X}$, but I think the discussion around this point could be much improved.", "was_revised": true, "final_justification": "The authors adequately addressed all of my concerns, as well as those of other reviewers.", "rebuttal": "We thank the reviewer for the positive feedback and for recognizing the novelty, originality, and significance of our work. We address the reviewer’s concerns below.\n\n> W1. Any statistical “test” that involves as much machinery as training a neural network is a bit dissatisfying nominally in that it cannot be easily applied “out of the box”.\n\n1. We use neural networks to parameterize the embedding and localization functions due to their flexibility in approximating a wide class of functions. **However, neural networks are not strictly necessary to our framework. One could replace them with alternative parametric families**, such as high-order polynomial function classes or other structured basis expansions. Since our divergence measure is based on the Sinkhorn distance, **it is, in principle, possible to estimate the embedding and localization functions analytically or via optimization techniques, without relying on neural network training**. While neural networks provide a convenient and expressive choice, our framework is general and compatible with other parameterizations that avoid training entirely.\n\n2. Moreover, **while we use neural networks in our experiments, this does not necessarily imply a lack of \"out-of-the-box\" applicability**. Across a variety of experiments with different values of $x$- and $\\theta$-dimensionality and varying posterior complexity, **we keep the neural network architecture fixed (3-layer MLPs with 256 hidden units per layer) and only make minor adjustments to hyperparameters** (see Section C.4). This consistency suggests that our method can be easily applied across a broad range of problems without the need for careful neural network design or hyperparameter tuning.\n\n3. We appreciate the insight behind this question and agree that it is worth emphasizing. In our revision, we will include a discussion on how our framework could be instantiated using alternative parametric function classes that eliminate the need for training a neural network.\n\n> W2. The motivation of the method is questionable in some areas: I would probably disagree with the assertion that $p(\\theta|x)$ and $q(\\theta|x)$ need to match across all conditioning inputs $x \\in \\mathcal{X}$ (line 22). If the data space $\\mathcal{X}$ is, for example, $\\{0,1\\}^{784}$, the space in which binarized MNIST digits would reside, we would only need these to match over a (practically) much smaller space. This is later implicitly accounted for by considering an expectation defined on $p(x)$, but I think the discussion around this point could be much improved.\n\n1. Thank you for the insightful comment. We agree that this point deserves a more careful discussion, and we will include it in our revision.\n\n2. The issue raised stems more from a **conceptual overgeneralization of the data space $\\mathcal{X}$, which is not a limitation of our method but rather a consequence of an impractical choice of $\\mathcal{X}$**. In simulation-based inference, $p(x)$ is defined through the generation process $p(\\theta)p(x|\\theta)$, meaning that the support of $p(x)$—and therefore the relevant domain of $\\mathcal{X}$—is induced by the generator $p(x|\\theta)$. For example, if $\\theta$ indexes MNIST digits and $p(x|\\theta)$ describes how images are generated, **then $\\mathcal{X}$ corresponds to the manifold of MNIST-like images, not the entirety of $\\\\{0,1\\\\}^{784}$**. In this context, it is meaningful and appropriate to require $p(\\theta|x) \\approx q(\\theta|x)$ for all $x$ in the support of $p(x)$.\n\n3. Nevertheless, your point is important—it highlights that the practical choice of $\\mathcal{X}$ should be considered carefully. We will revise the discussion to better clarify that our method targets alignment of posteriors over the support of the true data distribution, rather than over arbitrary or unrealistic regions of $\\mathcal{X}$.\n\n> Q1. One downside of the method is that small perturbations of $q(\\theta|x)$ are often OK; maybe even desirable (cf. line 103). For example, in the toy example, most practitioners would probably be thrilled if their variational posterior looked like panel (b) with the target being panel (a).\n\n1. We would not consider this sensitivity to small perturbations a downside of our method. Rather, it reflects a deliberate trade-off: **if one aims to achieve high statistical power, the test must be sensitive to even small deviations from the true posterior**. This is **an inherent tension in hypothesis testing**—it is a double-edged sword, not a flaw of our method.\n\n2. We fully understand the reviewer's concern that in practice, generative models often introduce acceptable approximations or slight deviations. **While our method may formally reject such approximate posteriors for a given significance level, the ACLD metric still provides a valid IPM-based distance between $q(\\theta|x)$ and $p(\\theta|x)$**. We further show this in Appendix C.3, where we use a diffusion model as a case study. **Although any non-zero perturbation level $\\alpha > 0$ technically leads to rejection, the magnitude of the *KS test statistic* increases monotonically with $\\alpha$ (see Figure 4)**. This behavior suggests that our method does more than issue binary accept/reject outcomes—it also offers a fine-grained measure of posterior closeness (by using KS test statistics).\n\nTherefore, even in applications where strict statistical matching is not the main goal, **our method can still serve as a useful diagnostic tool or distance metric to evaluate the quality of approximate inference procedures**.\n\n> Q2. In fact, the NPE target (forward KL divergence), is known to yield overdispersed posteriors, and this has even been argued to be desirable for downstream applications such as importance sampling. This should be explained or at least touched on, maybe with reference to some of [1][2][3][4][5]. \n\nThank you for pointing this out and for suggesting the relevant references—we will make sure to include and discuss them in our revision.\n\n1. While overdispersed posteriors may be desirable in certain downstream tasks such as importance sampling, it remains important to assess how closely an approximate posterior $q(\\theta \\mid x)$ matches the true posterior $p(\\theta \\mid x)$. Our test serves as a principled test tool for such evaluations, **ensuring that the approximation is not excessively far from the ground truth.**\n\n2. Even in settings where some degree of overdispersion is considered beneficial, **an approximate posterior that closely matches the true posterior is often ideal**. Passing our test indicates that $q(\\theta \\mid x)$ is sufficiently close to $p(\\theta \\mid x)$, which can lead to improved performance in downstream applications.\n\n> Q3. It’s not particularly novel to consider an expectation over instead of matching every single , which is impossible (as noted by the authors). Indeed, NPE targets the expected forward KL divergence in the first place. Some type of reference to this fact should be given, preferably with citations (see below).\n\nWe agree that evaluating expectations over $x$ is not a novel idea, but it is also not the main contribution of our work. We appreciate the suggestion and will incorporate this clarification into the paper, along with citations to the relevant prior work.\n\n> Q4. Does ${\\Theta}$ being compact suffice to satisfy the doubling condition? If so, maybe this could be added to line 150.\n\nYes, the compactness of $\\Theta$ is sufficient to ensure that the local doubling condition is satisfied. We will add this clarification in the revised version of the paper.\n\n> Q5.Theorem 1 seems pretty straightforward to me; I would recommend this be renamed a Proposition.\n\nSure. Thanks for your suggestion, we would change it to 'Proposition' in the revision.\n\n> The paper is light on citations; I would recommend citing some of the works below\n\nThank you for pointing these out. We will cite these literature in the revised version of the paper."}, {"review_id": "ff3Aluj8V8", "reviewer": "Reviewer_tRUQ", "summary": "The paper aims to develop a method to determine if a posterior distribution corresponding to data is approximately the same as a neural posterior estimate. The CoLT algorithm introduced uses a KS style statistic that can practically approximated to determine similarity of distributions.", "questions": "What is the significance of Theorem 1? The discussion on line 171 seems like it would be quite useful by elimiting the need for an exhaustive search over $x$, but Algorithm 1 and 2 seem to using entirely Theorem 2 and 3.", "limitations": "yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "While I am not overly familar with this area, the method introduced seems to be very good. It seems quite novel and performs well on the experiements presented. \n\n\nAll problems I was able to find are technical in nature. \n\nFirst some minor mistakes\n- Line 144: $\\phi$ is not a metric. I guess what is meant is that it induces a metric through the euclidean norm as is done on line 718. Although $d_\\phi$ need actually be a metric if $\\phi$ is not injective. Are embeddings here defined to be injective? If so, then it doesn't seem that theoretical results can be applied to the case where $\\phi$ is a neural network.\n - Line 149: What is $D$? Presumably $\\Theta\\subseteq {\\mathcal R}^{D}$.\n - Line 150: It is said that it is sufficient for $\\phi$ to be Lipschitz continuous, but appeandix A only shows that it is sufficient that $\\phi$ is bi-Lipschitz. I also can't find the condition that the balls of $\\phi$ live inside of Euclidean ellipsoids of uniformly bounded eccentricity in appendix A. This condition feels false as stated and probably needs more explaination. What if $\\Theta$ is an simply an ellipse? All balls will then trivially be inside an ellipse\n - Theorem 1, line 162: is it is unclear if measurability is WRT the Euclidean function on Theta, or on the metric $d_\\phi$.\n- Line 729: I can't see how this condition holds for all $\\theta$ and for $R$ sufficently small. I would have guessed that this is an application on the out of line equation on line 720 where this condition holds for a.e. $\\theta$ and for all $R$.\n - Line 730: If $\\Theta$ has an isolated point or if $\\phi$ is discontinuous, then I don't think that $m(B_\\phi(\\theta,R))>0$.\n - Line 776: What is the mapping $\\alpha\\mapsto R_{\\theta_l(x)}(\\alpha)$ a bijection into? I would guess that the structure of $\\Theta$ and $\\phi$ would change the the size of balls and hence the range of values $\\alpha\\mapsto R_{\\theta_l(x)}(\\alpha)$ could possibly take. From line 778 I'm guessing that it is supposed to be bijective into the space of all possible distances to $\\theta_l(x)$.\n\nThere are some mistakes which seem fairly major to me \n- line 728: I can't find the Lebesgue differentiation theorem for doubling measures in [16]. I'd assume a similar result to theorem 1.8 in \"Lectures on Analysis on Metric Spaces\" by Juha Heinonen, or the Lebesgue differentiation theorem on page 81 of https://homepages.uc.edu/~shanmun/book.pdf is being applied. In both these theorems the measure is defined over a metric measure space, but as stated above $d_\\phi$ is not a metric without some type of injectivity condition on $\\phi$. It will however be a pseudo-metric, but I doubt that the Lebesgue differentiation theorem holds for pseudo-metrics. I'd also assume there is an implicit assumption that $\\phi$ is continuous so that the Lebesgue measure can be defined on $(\\Theta, d_\\phi$).\n - Line 733: I can't see why the inner integral be 0. What is assumed is essentially that the inner integral is be 0 on average. This doesn't seem to imply that the integral must be equal to 0 a.e. If $d_x$ where always positive (or always negative) this true, but it seems that $d_x$ will essentially always not be sign-definite. \n - Line 734: Invoking lemma 1 is not valid here as R is fixed and lemma 1 requires the integral equality to hold for all R. I think a sequence of $R_n\\to 0$ needs to be taken. \n - Line 732: It doesn't seem that the fact that the nested integral is =0 for all measurable $\\theta_l$ used. The proof seems to depend only on there existing only a single $\\theta_l$ for which the integral is 0. This assumption seems far too weak for the result to hold. The proof doesn't seem to be following the intuitive idea given in line 165.\n - Line 776: I can't see why $\\alpha\\mapsto R(\\alpha)$ is an injection if $\\phi$ is not injective. It seems that possible that for many $\\alpha$ $R(\\alpha)$ would be infinite as the infimum defining $R$ would be the infimum over $\\emptyset$. It seems like small changes in $r$ can lead to large changes in the size of balls in $d_\\phi$. The doubling condition might save this but this is not obvious to me. A map like $\\phi(x)=0$ seems to be a counter-example. \n\nThe paper is quite well written. There are also some minor formatting mistakes in the appendix \n - Line 721, 730: To be consistent with the notation below and above, I think B should be defined as $B_\\phi$.\n - Line 733: I think the wrong equation is cited. There is no integral in equation 1.\n - Line 746: I think $\\theta^\\star$ should be $\\theta^*$. Similar problem with the equation below line 748\n - Line 775: I think notation is being mixed. $U_q(\\theta^* |x)$ should be $U_{q,x}(\\theta^* )$. If not, then I don't know what $U_q(\\theta^* |x)$ is", "was_revised": true, "final_justification": "All issues brought have been adressed in the rebuttle. I see no more major techincal problems with the paper.", "rebuttal": "We thank the reviewer for their insightful questions and apologise for any confusion. We agree that we need to be clearer and more precise about regularity conditions. The two main issues are the conditions of theorem 1, and the proof itself.  With the clarifications below (to be incorporated into the next version), we hope you agree that the main results from the paper are correct and precisely stated.\n\n**We first address your concerns about the metric $d_\\phi$**. You are correct that $\\phi$ is not a metric, but induces a metric $d_\\phi$ under certain assumptions. We apologise for the lack of precision. You are also correct that $\\phi$ must be injective. Otherwise $d_\\phi$ can fail the requirement that $\\theta_1 = \\theta_2 \\iff d_\\phi(\\theta_1, \\theta_2) = 0$. We apologise for not including this condition explicitly, which created confusion. Clearly if $\\phi$ is bi-Lipschitz (which we do discuss explicitly), then it is necessarily injective, and also induces a $d_\\phi$ that satisfies the doubling condition.  But it is clearer to state the doubling-condition requirement on the metric without explicitly referring to $\\phi$, and then to discuss separately how we might construct a metric satisfying that condition using a trainable neural network.\n \n In revision, we will state the regularity conditions of Theorem 1 precisely and include a clear discussion of what trainable metrics meet the conditions. In our original approach a bi-Lipschitz $\\phi$ is enough.  There are several neural network architectures that work; normalizing flows and RealNVP lead to invertible mappings. In practice, most networks with injective activations that do not have a \"bottleneck\" layer that reduces the dimension of the input typically satisfy injectivity in practice.  But other approaches also work.  For example, we can use a deep kernel-based characterization of distance as in [1]. For $\\epsilon>0$, we might define the deep kernel-based distance as $d_k(\\theta, \\theta') = \\sqrt{2(1 - k_\\epsilon(\\theta, \\theta'))}$ where $k_\\epsilon(\\theta, \\theta') = [(1-\\epsilon)k(\\phi(\\theta), \\phi(\\theta')) + \\epsilon]q(\\theta, \\theta')$, where $k$ and $q$ are Gaussian RBF kernels and $\\phi$ is a Lipschitz continuous function of order $L$ (parameterized by neural networks). [1] show that $k_\\epsilon$ is a characteristic reproducing kernel and hence the corresponding Hilbert space mapping defined by $k_\\epsilon$ is injective. In our revised manuscript, we will include a proof that for any Lipschitz continuous function $\\phi$, the metric $d_k$ satisfies a **local doubling condition**, meaning that for any fixed radius $R_0 > 0$, there exists a constant $C_{R_0} > 0$ such that for all $\\theta \\in \\Theta$ and all radii $r \\in (0, R_0]$, $\\mu(B_k(\\theta, 2r)) \\le C_{R_0} \\mu(B_k(\\theta, r))$, where $\\mu$ is the Lebesgue measure. If $\\Theta$ is compact, the metric is uniformly doubling trivially. But even for locally doubling spaces, the Lebesgue differentiation theorem holds, as its proof hinges on a Vitali-type covering theorem, which we can show still applies under the weaker locally doubling condition.\n\n [1] Liu, Feng, et al, \"Learning deep kernels for non-parametric two-sample tests.\"\n \n> Concerns about the proof of theorem 1.\n\n Thank you for bringing your concerns to our attention. Upon close inspection, our proof had a gap. Here we briefly explain how to rectify this and give a fully correct proof.  The condition\n$$\n\\int_{\\mathcal{X}} p(x)\\Bigl[\\int_{B(\\theta_l(x),R)} d_x(\\theta)d\\theta\\Bigr]dx = 0\n\\quad\\text{for all measurable }\\theta_l\\colon\\mathcal{X}\\to\\Theta\n$$\n\nsays that **no matter how we choose the ball center as a function of $x$**, the $p$–weighted average of the localized discrepancy vanishes. If we only allowed **constant** selectors, cancellations across $x$ could hide nonzero regions of the inner integral; the key of the proof is to adapt the center to each $x$ so that cancellations cannot occur. This mirrors many classic results in variational analysis and stochastic control, which confront similar problems of converting averaged statements into pointwise conclusions. There the use of measurable selectors is a standard device that aligns a choice variable with the pointwise extremum of an integrand.\n\nDue to space constraints we can only sketch the proof. We define $F_R(x,\\theta)=\\int_{B(\\theta,R)}d_x(\\theta')d\\theta'$, and for any $R$, define the envelope $G_R(x)=\\sup_{\\theta}|F_R(x,\\theta)|$. Now fix $R>0$. The goal is to rule out any cancellation across $x$ by\naiming the center at the worst local discrepancy. To do this in a\nmeasurable way, we construct a countable dense set $\\\\{\\vartheta_k\\\\}\\subset\\Theta$\nand, for each $n\\in\\mathbb{N}$, choose a measurable *$\\varepsilon$-maximizer*\n$\\theta_{R,n}(x)\\in\\\\{\\vartheta_k\\\\}$ satisyfing\n\\begin{equation*}\n\\bigl|F_R\\bigl(x,\\theta_{R,n}(x)\\bigr)\\bigr|\n\\ge G_R(x)-\\tfrac{1}{n}\n\\qquad\\text{for all }x.\n\\end{equation*}\nBecause the hypothesis holds for **every** selector, it holds for these\n$\\theta_{R,n}$ and also for analogous selectors that nearly maximize $-F_R$.\nThis yields the **inequality**\n$$\n\\int_{\\mathcal{X}} p(x)G_R(x)dx\n\\le \\frac{2}{n} \\int_{\\mathcal{X}} p(x)dx\n\\qquad \\text{for each } n.\n$$\n\nNow pass to the limit $n\\to\\infty$. The right-hand side tends to $0$,\nwhile the left-hand side is a nonnegative number (it is the integral of\na nonnegative function). Hence\n\\begin{equation*}\n\\int_{\\mathcal{X}} p(x)G_R(x)dx = 0.\n\\end{equation*}\nIntuitively, this says: even after we **optimize** the center to expose\nthe largest local discrepancy at each $x$, the weighted average of those\nmaximal discrepancies is still zero. Since $p(x)>0$ almost everywhere and\n$G_R\\ge 0$, this forces $G_R(x)=0$ for $p$-a.e. $x$; otherwise there would\nexist $\\varepsilon>0$ and a set $A$ with $\\int_A p>0$ on which $G_R\\ge\\varepsilon$,\nleading to $\\int pG_R\\ge \\varepsilon\\int_A p>0$, a contradiction.\nThus, for such $x$ and this fixed $R$,\n\\begin{equation*}\n\\sup_{\\theta\\in\\Theta}|F_R(x,\\theta)|=0\n\\quad\\Longrightarrow\\quad\nF_R(x,\\theta)=0\\ \\ \\text{for all }\\theta\\in\\Theta.\n\\end{equation*}\nFinally, pass from fixed radii to points by shrinking the balls and apply the Lebesgue Differentiation Theorem.\n\n> Line 149: What is $D$?\n\nYou are correct that $D$ refers to the dimensionality of $\\theta$. We will clarify this point.\n\n> Line 150: It is said that it is sufficient for $\\phi$ to be Lipschitz continuous... What if is an simply an ellipse? All balls will then trivially be inside an ellipse\n\nYou are correct that we need bi-Lipschitz. We will state this clearly in the revised version, and we thank the referee for the opportunity to clarify. Our mention of an ``ellipsoidal condition'' was a hand-wavy reference to the geometric requirement that the metric $d_\\phi$ must not exhibit unbounded local distortion. This intuition is formalized precisely by the condition that the embedding $\\phi$  be bi-Lipschitz, which is the essential requirement of our original approach. We were simply trying (and failing) to articulate the idea that the bi-Lipschitz property ensures that the infinitesimal balls under $d_\\phi$ correspond to Euclidean ellipsoids with uniformly bounded eccentricity. We will remove references to the ellipsoidal conditional and make clear that one way to satisfy the theorem's conditions is with a bi-Lipschitz $\\phi$.\n\n> Theorem 1, line 162: is it is unclear if measurability is WRT the Euclidean function on Theta, or on the metric $d_\\phi$.\n\nWe will clarify that it is measurable on $\\mathcal{X}$. We assume $x \\in \\mathbb{R}^d$ and that $\\theta_{l}(x)$ is a measurable function.\n\n> Line 729: I can't see how this condition holds for all $\\theta$ and $R$ for sufficently small. I would have guessed that this is an application on the out of line equation on line 720 where this condition holds for a.e. $\\theta$ and for all $R$ .\n\nYou are correct and we will clarify this point in the revision.\n\n> Line 730: If $\\phi$ is discontinuous...\n\nIf $\\phi$ is bi-Lipschitz, it is trivially continuous and the problem does not arise.\n\n> Line 776: What is the mapping $\\alpha\\to R_{\\theta_l(x)}(\\alpha)$ a bijection into? I would guess that the structure of $\\Theta$ and $\\phi$ would change the the size of balls and hence the range of values $\\alpha\\to R_{\\theta_l(x)}(\\alpha)$ could possibly take. From line 778 I'm guessing that it is supposed to be bijective into the space of all possible distances to $\\theta_l(x)$.\n\nYes, you are correct. It is a bijection into the space of all possible distances to $\\theta_l(x)$. We will clarify this in the revised manuscript.\n\n> line 728: I can't find the Lebesgue differentiation theorem for doubling measures in [16]. I'd assume a similar result to theorem 1.8 in \"Lectures on Analysis on Metric Spaces\" by Juha Heinonen...\n\nYou are right and we have corrected the reference. Per the discussion above, our bi-Lipschitz condition on $\\phi$ is sufficient to guarantee that $d_\\phi$ is a metric satisfying the doubling condition. \n\n> Line 776: I can't see why $\\alpha\\to R(\\alpha)$ is an injection if $\\phi$ is not injective. It seems that possible that for many $\\alpha$, $R(\\alpha) $would be infinite as the infimum defining would be the infimum over $\\emptyset$. It seems like small changes in $r$ can lead to large changes in the size of balls in $d_\\phi$. The doubling condition might save this but this is not obvious to me. A map like $\\phi(x)=0$ seems to be a counter-example.\n\nPer the earlier discussion, you are right that injectivity is required, and guaranteed by the bi-Lipschitz condition. \n\n> Line 721,730,733, 746,775\n\nThank you for pointing out these formatting mistakes, and that the equation should be the first equation in Theorem 1. You are right, and we will correct them in the revision."}]

Author rebuttal:

Meta-review: The paper proposes a method to detect discrepancies between approximate neural posteriors and the true ones. The method, conditional localization test (CoLT), trains localization functions to find neighborhoods of such discrepancy, which is supported by theoretical contributions regarding the possibility of this procedure. The latter is also empirically corroborated. These constitute the main strengths of the paper, a rigoros method with theoretical and empirical support for an important problem that has the potential to impact many other works. The present main weakness of the paper may be that it can be challenging to apply to a broader set of problems beyond the simple experiments shown in the paper. This seems to primarily be due to the fact that it can be very difficult to choose $\mathcal{X}$ appropriately in practical problems. 

Overall, the strengths of the paper clearly outshine all the weaknesses. All reviewers rate the paper very positively, with a  few major concerns having being resolved in the rebuttal. One such concern was the practicality of the method to e.g. images, and the entailed discussion on dimensionality of the input. The authors have agreed to resolve this through further discussion in the paper and the AC recommends to include it in the camera ready. The other main concern was related to several assumptions being insufficiently discussed in the paper writing, which has led to reviewer tRUQ initially questioning the validity of some of the derivations. Through engaged discussion, these concerns have been resolved and additional explanations will be provided in the paper. The AC agrees with all reviewers and believes this paper is a valuable contribution to a meaningful problem that many might care about in the community.

The AC recommends to accept the paper with a spotlight.
Decision: Accept (spotlight)