Title: CoLT: The conditional localization test for assessing the accuracy of neural posterior estimates
Venue: NeurIPS 2025
Accepted: yes
Keywords: Statistical Hypothesis Testing; Neural Posterior Estimation; Likelihood-Free Inference; Bayesian Methods; Simulation-Based Inference
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

Abstract

We consider the problem of validating whether a neural posterior estimate q ( θ | x ) is an accurate approximation to the true, unknown true posterior p ( θ | 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 ( θ | x ) and q ( θ | 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 θ 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 θ ∼ p ( θ | x ) from the true posterior is observed for each conditioning input, but where the neural posterior q ( θ | 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.

1 Introduction

This paper proposes a new method for determining whether two conditional distributions p ( θ | x ) and q ( θ | x ) are equal, or 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, or NPE, which is a rapidly growing area of simulation-based inference. Here θ represents the parameter of a scientific model with prior p ( θ ) , while x ∼ p ( x | θ ) represents data assumed to have arisen from that model. In NPE, we simulate data pairs ( θ, x ) drawn from the joint distribution ( x, θ ) ∼ p ( x, θ ) ≡ p ( θ ) p ( x | θ ) . A conditional generative model-such as a variational autoencoder [1], normalizing flow [2], diffusion model [3, 4, 5, 6], or flow-matching estimator [7]-is then trained on these ( x, θ ) pairs to approximate p ( θ | x ) with a learned distribution q ( θ | x ) . The problem of neural posterior validation is to assess whether the learned q is a good approximation to the true p .

This setting poses challenges not present in simpler problem of testing for equality of unconditional distributions, with no x . For one thing, we must verify that q ( θ | x ) approximates p ( θ | x ) not merely for a single given x , but consistently for all x , without having to explicitly consider all possible x

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 a single "real" sample θ ∼ p ( θ | x ) for each x , yet we can generate an arbitrary number of "synthetic" samples ˜ θ ∼ q ( θ | x ) by repeatedly querying our NPE model. Any successful method for assessing distributional equivalence of p and q 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 a simple rank-based statistic for each margin of q ( θ | x ) , but this provides only a 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 on the choice of a (non-trainable) probability distribution to generate "reference" points that are needed to perform the diagnostic, and the method can perform poorly under a suboptimal choice of this distribution. Finally, the classifer two-sample test, or C2ST [10], involves training a classifier to distinguish whether a given θ 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 ability to effectively learn a global decision boundary over X and Θ simultaneously. 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 θ from the true posterior at a given x . This is often impractical, as in many settings we only have access to a single θ ∼ p ( θ | x ) at a given x .

Our contributions. Our paper addresses these shortcomings with a principled and efficient approach, called the Conditional Localization Test (CoLT), for detecting discrepancies between p and q . CoLT is based on the principle of measure theoretic distinguishability: intuitively, if two conditional densities p ( θ | x ) and q ( θ | x ) are unequal, they must exhibit a 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 θ l : X → Θ that adaptively selects the point θ l ( x ) where, for a given x , p and q are maximally different in the mass they assign to a neighborhood of θ l ( x ) . Intuitively, a neural network that learns a smooth mapping θ l ( x ) should be well suited for this task: if two conditioning inputs x and ˜ x are close, we might reasonably expect that any differences between p and q would manifest similarly (i.e. in nearby regions of θ space) for both x and ˜ x . 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-theoretic distinguishability is well established, and so one might fairly ask: why has it not been widely exploited in machine learning as a 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 q are most readily distinguishable. To address this, we use a trainable embedding function φ that maps points from the parameter space Θ into a latent Euclidean space, where distances can better reflect the concentration of probability mass. We then assess mass equivalence over Euclidean balls in this latent space, i.e. over balls B φ ( θ, R ) = { θ ′ ∈ Θ : ‖ φ ( θ ′ ) -φ ( θ ) ‖ 2 ≤ 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-Euclidean metric balls, naively training a localization function θ l ( x ) would seem to require repeatedly sampling θ 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 θ l can be done far more efficiently. The essential idea involves using a single observed draw from p ( θ | x ) to anchor our comparison of whether the conditional mass of q ( θ | x ) aligns with p ( θ | x ) , in expectation over x . This single draw, combined with the localization function θ l , can be used to carefully construct a one-dimensional ball probability rank statistic that is uniformly distributed if and only if p and q agree on all local neighborhoods around θ l ( x ) . We rigorously construct this rank statistic, and we show how it leads to a practical

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optimization algorithm for θ l ( x ) . Moreover, the rank statistic naturally induces a valid integral probability metric (IPM), offering a 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-art methods across a 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.

A toy example. To provide an initial demonstration of CoLT's effectiveness, we begin with a toy example. Panel A of Figure 1 shows p ( θ | x ) as living on a structured manifold, with branches A (bottom left) and B (top right) representing distinct regions of probability mass, as introduced in [11]. We sample a conditioning input as x ∼ N (0 , 1) , with the true conditional distribution defined as:

Our goal here is to assess whether a method can reliably detect even small perturbations of p . This example, while simple, effectively targets a common failure 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 ˜ θ ∼ q ( θ | x ) by adding a small amount of isotropic Gaussian noise to "correct" samples: that is, ˜ θ = θ + e , where θ is a draw from p ( θ | x ) and each component of e has standard deviation 0 . 01 · α . We then varied α , which controls the degree of mismatch between p and q , and we tested the power of CoLT versus TARP, SBC, and the C2ST for each α . The nominal Type-I error rate was set to 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 α = 1 . 5 (Panel B), the samples from q fall very slightly off the correct manifold. CoLT can reliably detect this difference (power = 0 . 877 ), while C2ST failed entirely (power = 0 . 000 ). At a larger value of α = 4 . 0 (Panel C), CoLT achieves perfect power (1.000), whereas C2ST only reaches power of 0.065. Panel D shows that, while performing a bit better than C2ST, neither TARP nor SBC are competitive with CoLT at any α . These results highlight our method's performance advantage even in scenarios where the posterior lives on a structured manifold, and the discrepancy between p and q is reasonably small. We also emphasize that CoLT doesn't merely detect the difference; as our theory shows, it can also quantify the difference via an integral probability metric.

2 Theoretical Results

In this section, we present our main theoretical results; all proofs are given in the Appendix. Throughout, we denote the Lebesgue measure by m ( · ) and use dθ to represent Lebesgue integration. We also use the shorthand notation q ( θ | x ) = p ( θ | x ) , or simply p = q , to denote that q ( θ | x ) = p ( θ | x ) for almost every ( θ, x ) ∈ Θ ×X . Throughout, we assume that p and q are absolutely continuous with respect to Lebesgue measure for all x .

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 θ l , and the local neighborhood around θ l , where the mass discrepancy between p and q is as large as possible. If the largest such discrepancy is 0, the two distributions are equal.

Taken at face value, however, the localization principle seems deeply impractical. First, if we wish to conclude that p ( θ | x ) = q ( θ | x ) for all x , it seems that we would need to apply the principle pointwise over a grid of x -values. Second, for each x , we would need to search for the point θ l that maximizes the discrepancy in local mass between p and q , if one exists for that x . Finally, we would need to draw many samples from both p ( θ | x ) and q ( θ | x ) to obtain reliable Monte Carlo integrals. The sheer number of evaluations needed-across many x -values, many candidate θ l -locations per x , and many Monte Carlo samples per ( θ l , x ) pair-renders this naïve approach not just intractable, but nestedly intractable.

Luckily, we can do much better than the naïve approach. In fact, our subsequent results can be thought of as peeling back these layers of intractability one at a time.

We begin with a key definition. Specifically, we consider balls of the form B φ ( θ, R ) = { θ ′ ∈ Θ : ‖ φ ( θ ′ ) -φ ( θ ) ‖ 2 ≤ R } , where φ is an embedding function. By defining neighborhoods through d φ , we can shape our regions of comparison to better reflect meaningful differences in probability mass. The following imposes a mild, but useful, geometric regularity condition on the metric φ .

Definition 1 (Doubling Condition) . Let Θ be a set equipped with a map φ and let m be a measure on Θ . For each θ ∈ Θ ⊆ R D and R > 0 , define the φ -ball

We say that φ satisfies the doubling condition with respect to m if there exists a constant C > 0 such that for all θ ∈ Θ and all R > 0 ,

Intuitively, this condition ensures that the metric balls B φ do not distort the underlying geometry of R D too severely, for instance by creating regions of infinite density or measure.

A straightforward sufficient condition for this is that the embedding φ be bi-Lipschitz of any order, which guarantees the above global doubling condition. However, a strict bi-Lipschitz map is not necessary. As a more flexible and practical alternative, we can define φ ( θ ) = k ξ ( θ, · ) as a deepkernel embedding [12], which uses any Lipschitz encoder ξ : R D → R m to extract features. The corresponding kernel-based distance, given by ‖ φ ( θ ′ ) -φ ( θ ) ‖ 2 = √ k ξ ( θ, θ ) + k ξ ( θ ′ , θ ′ ) -k ξ ( θ, θ ′ ) , yeilds a local doubling condition, defined in Appendix A. This weaker, local condition is sufficient for our following localization result to hold. We provide a detailed proof and discussion for both the bi-Lipschitz and deep-kernel cases in Appendix A.

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-mass condition for each x with a weaker condition that involves averaging over x . We formalize this idea in terms of a localization function θ l ( x ) : X → Θ , which identifies the most informative localization point based on x . Intuitively, θ l ( x ) serves as a witness to any potential discrepancy between p ( θ | x ) and q ( θ | x ) .

Theorem 1 (Conditional localization) . Let p ( θ | x ) and q ( θ | x ) be defined as before, and define the difference function d x ( θ ) = p ( θ | x ) -q ( θ | x ) . Let d φ : Θ × Θ → R + be the distance function, in induced by the embedding map φ , satisfying the doubling condition with respect to Lebesgue measure. Let B φ ( θ l ( x ) , R ) denote the φ -ball of radius R centered at θ l ( x ) . Assume further that p ( x ) > 0 is a density on X which is strictly positive almost everywhere.

If, for every measurable function θ l : X → Θ and every R > 0 , we have

then d x ( θ ) = 0 for almost every ( x, θ ) in X × Θ .

Afull proof is provided in Appendix B. The sketch is as follows: The theorem's assumption-that the average discrepancy over all x is zero-is challenging because positive and negative discrepancies

could cancel. However, a crucial feature of the theorem is that the center of the metric ball, θ l ( x ) , is allowed to depend on x via a localization map. The proof uses a measurable selection argument to construct this adversarial localization function θ l ( x ) that, for each x , intentionally centers the ball B φ 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 x , which implies the discrepancy is zero for all balls. From this, the Lebesgue Differentiation Theorem [13]-which applies due to the doubling condition-allows us to conclude that if the average difference over all shrinking balls is zero, the pointwise difference p ( θ | x ) -q ( θ | x ) must itself be zero almost everywhere. Moreover, p = q implies that the supremum of

over all measurable choices of θ l ( · ) and all R > 0 , must be 0. This gives us a natural target for optimization over the choice of the localization function θ l ( x ) .

2.2 The ball probability rank statistic: a practical condition for mass equivalence

Theorem 1 eliminates the need for an exhaustive search over x . But its direct application still appears to require many draws from both p ( θ | x ) and q ( θ | x ) to verify the equality of mass over metric balls. Testing this condition via Monte Carlo would typically involve repeatedly sampling θ from both distributions at the same x and comparing their local integrals. This remains computationally demanding even in principle. Moreover, in the typical setup where this methodology might be applied, the situation is asymmetric: p ( x ) and p ( θ | 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 ( θ | x ) . By contrast, q ( θ | x ) represents a (conditional) generative model that we can query arbitrarily many times for a given x . A practical formulation must leverage this structure by treating the single "true" θ draw as an anchor and evaluating whether the conditional mass of q aligns with p in expectation over x .

Our next result establishes precisely this adaptation, ensuring that the comparison suggested by Theorem 1 can be done feasibly. The basic idea is as follows: we can draw a random sample ( θ ∗ , x ) ∼ p ( θ, x ) , compute the localization point θ l ( x ) , and let the radius be implicitly determined as R ( θ ∗ ) = d φ ( θ l ( x ) , θ ∗ ) . 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 x to lighten the notation.

Theorem 2. Let p and q be defined as above. Fix a reference point θ l ∈ Θ , and define the metric ball

For any θ ∗ ∈ Θ , define the ball probability rank under q as

Then, the condition that p and q assign the same probability to all balls centered at θ l , i.e.,

is equivalent to the statement that, when θ ∗ ∼ p , the random variable U q ( θ ∗ ) is uniformly distributed on [0 , 1] . That is, checking whether, for all choices of θ l , U q ( θ ∗ ) ∼ Unif (0 , 1) under θ ∗ ∼ p is both necessary and sufficient for p = q .

Intuitively, if p and q differ, then there must exist some point θ l and some radius R for which the two distributions assign different mass to the ball B ( θ l , R ) . This mismatch causes the distribution of U q ( θ ∗ ) to deviate from uniformity when θ ∗ ∼ p . Conversely, if U q ( θ ∗ ) ∼ Unif (0 , 1) under θ ∗ ∼ p for every choice of θ l , then p and q must agree on the mass of all such balls, and hence be identical. Thus taken together, Theorems 1 and 2 collapse a daunting, high-dimensional equality-of-mass requirement into a one-dimensional uniformity condition that can serve as the basis for a tractable optimization problem.

2.3 From local-mass uniformity to an IPM

Theorem 2 shows comparing the ball-probability rank statistic U q,x to a uniform distribution gives us a test for whether p = q . The next result shows that, once we optimize over every allowable localization map θ l , 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 (ACLD). Concretely, let

the class of indicator functions of metric balls whose centers depend on x . The corresponding IPM is

Our next theorem connects this distance to the ball probability rank statistic from Theorem 2.

Theorem 3 (Ball-probability IPM) . Let p ( θ | x ) and q ( θ | x ) be absolutely continuous conditional densities on a common parameter space Θ for x ∈ X , and suppose p ( x ) > 0 a.e. on X . For x -dependent) center θ l ( x ) ∈ Θ and for the metric d φ ( θ, θ ′ ) = ‖ φ ( θ ) -φ ( θ ′ ) ‖ 2 induced by an embedding φ : Θ → R m satisfying the doubling condition, define

Now let ˜ d ( p, q ) be the worst-case Kolmogorov distance of U q,x ( θ ∗ ) , averaged over x , from the Uniform distribution:

Then

The theorem establishes that the largest possible deviation from uniformity that one can provoke in U q,x , by freely choosing the localization function, embedding, and ball radius, is numerically identical to an IPM built from indicator balls. Hence training the localization network θ l ( x ) to maximise the distance between U q,x 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 ( θ | x ) has passed the full mass-equivalence test implied by Theorem 1. Conversely, if U q,x is not uniformly distributed, then its empirical KS distance to U (0 , 1) gives us both a p -value based on the classical KS test, and estimates a distance between p and q .

3 The CoLT Algorithm

The key insight from Theorem 2 is that searching for an embedding φ and localization function θ l ( x ) that maximally distort the ball probability rank statistic U q,x 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 φ and the localization function as neural networks, θ l ( x ; ψ ) with learnable parameters ψ . Our strategy is roughly as follows:

Generate a rank statistic: Draw a minibatch of "anchor" points ( θ ∗ i , x i ) B i =1 from p ( θ, x ) , the true joint distribution. By construction, each ( θ ∗ i | x i ) has conditional distribution p ( θ | x i ) . For each anchor point i , sample M synthetic draws { ˜ θ ij } j M =1 from q ( θ | x i ) , and compute the empirical ball probability rank statistic:

Measure non-uniformity: As a loss, we use a negative divergence from a uniform distribution, L ( ψ, φ ) = -D ( ˆ U i ( ψ, φ ) , Uniform ) . We discuss the choice of divergence below.

Optimize: Gradient descent is applied to the loss function. If p = q , optimization will stall, as no choice of ( φ, θ l ( x ; ψ )) will yield substantial deviation from uniformity. Otherwise, the optimizer finds a localization map that exposes the failure of q .

This approach is detailed in Algorithm 1 (training phase) and Algorithm 2 (testing phase). We first apply Algorithm 1 to train the embedding network φ and localization network θ l , aiming to maximize the discrepancy between the empirical ˆ U i values and the uniform distribution. Then with the trained networks and a test set of { ( θ i , x i ) } , we compute a test statistic and corresponding p -value using the one-sample Kolmogorov-Smirnov (KS) test in Algorithm 2.

Algorithm 1 Conditional Localization Test (CoLT): Training Phase

Algorithm 2 Conditional Localization Test (CoLT): Testing Phase

We make three remarks about this algorithm. First, because U i involves an indicator function, gradients cannot propagate directly; we therefore use the Straight-Through 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 a wide range of transformations. Moreover, neural networks are typically Lipschitz-continuous under mild conditions [15], which ensures that the doubling condition (Definition 1) is satisfied; see Appendix A. Alternatively, a fixed, non-trainable form of φ can be specified, and our theoretical guarantees will still hold, but power may be reduced. For example, setting φ as the identity reduces d ( · , · ) to the /lscript 2 distance.

Third, in Algorithm 1, various divergence measures can be used to quantify the discrepancy between the empirical distribution of rank statistics U i and the uniform distribution. While the Kolmogorov-Smirnov (KS) distance is a natural choice motivated directly by our theory, it 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-regularized version of Wasserstein distance that retains geometric sensitivity while offering a smooth objective. Empirically, 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. At 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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4 Experiments

Benchmark tasks. To evaluate CoLT against established NPT methods, we use a suite of benchmark tasks introduced by [17] 2 (details in Appendix C.2). Each benchmark defines a reference posterior p ( θ | x ) , then introduces a family of perturbed alternatives q ( θ | x ; α ) , where the scalar parameter α ≥ 0 controls the severity of deviation. As α increases, so does the discrepancy between p and q , allowing us to generate smooth performance curves that quantify the sensitivity of each NPT method.

We evaluate CoLT on two such benchmark families. The first is based on multivariate Gaussian posteriors with data-dependent mean and covariance. Specifically, we sample x ∼ N ( 1 m , I m ) and define

where W 1 ∈ R s × m and W 2 ∈ R s × 1 are fixed matrices constructed from i.i.d. Gaussians, and Σ is a Toeplitz matrix with entries Σ ij = ρ | i -j | , using ρ = 0 . 9 . The alternative q ( θ | x ; α ) is then constructed by applying structured perturbations either to µ x or Σ x , as 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 2, 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, θ := f ( ˜ θ ) ≡ A h ( B ˜ θ ) , where ˜ θ ∼ N ( µ x , Σ x ) , h is a coordinate-wise sine nonlinearity, and A ∈ R d × d , B ∈ R d × s are fixed matrices. This creates a posterior distribution p ( θ | x ) concentrated on a smooth, curved manifold of intrinsic dimension s in R d . 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, we evaluate two variants: CoLT Full, where both the embedding network φ and the localization network θ l are jointly optimized; and CoLT ID , where φ is the identity and only the localization network θ l is trained. We assess both Type I error (at α = 0 ) and statistical power (for α > 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 θ from q ( θ | x ) for each x 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 a KS test against the uniform distribution.

In both benchmark families, we vary the input, parameter, and latent dimensions ( m,s,d ) and report power as a function of α . We sample 100 pairs { ( θ i , x i ) } from the true joint distribution p ( θ | x ) p ( x ) , along with 500 samples from q ( θ | x ) for each corresponding x during training. After training, we evaluate a method's power by sampling 200 additional batches with the same sampling budget. For

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Dim(x)=3,Dim（0）=3

Dim(x)=3, Dim(θ)=3

1.0

1.0

0.7

0.8

0.8

0.6

0.8

0.6

0.6

0.5

0.4

0.6

CoLT Full

CoLT ID

C2ST

0.4

0.4

0.3

0.4

SBC

TARP

0.2

0.2

0.2

0.2

0.1

0.0

0.0

0.0

0.0

0.00

0.25

0.50

0.75

1.00

1.25

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.0

0.1

0.2

0.3

0.4

α

α

α

α

all methods, we set a nominal Type-I 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 benchmark families and four specific perturbation types: covariance scaling, anisotropic covariance distortion, heavy-tailed deviations via t -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 balls-performs 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 to 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 or exceeds the performance of other methods. These results highlight an important inductive bias: while fixed Euclidean balls suffice for flat posteriors, learned embeddings are crucial for detecting structured mismatch on 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 application to diffusion-based generative posteriors, and in Appendix D, which includes expanded results across more perturbation types and dimensional configurations. We provide the code at https://github.com/TianyuCodings/CoLT .

Discussion and limitations. Our theoretical and empirical results establish CoLT as a principled and practical approach for detecting local discrepancies between conditional distributions, with stateof-the-art performance compared to existing methods. But CoLT does have limitations. The method relies on learning both a localization function θ l ( x ) and an embedding φ ( θ ) , introducing inductive

bias through architectural and optimization choices. If either component is underparameterized 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 CoLT yields a continuous IPM-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-world 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 a clear, extensible testbed, probing a number of common failure modes of NPE methods.

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A Notes on φ

Bi-Lipschitz condition : The key requirement on φ is that it satisfies the doubling condition with respect to Lebesgue measure. One sufficient condition for this to hold is that φ be a bi-Lipschitz function, where there exist constants C 1 , C 2 such that, for all θ 1 , θ 2 ,

If this condition holds, then the doubling condition holds with doubling constant C = ( C 2 C 1 ) D > 0 . To see this, observe that for such a φ , the metric balls satisfy

Then we have

Hence, from it follows that

which shows that m ( B φ ( θ, R ) ) scales like R D up to a constant factor.

Deep kernel-based distances : We also propose using the following the deep kernel-based distance [12]:

where the infinite dimensional embedding φ ( θ ) = k ξ ( θ, · ) ∈ H k is defined through the following deep kernel:

for /epsilon1 ∈ (0 , 1) , RBF Gaussian kernels k 1 ( · , · ) and k 2 ( · , · ) given by k i ( x, y ) = exp ( ‖ x -y ‖ 2 2 σ 2 i ) D m

and any Lipschitz embedding function ξ : R → R . The kernel k ξ ( · , · ) is known to be a characteristic kernel [12], and hence defines a valid distance metric. The following lemma shows that the kernel-based distance defined in (2) satisfies a local-doubling condition , i.e., for any fixed radius M > 0 , there exists a constant C M > 0 such that for all θ ∈ Θ and all radii r ∈ (0 , M ] , we have m ( B φ ( θ, 2 r )) m ( B φ ( θ,r )) ≤ C M .

Lemma 1. Let ξ : R D → R m be a Lipschitz continuous function with constant L 2 . The distance metric d φ defined in (2) satisfies the following:

(Global Upper Bound) There exists a constant C 1 > 0 such that for all θ, θ ′ ∈ R D ,

(Local Lower Bound) For any M > 0 , there exists a constant C M > 0 such that for all θ, θ ′ ∈ R D with ‖ θ -θ ′ ‖ ≤ M ,

Hence, d φ is locally doubling as defined above.

We defer the proof to Appendix B.5. The result of above lemma is critical as satisfying the localdoubling condition makes (Θ , d φ , m ) a Vitali measure space [13, Theorem 3.4.3], which suffices for the Lebesgue Differentiation Theorem to hold and makes our Theorem 1 applicable.

B Proofs

B.1 A preliminary lemma

To prove Theorem 1 we first need the following lemma, which adapts standard measure-theoretic results to the case of a non-Euclidean metric based on an embedding function that satisfies the doubling condition in Definition 1.

Lemma 2. Let p ( θ | x ) and q ( θ | x ) be defined as above, and let φ be an embedding function that induces a metric d φ on Θ , defined as

Further, assume that φ satisfies the doubling condition (1) with respect to Lebesgue measure.

Suppose that for almost every θ l ∈ Θ , we have

for all metric balls B ( θ l , R ) , defined as

Then p ( θ | x ) = q ( θ | x ) almost everywhere ( θ ).

Proof. Define the difference function

The goal is to show that d x ( θ ) = 0 almost everywhere in Θ using the given integral condition.

Because φ is assumed to satisfy the doubling condition with respect to Lebesgue measure, we have for some C > 0 that

Now since d x ( θ ) is locally integrable (as it is a difference of probability densities), we apply the Lebesgue Differentiation Theorem for doubling measures [18], which implies:

However, by assumption, we know that for all θ and all sufficiently small R ,

Since m ( B ( θ, R )) > 0 , dividing by the Lebesgue measure of the ball and taking the limit yields:

Since d x ( θ ) = 0 a.e., it follows that p ( θ | x ) = q ( θ | x ) almost everywhere in Θ .

B.2 Proof of Theorem 1

Let B ( θ, R ) be a d φ -ball with radius R . Assume the doubling property: ∃ C ≥ 1 such that

and that ball boundaries have m -measure zero.

Let X be an open subset of R k with Borel measure µ . Suppose

p : X → (0 , ∞ ) is a continuous probability density;

( x, θ ) ↦→ p ( θ | x ) and ( x, θ ) ↦→ q ( θ | x ) are continuous in x and belong to L 1 loc (Θ) for every x ;

for every measurable function θ l : X → Θ and every R > 0 ,

Then we show that p ( θ | x ) = q ( θ | x ) for µ ⊗ m -a.e. ( x, θ ) .

Proof. The proof proceeds in two main steps. First, we fix R > 0 and use a measurable selection argument to show that the inner integral is zero for almost every x . Second, we apply the Lebesgue Differentiation Theorem to show that the integrand itself must be zero almost everywhere.

Step 1: Show ball integrals are zero for a.e. x . Let d x ( θ ) = p ( θ | x ) -q ( θ | x ) . For a fixed R > 0 , define the ball integral

The function F R ( x, θ ) is a Carathéodory function: that is, it is measurable in x for each fixed θ (by Fubini's theorem, as d x is continuous in x ) and continuous in θ for each fixed x (by the Dominated Convergence Theorem, as d x ∈ L 1 loc and m ( ∂B ( θ, R )) = 0 ).

Define the pointwise suprema:

Since Θ ⊂ R D is separable, let D = { ϑ k } k ≥ 1 be a fixed countable dense subset. Because θ ↦→ F R ( x, θ ) is continuous and D is dense, the suprema over Θ are equal to the suprema over D . Specifically, let ˜ S + R ( x ) := sup k ≥ 1 F R ( x, ϑ k ) and ˜ S -R ( x ) := sup k ≥ 1 ( -F R ( x, ϑ k )) . These are measurable functions, since they are countable suprema of measurable functions, and we have S + R ( x ) = ˜ S + R ( x ) and S -R ( x ) = ˜ S -R ( x ) for all x .

Now, we construct measurable ε -maximizing selectors. Fix n ∈ N . Define:

These minima are well-defined and finite. By the definition of the supremum, for any ε = 1 /n > 0 , the set of indices k satisfying the condition is guaranteed to be non-empty. By the well-ordering principle, a non-empty subset of N has a minimum. The maps x ↦→ k ± n ( x ) are measurable, as the sets { x : k + n ( x ) = k } are formed by measurable comparisons. Thus, the selectors θ + R,n ( x ) := ϑ k + n ( x ) and θ -R,n ( x ) := ϑ k -n ( x ) are well defined and measurable.

By construction, these selectors satisfy:

Applying the theorem's hypothesis with θ l = θ + R,n gives:

Similarly, applying the hypothesis with θ l = θ -R,n gives:

Since G R ( x ) ≤ S + R ( x ) + S -R ( x ) = ˜ S + R ( x ) + ˜ S -R ( x ) , we have

This must hold for all n , and so we must have ∫ X p ( x ) G R ( x ) dx = 0 . Since p ( x ) > 0 and G R ( x ) ≥ 0 , we must therefore have G R ( x ) = 0 for p -a.e. x . This means that for p -a.e. x , we have F R ( x, θ ) = 0 for all θ ∈ Θ .

Step 2: Apply the Lebesgue Differentiation Theorem. From Step 1, we know there is a set X 0 ⊂ X with p ( X 0 ) = 1 such that for any x ∈ X 0 , ∫ B ( θ,R ) d x ( θ ′ ) dθ ′ = 0 for all R > 0 and all θ ∈ Θ .

The Lebesgue Differentiation Theorem for doubling spaces states that for any function f ∈ L 1 loc (Θ , m ) ,

For any fixed x ∈ X 0 , the function θ ↦→ d x ( θ ) is in L 1 loc (Θ) by hypothesis. Applying the theorem gives:

for m -a.e. θ ∈ Θ .

Since this holds for every x ∈ X 0 where µ ( X \ X 0 ) = 0 (as p is a density for µ ), we have d x ( θ ) = 0 for µ ⊗ m -a.e. ( x, θ ) . This means p ( θ | x ) = q ( θ | x ) for µ ⊗ m -a.e. ( x, θ ) .

Remarks. We conclude with two remarks.

Necessity of quantifying over all measurable selectors. The assumption that the inner integral vanishes for every measurable selector θ l : X → Θ is crucial. If the condition held only for constant maps θ l ( x ) ≡ θ 0 , it would assert only that the x -average of the localized integrals vanishes:

This would allow for cancellations across x and would not imply that the inner integral vanishes pointwise in x . The logic of the proof requires the freedom to vary the center of the ball adaptively with x to prevent these cancellations.

Continuity in x can be relaxed. The theorem and proof remain valid if the maps x ↦→ p ( θ | x ) and x ↦→ q ( θ | x ) are merely measurable rather than continuous, provided that the resulting function F R ( x, θ ) = ∫ B ( θ,R ) d x ( θ ′ ) dθ ′ is a Carathéodory function (measurable in x , continuous in θ ). This condition holds under weaker assumptions than continuity, such as joint measurability of ( x, θ ) ↦→ d x ( θ ) . The continuity-inx assumption is a straightforward condition that guarantees this property, which is all that is needed for the measurable selection argument to succeed. Moreover, the measurability of the set N = { ( x, θ ) : x ∈ X 0 , θ ∈ Θ c x } , where Θ c x is the set of non-Lebesgue points for the function d x ( θ ) = d ( x, θ ) , follows from the joint measurability of d x ( θ ) .

B.3 Proof of Theorem 2

We first need the following lemma.

Lemma 3. Let (Θ , d φ ) be a metric space, and let θ l ∈ Θ be fixed. Define the function

Now let q be a probability measure on Θ . For any θ ∗ ∈ Θ , define the ball probability rank of θ /star under q as

Then, if we also have that θ ∗ ∼ q , then the random variable U q ( θ ∗ ) is distributed as Uniform (0 , 1) , i.e.,

Proof. Define F R as the cumulative distribution function (CDF) of the random variable R q ( θ ) = d ( θ l , θ ) , where θ ∼ q , i.e.,

By definition of U ( θ ∗ ) , we have

But by assumption, we have θ /star ∼ q . Accordingly, R ( θ ∗ ) is itself a random variable drawn from the distribution whose CDF is F R , it follows from the probability integral transform that for any localization point θ l ,

Thus, U q ( θ ∗ ) ∼ Uniform (0 , 1) , completing the proof.

The key observation from Lemma 3 is that the probability mass assigned by p to the ball of this radius, centered at θ l , follows a uniform distribution when θ ∗ ∼ p . Thus, if equation (3) (which states that U ( θ ∗ ) ∼ Uniform (0 , 1) ) holds for all possible choices of θ l , then the conditional distributions p ( θ | x ) and q ( θ | x ) must be identical. Intuitively, this is because the process of drawing θ ∗ and measuring probability mass within its corresponding ball implicitly tests equality of mass across all possible radii in a structured way. If the distributions p and q were different, there would exist some localization point θ l where the resulting uniformity condition fails, revealing a discrepancy in their induced probability measures.

With this lemma in place, we can now prove Theorem 2.

Proof. ( ⇒ ) Suppose that p ( B r ) = q ( B r ) for all r ≥ 0 . Consider the cumulative distribution function (CDF) of the distance variable R ( θ ∗ ) = d ( θ l , θ ∗ ) , when θ ∗ ∼ p :

Similarly, under θ ∗ ∼ q , the corresponding CDF is

By assumption, these two CDFs are identical, i.e., F p ( r ) = F q ( r ) for all r . Now, by the definition of U q ( θ ∗ ) ,

Since F q = F p , we obtain

From Lemma 3, we know that F p ( R ( θ ∗ )) ∼ Uniform (0 , 1) when θ ∗ ∼ p , which implies that U q ( θ ∗ ) ∼ Uniform (0 , 1) under θ ∗ ∼ p . Thus, the distributions of U q ( θ ∗ ) under p and q must be identical.

( ⇐ ) Now suppose that U q ( θ ∗ ) ∼ U p ( θ ∗ ) . Then, for any u ∈ [0 , 1] ,

Rewriting in terms of the CDFs, this implies

By the probability integral transform, since F q ( R ( θ ∗ )) ∼ Uniform (0 , 1) under both p and q , it follows that F q ( R ( θ ∗ )) = F p ( R ( θ ∗ )) in distribution. This means that F p = F q , implying

Thus, the probability assigned to each metric ball is identical under p and q .

B.4 Proof of Theorem 3

From Lemma 3, we have that U q ( θ ∗ | x ) follows a uniform distribution when θ ∗ ∼ q ( θ ∗ | x ) , thus satisfying

Next, define the radius R θ l ( x ) ( α ) as follows

Since q ( θ | x ) is absolutely continuous with respect to the Lebesgue measure, the mapping α ↦→ R θ l ( x ) ( α ) is a bijection. Additionally, by definition of U q ( θ ∗ | x ) , it follows that

Combining these observations, we obtain:

B.5 Proof of Lemma 1

Proof. Let u = ‖ θ -θ ′ ‖ 2 2 σ 2 2 and v = ‖ φ ( θ ) -φ ( θ ′ ) ‖ 2 2 σ 2 1 . Since θ, θ ′ ∈ R D , u ≥ 0 and v ≥ 0 . The squared distance is given by

Proof of the Global Upper Bound We can rewrite the expression for the squared distance as:

For any z ≥ 0 , the standard inequality 1 -e -z ≤ z holds. Furthermore, since /epsilon1 ∈ (0 , 1) , e -u > 0 , and 1 -e -v ≥ 0 , the second term is non-positive. We can therefore bound the expression:

Substituting the definitions of u and v yields:

By the Lipschitz assumption on φ , we have ‖ φ ( θ ) -φ ( θ ′ ) ‖ ≤ L 2 ‖ θ -θ ′ ‖ , which implies:

Taking the square root provides the global upper bound with the constant C 2 = √ 1 σ 2 2 + L 2 σ 2 1 .

Proof of the Local Lower Bound We rewrite the term 1 -k ξ ( θ, θ ′ ) :

Since 1 -/epsilon1 > 0 , e -u > 0 , and 1 -e -v ≥ 0 , the second term is non-negative. This allows us to bound the expression below:

This provides a lower bound on the squared distance:

To establish a linear relationship between d φ ( θ, θ ′ ) and ‖ θ -θ ′ ‖ , we analyze the function g ( z ) = 1 -e -z z for z > 0 . By L'Hôpital's rule, lim z → 0 + g ( z ) = 1 . The function g ( z ) is continuous and strictly positive on any compact interval [0 , Z max ] . By the Extreme Value Theorem, it must attain a minimum value c min > 0 on this interval. Therefore, for all z ∈ (0 , Z max ] , the inequality 1 -e -z ≥ c min · z holds.

Let us restrict our domain to a bounded set where

‖

θ

-

θ

′

M

for some constant

M >

0

. This implies that u = ‖ θ -θ ′ ‖ 2 2 σ 2 ≤ M 2 2 σ 2 = Z max. On this domain, we can apply the linear inequality derived above:

Taking the square root gives the local lower bound d φ ( θ, θ ′ ) ≥ C 1 ,M ‖ θ -θ ′ ‖ with the constant C 1 ,M = √ c min σ 2 , which depends on M through c min.

This yields the local-doubling condition with constant C M = ( C 2 C 1 ,M ) D , completing the proof.

C Experiments Details

C.1 Toy Example

We construct a synthetic ground-truth data distribution in R 2 by defining a Gaussian mixture model (GMM) whose components are procedurally placed according to a recursive branching process. We use the code from paper [11] whose generation process can be found in https://github.com/ NVlabs/edm2/blob/main/toy_example.py . To maintain the completeness of our paper, we include the generation process here.

Gaussian Mixture Representation

The base distribution is modeled as a weighted sum of multivariate Gaussian components:

φ k ∈ R + are normalized mixture weights,

µ k ∈ R 2 are the component means,

Σ k ∈ R 2 × 2 are the component covariance matrices,

σ ∈ R + where we set σ = 1e -2 for p ( θ | x ) and (1 + α ) · σ for q ( θ | x ) .

Each component is assigned a weight and covariance that decays with tree depth, producing finer-scale detail at 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 at the same root origin and with distinct initial angles (e.g., π/ 4 and 5 π/ 4 ).

At each recursion level (up to depth 7), a branch is extended in a given direction, and eight Gaussian components are placed uniformly along the branch.

where:

Each component's mean is computed by interpolating along the current direction vector, and the covariance is anisotropically scaled to align with the branch'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:

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 on the experiments in Section 4. For these benchmarks, we construct a ground-truth conditional distribution by first simulating latent Gaussian variables with x -dependent means and variances:

where ˜ θ ∈ R s , W 1 ∈ R s × m , and W 2 ∈ R s × 1 are fixed weight matrices with standard normal entries. The matrix Σ ∈ R s × s is a fixed Toeplitz correlation matrix, with entries Σ ij = corr | i -j | and corr = 0 . 9 to simulate strong structured correlations.

In all cases we sample conditioning inputs x ∼ N ( 1 m , I m ) .

C.3 Diffusion Training and Sampling Procedure

Diffusion Model To further evaluate our method in the context of generative posterior estimation, we construct a benchmark based on a diffusion model. We begin by sampling x ∼ Unif [1 . 5 π, 4 . 5 π ] and defining θ = ( x cos( x ) , x sin( x )) , which induces a highly nonlinear and non-Gaussian posterior structure. We then train a diffusion model to approximate this distribution. During evaluation, we generate samples via reverse diffusion and treat the output after 20 reverse steps as a notional groundtruth posterior (Figure 4a)-not because it is the true target distribution, but because it represents the best available approximation produced by the model. Outputs from earlier steps ( 20 -α ) serve as degraded approximations to this endpoint. This yields a generative-model-based, monotonic perturbation scheme parameterized by α (Figures 4b and 4c).

Beyond simple rejection, our method provides fine-grained quantitative insight: we use the test statistic t from Algorithm 2 to measure how close each approximate posterior (from fewer reverse steps) is to the reference posterior (20-step output). This allows us to quantify posterior degradation as a function of reverse diffusion progress. As shown in Figure 4d, the test statistic increases monotonically with α , reflecting growing divergence from the true posterior.

As described in Section 4, we sample x ∼ Unif [1 . 5 π, 4 . 5 π ] and define θ = ( x cos( x ) , x sin( x )) . This produces a nontrivial two-dimensional manifold for posterior inference.

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We generate 32,768 samples to train a diffusion model using 50,000 training epochs and a learning rate of 1 × 10 -4 . The diffusion model is trained and sampled following the implementation provided in EDM [19].

For posterior approximation, we set the total number of reverse diffusion steps to 20, treating the output at step 20 as the ground-truth posterior. To construct approximate posteriors at varying levels of fidelity, we also record intermediate samples from reverse steps 15 through 19. These intermediate outputs serve as 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( x ),dim( θ )): (3 , 3) , (10 , 10) , (50 , 10) , and (100 , 100) . We denote N as the number of sample pairs { ( θ i , x i ) } drawn from the true joint distribution p ( θ | x ) p ( x ) , and K as the number of samples drawn from the estimated posterior q ( θ | x ) for each conditioning value x .

In the CoLT Full setting, we utilize a distance embedding network φ with input dimension equal to dim( θ ) and output dimension set to dim( θ ) . Although alternative output dimensions for φ may potentially improve performance, we fix the output dimension to dim( θ ) to avoid additional hyperparameter tuning and ensure a fair comparison across settings.

All neural networks in our method (including φ , θ l 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 α tested for each experiment type, along with the sampling and training hyperparameters for both CoLT and C2ST.

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 to increase the curvature of θ . The network consists of a fully connected layer (' torch.nn.Linear ') with input dimension equal to the dimension of θ , a hidden layer of

size 128, followed by an element-wise sine activation, and another linear layer mapping from 128 to the original dimension of θ . The weights of the linear layers are initialized using PyTorch's default random initialization.

To compute the curvature, we 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 a straight line, with a 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 θ space, allowing us to study the performance of methods under non-Euclidean geometries.

D Additional Experimental Results

D.1 Tree Task

We present additional visualizations for the toy tree-structured posterior under various levels of perturbation α . As α increases, the sampled points become increasingly dispersed and less concentrated around the underlying structure. The shaded region indicates the true posterior manifold corresponding to α = 0 .

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D.2 Perturbation

In this section, we present additional results across varying dimensions of ( x, θ ) for different tasks, evaluated under multiple perturbation magnitudes α in the non-curvature setting. The results demonstrate the robustness of CoLT and baseline methods under a 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 x entirely: q ( θ | x ) = p ( θ ) , i.e., the posterior estimate is simply the prior distribution of θ . This scenario serves as an important pathological case, as it has been shown to cause both SBC and TARP (with random reference points) to fail-these methods are unable to detect the distributional discrepancy between q ( θ | x ) and the true posterior p ( θ | 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 ( θ | x ) = p ( θ ) . This case is particularly challenging, as both TARP and SBC fail to detect the resulting distributional discrepancy.

In contrast, our proposed methods-CoLT ID and CoLT Full-successfully detect this violation 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.

In addition to reporting the statistical power of each method in Table 3, we provide their corresponding Type I error rates in Table 4. Since the p -value threshold is set to 0.05, all methods successfully control the Type I error within the expected range, indicating that none falsely reject the null hypothesis under the correctly specified posterior.

D.4 Diffusion Sampling Results

In this section, we provide additional visualizations related to the diffusion-based posterior approximation. Figure 7 illustrates the underlying data manifold used to train the diffusion model, as 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 a power and Type I error curve that quantifies how the performance of our method changes with respect to the number of reverse steps. As expected, the statistical power increases as the number of reverse steps approaches 20, while Type I error remains well-controlled.

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In this section, we 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 a more comprehensive understanding of the method's performance across the design space.

Model Capacity. As long as the localization and embedding networks possess sufficient representational 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.

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To further assess the effect of network depth, we conducted an ablation study under mean-shift perturbations with α = 0 . 2 and α = 0 . 3 . The results, presented in Table 5, show that reducing the number of layers slightly weakens statistical power, while deeper networks yield marginal gains. Nevertheless, the overall performance remains within a similar range, suggesting diminishing returns beyond moderate depth. Although a full architecture search is beyond our current scope, we expect further improvements with more refined architectural design.

Table 5: Effect of MLP depth on the CoLT-Full performance under mean-shift perturbation with different α . 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, as shown in Table 6, in the kurtosis adjustment task under t -distribution perturbations with α = 0 . 2 and α = 0 . 3 , the Sinkhorn divergence achieves the best performance. We attribute this to its smoother loss landscape, which facilitates optimization [16].

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe 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.\n\nSTRENGTHS AND WEAKNESSES:\nWhile 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\n\nQUESTIONS:\nWhat 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.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-30"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe 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.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\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.\n\nQUESTIONS:\n- 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.\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe 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.\n\nSTRENGTHS AND WEAKNESSES:\n**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.\n\nQUESTIONS:\n1. 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?\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 4, "DATE": "2025-07-07"}