Density Ratio Estimation with Conditional Probability Paths
Authors: Hanlin Yu
Arto Klami
Aapo Hyvarinen
Anna Korba
Omar Chehab
Venue: ICML 2025 poster (2025)
Keywords: score matching
density ratio estimation
probability paths
Primary area: probabilistic_methods

Abstract: Density ratio estimation in high dimensions can be reframed as integrating a certain quantity, the time score, over probability paths which interpolate between the two densities. In practice, the time score has to be estimated based on samples from the two densities. However, existing methods for this problem remain computationally expensive and can yield inaccurate estimates. Inspired by recent advances in generative modeling, we introduce a novel framework for time score estimation, based on a conditioning variable. Choosing the conditioning variable judiciously enables a closed-form objective function. We demonstrate that, compared to previous approaches, our approach results in faster learning of the time score and competitive or better estimation accuracies of the density ratio on challenging tasks. Furthermore, we establish theoretical guarantees on the error of the estimated density ratio.

Density Ratio Estimation with Conditional Probability Paths
Hanlin Yu Arto Klami'
Aapo Hyvärinen
Anna Korba²
Omar Chehab
Abstract
Density ratio estimation in high dimensions can be reframed as integrating certain quantity, the time score, over probability paths which interpo- late between the two densities. In practice, the time score has be estimated based on samples from the two densities. However, existing meth- ods for this problem remain computationally ex- pensive and can yield inaccurate estimates, In- spired by recent advances in generative modeling, we introduce novel framework for time score es- timation, based on conditioning variable. Choos- ing the conditioning variable judiciously enables closed-form objective function. We demonstrate that, compared to previous approaches, our ap- proach results in faster learning of the time score and competitive or better estimation accuracies of the density ratio on challenging tasks. Further- more, we establish theoretical guarantees on the error of the estimated density ratio.
Figure 1: Densities are shown in blue. Left: bi-modal probability path transitioning from Gaussian distribution (t 0) to a mixture of Diracs = 1). This path is esti- mated using "time scores", which are not available in closed form in general: they are depicted by arrows, with magni- tudes ranging from low (gray) to high (red). Right: A useful decomposition of the probability path and time scores is obtained by conditioning on final data point. The ensuing conditional density Gaussian, and thus, the ensuing con- ditional time scores analytically tractable. We propose to use this decomposition to estimate the "time scores".
1. Introduction
Estimating the ratio of two densities is a fundamental task in machine learning, with diverse applications (Sugiyama ett al. 2010). For instance, by assuming that one of the densities is tractable, often a standard Gaussian, we can construct an estimator for the other density by estimating their ratio (Gutmann & Hyvärinen| 2012] Gao et al., 2019 Rhodes et al.) 2020 Choi et al., 2022). It also possible to consider scenario where both densities not tractable. As noted by previous works (Choi al. 2022). density ratio estimation finds broad applications across machine learning, from mutual information estimation (Song & Er-
The seminal work by Gutmann & Hyvärinen (2012) pro- posed learning objective for estimating the ratio of two densities, by identifying from which density a sample is drawn. This can be done by binary classification. However, their estimator has a high variance when the densities have little overlap, which makes impractical for problems in high dimensions (Lee et al. 2023
Chehab al. 2023b).
To address this issue, Rhodes et al. (0200 proposed connect- ing the two densities with a probability path and estimating density ratios between consecutive distributions. Since two consecutive distributions are "close" to each other, the sta- tistical efficiency may improve at the cost of increased com- putation, as there are multiple binary classification tasks to solve. Choi et al. (2022) examined the limiting case where the intermediate distributions become infinitesimally close. In this limit, the density ratio converges to quantity known as the time score, which is learnt by optimizing a Time Score Matching (TSM) objective. While this limiting case leads to empirical improvements, the TSM objective is computationally inefficient to optimize, and the resulting
mon 2020), generative modelling (Goodfellow et al. 2020) importance sampling (Sinha et al.] 2020)) likelihood-fii inference (Izbicki et al. 2014) to domain adaptation (Wang
al. 2023).
'University of Helsinki, Finland FENSAE, CREST, IP Paris, France. Correspondence to: Hanlin Yu <hanlin yu@ helsinki.fi>.
Proceedings of the ndnd International Conference Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

---

Density Ratio Estimation with Conditional Probability Paths
estimator may be inaccurate. Moreover, it is unclear what are the theoretical guarantees associated with the estimators.
Continuous label More recent developments relax the label that it is continuous t E [0, Now, conditioning on defines intermediate distributions p(x [t), equivalentyy noted pa (x),, along a probability path that connects po to PL
In this work, we address these limitations. First, in Section [ we introduce a novel learning objective for the time score, which we call Conditional Time Score Matching (CTSM). is based on recent advancements in generative modeling [Vincent 2011 Pooladian et al., 2023 Tong et al., 2024a), which consider probability paths that are explicitly decom- posed into mixtures of simpler paths, and where the time score is obtained in closed form. We demonstrate empiri- cally that the CTSM objective significantly accelerates opti- mization in high-dimensionn settings, and is several times faster compared to TSM.
Then, the following identity is used (Choi et al. 2022)
(x() = log
0, log Pa (x)dtt,
Px(x)
or its discretization in time (Rhodes et al. 2020).
Probability path We next consider a popular use-case, where po a Gaussian and pi is the data density (Rhodes) et al. 2020 Choi et al. 2022); since po is known analyt- ically, the ratio of the two provides directly an estimator for P1- In practice, one can construct probability path where the intermediate distributions can be sampled from but their densities cannot be evaluated. This is because the probability path is defined by interpolating samples from Po and p1. There are multiple ways to define such interpo- lations (Rhodes et al. 2020 Albergo & Vanden-Eijndee 2023), which we will further discuss in Section 4 A widely used approach is the Kariance-Pressrvvngg (VP) probability path, which can be simulated by (Song et al. 202 lb Lipman
Second, in Section 4 we modify our CTSM objective with number of techniques that are popular in generative model- ing (Song et al.) 2b Choi al. 2022 Tong et al. (224a) to ease the learning. In particular, we derive a closed-fome weighting function for the objective, as well as vectorized version of the objective which we call Vectorized Condi- tional Time Score Matching (CTSM-v). Together, these modifications substantially improve the estimation of the density-ratio in high dimensions, leading stable estima- tors and significant speedups.
et al. 2023 Choi et al. 2022)
Third, in Section [ we provide theoretical guarantees for density ratio estimation using probability paths, addressing a gap in prior works (Rhodes et al. 2020 Choi et al. 2022).
x
of x1 + V - af Xo.
(3)
where Xy ~ (O., X1 ~ pr follows the data distribution,
2. Background
time is drawn uniformly L~ 7[00, 1] and ar E [0, 1] is a
positive function that increases from to 1. By conditioning on t, we obtain densities pt (x(
Our goal is to estimate the ratio between two densities Po and pl. given samples from both. We start by defining distribution over labels t and data points x.
pol Vi-o? vi-oz Vi-0;
- Pi ( that cannot be computed in closed-form, given that the density P1 is unknown and that the convolution (1)
requires solving a difficult integral.
f(x,t)= = p(t)p(x(1)
Estimating the time score Importantly, the identity in Eq. 2 requires estimating the time score O log pe (x),
constructed such that we recover Po and p1 for t = 0 and t = 1 respectively. We next show how several relevant methods can be viewed as variations on this formalism.
which is the Fisher score where the parameter is the label t. It can also be related the binary classifier between two infinitesimally close distributions pt and pt+ds (Choil et al. 2022 Proposition 3). Formally, this time score can be approximated by minimizing the following Time Score Matching (TSM) objective
Binary label Fundamental approaches to density-ratie es- timation consider a binary label t E {0, 1). Among them, Noise Contrastive Estimation (NCE) is based on the obser- vation that the density ratio is related to the binary classifier p([[æ) (Gutmann & Hyvärinen| 2012 Eq. 5). NCE esti- mates that classifier by minimizing& a binary classification loss based on logistic regression, computed using samples drawn from Pu and P1. In practice, using NCE is challeng- ing when Po and pı are "far apart". In that case, both the binary classification loss becomes harder to optimize (Liu) et al. 2022) and the
LTSM (0) Ep(t,x) [A(t log (xx) so(x, +)) (4)
where X(t) is any positive weighting function. This objec- tive requires evaluating the time score ò log px)). How- ever, as previously explained, the formula for the time score is unavailable because the densities Pt, while dell-defined, are not known in closed form.
of its minimizer de- terioriates (Gutmann & Hyvärimen| 2012
Lee et al. 2023
To make the learning objective in Eq. 4 tractable, an insight from Hyvärinen (005) led Choi al. 2022); Williams et al.
Chehab et al. 2023a b).
2

---

Density Ratio Estimation with Conditional Probability Paths
(2025) to rewrite it using integration by parts. This yields
Note that the regression target is given by the time score of the conditional distribution, O log Px 2). The reformula- tion is justified by the following theorem:
LTSM (0) px (x) (x, 0)] - 2B (x) (x, +)}+ Ep(tx) (20, 8) + (x t) + (xt)so(x, t)?)
(5)
Theorem (Regressing the time score) The TSM loss Eq. 1
which no longer requires evaluating the time score O, Po (x). However, this approach has one clear com- putational drawback: differentiating the term (x t) in the loss Eq. 5 involves using automatic differentiation twice first in and then in 0 which can be conssmnnnn (we verify this in Section 6 This motivates us to find better ways of learning the time score.
and CTSM loss Eq. are equal, up to an additive constant.
The proof can be found in Appendix D.2 This new ob- jective is useful, as it requires evaluating the time score of the tractable distribution Pe pe(x 22) instead of the intractable distribution pr (x). By minimizing this objective, the model so t) learns to output O, log p. (x). A similar observation was made in De Bortoli et al. (2022 Appendix I.3.), how- ever they did not translate this observation into the CTSM objective and use it for learning. Furthermore, their set- ting was more restrictive, as the conditioning variable was specifically chosen to be X1.
3. Novel Objectives for Time Score Estimation
In this section, we propose novel methods to estimate the time score.
3.1. Basic Method
3.2. Vectorized Variant
Augmenting the state space First, we rewrite Eq. Aso that is tractable. The idea to further augment the state space to (x, t, z) by introducing a conditioning variable z, as in related literature. Thus, we extend the model from Eq.
We propose a further objective for learning the time score, called Vectorized Conditional Time Score Matching (CTSM- v). The idea is that we can easily vectorize the learning task, by forming joint objective over the D dimensions. The intuition is that the time score can be written as sum of autoregressive terms, and that we learn each term of the sum instead of the final result only. We verify in section 6 that this approach empirically leads to better performance. Formally, define the vectorization of the conditional time score as the result of stacking its components as
into
p(x, t, = p())p(()ppx)))))))) (t, z),
(6)
such that the intermediate distributions p(x [4, 2) now conditioned on z - can be sampled from and evaluated. We remark that this insight is shared by previous research in score matching Vincent (2011) and flow matching (Lipman Pooladian et al. 2023 Tong et al. 2024a).
et al. 2023
vec(O, log (x 2)) = [0e log x** > (9)
Consider for example Eq. 3 By choosing to condition on
The time score is then obtained by summing these compo- nents. Our vectorized objective is given by
z = X1. we get a flomed-form p(x [t, z) = NN(x; a z, (1 aT) In this example, z is a sample of "raw" data (for ex- ample, real observed data) while is a corrupted version of data, and / controls the corruption level, ranging from 0 (full corrpttion) to 1 (no corruption), as in Vincent I201I). In the following, we explain how to relate the descriptions of the
(CIsM.(()) Ep(t,x.x) [A(t)]lvec(d, log (x so (x, [)]] (10)
intractable marginal probability path (x) to descriptions of the tractable conditional probability path pt px(x (2).
Theorem 2 (Regressing the vectorized time score) The CTSM-v objective Eq. [0] is minimized when the sum of the entries of the score network equals the time score.
Tractable objective for learning the time score As a result of Eq. 6 we relate the time scores, obtained with and
This is proven in Appendix D.2
By minimizing
without conditioning on (derivations are in Appendix D.I
this objective, the model sg (x, t) learns to output [Ep.(x/x))0 log lo tified in the next Theorem 3 The original time score can be obtained from the learnt vec (x. t) by summing all the entries. Further, while the components of the regression target are formally given by [a, log (c x 2)),T([.o)) for commonly used probability paths like the VP path, the dependency on x is dropped. We remark that Meng et al. (2020) proposed autoregressive score matching which shares similar spirit, albeit for the purpose of training scalable au- toregressive models and based on Stein score.
2))l ke |ig this is further jus-
Bo = Ex,(x x) [ log Pt (x| z)]
(7)
and exploit this identity to learn the time score, by plug- ging Eq. intoo the original loss in Eq. 4 This way, we can reformulate the intractable objective in Eq. iinto a tractable objective which we call the Conditional Time Score Match- ing (CTSM) objective
LCTSM(0) (0) = Fxx.....)) (O) logpe(x(2)) so(x.)))
3

---

Density Ratio Estimation with Conditional Probability Paths
3.3. General Framework
we consider two design choices for both of these learning objectives the conditional probability path pc(x 2) and the weighting function X(t).
We next show that our learning objectives, i.e., both the con- ditional time score matching one and the vectorized variant, are actually special cases of more general framework.
Choice of probability path Our regression objectives require computing the time score and its vectorization of a conditional density that is analytically known. One natural choice is a Gaussian (r 2) = ~x(x; [2,(2), ki 1) [Lipman) et al. 2023). so that the conditional time score is obtained in closed form. We specify popular choices of z. (2( hi
Just as we related the marginal and conditional time scores, O, logpt Pt. (x) and Or log pa (x(x 2) in Eq. 7 let us now con- sider the same identity for general, vector or scalar valued functions gxx, t) and f(x, 1, z), where t [0, 1]
g(x, 1) = [x) [[(x (,
(11)
in Appendix [
By analogy to previous paragraphs, we call the functions g and f, "marginal" and "conditional"" We consider the sce- nario where g(x. t) is intractable, yet f(x, t, z) is tractable. Similarly, we obtain theorem that states that regression problem over the "marginal" function g(x,t) t) can be re- formulated as a regression problem over the "conditional" function fxx, z), thus resulting in a tractable training ob- jective.
In particular, previous works on density ratio estima- tion [hodes et al. (0200) Choi et al. (2022) focused on
the VP probability path Eq. 3 which is also popular in the
literature of diffusion models (Sohl-Dickstein et al., [015]
Ho et al. 2020] Song et al.) (021b). By conditioning Eq.
X1, we obtain the conditional densities
[x z) = N (1 (L
(14)
Theorem 3 (Regressing a function) Consider vector or scalar valued functions f(x, t| z) and g(x.t) = Epttz (ootz\x)[f(f) [x) 1) 2)]. Then, the following two loss functions are equal up to an additive constant that does not depend on 0:
The conditional time score is
O₂ lo (x
D - a
||c||
(15)
1 oz
cl x1x1,
(16)
VI af
-00-ssssxx.0 (r(0)= = Ep(t.xx) [A(t)]((xx f|z) - (x( -x
(12)
where e
X-axx: Vi-a3
Finally, the vectorized conditional time
(0) = Ep(t,)) A(t)lg((x)) 1)- - so(x, {
(13)
score is
We prove this result in Appendix D.2 Our Theorem I a special case when f(x, 1 z) = o, lo 2) and g(x,t) t) O, log (x). Similarly, our Theorem - is
vec (oglog((x = 0:04
araz
(17)
1 a
1
E ea, X1,
(18)
special case when f(x, l|2) = ve po 2)) and g(x, t) (x|x) (x) [f(x t)].
of
where the square and the product are element-wise opera- tions. More discussion can be found in Appendix B.I
Versions of Theorem 3 appear multiple times in the liter- ature, yet they have always been stated for specific fune-
tions g that are Stein scores 0z log (x) (Vincent 2011
Choice of weighting function The cost function in 8 com- bines multiple regression tasks, indexed by t. into a single objective, representing multi-task learning problem. A practical challenge is determining how to weigh the differ- ent tasks (Ruder 2017 Rhodes et al. 2020).
Song et al. 2021b) or velocities that generate the proba-
bility path (Lipman et al.) 2023] Pooladian et al. 2023
Tong et al.
2024a). For example, in |Vincett (2011)
f(x, { 2) 0x log le (z) and p(t) is a Dirac. In Tong
et al. 2024a), f(x,t) z) = (x(x|22 22) which is a velocity such that the solution to the ordinary differential equation x; = ve(x 22) has marginals (x( z). To our knowledge, has not been stated for general functions, whose output may have any dimensionality, nor has it been applied time scores or vectorized time scores, we do.
Some approaches estimate a weighting function during train- ing (Kendall et al.; 2017: Nichol & Dhariwal 2021 Choi et al. proximation which does not depend on the parameter (Song) et al. 2021b Tong et al.) 2024b). We follow the latter approach and draw inspiration from the diffusion models literature (Ho et al., 2020 Song et al., 2021b), where it is common to choose as weighting function
2022 Kingma & Gao 2023). while others use an ap-
4. Design Choices
In the previous section, we derived two novel and tractable learning objectives for the density ratio of two distribu-
X(t) x E Ex(x,) [110x log pt |x|1))))
(19)
tions, CTSM Eq. X and CTSM-v Eq. 10 In this section,

---

Density Ratio Estimation with Conditional Probability Paths
which is also the default weighting scheme from Choi et al. (022). It was derived for estimating the Stein score Oge log Px z) (Song et al. 2021b), and we refer to this weighting scheme as Stein score normalization. We show in Appendix
5. Theoretical Guarantees
In this section, we provide theoretical guarantees on the density estimated by CTSM or CTSM-v. All proofs are included in Appendix D
B that it simplifies to A(t) x kip
In practice, we can approximate Eq. 2 as
However, as the name and the equation itself suggest, Stein score normalization is derived based on Stein score, thus not directly relating to the time score. One benefit of Stein score normalization is that its scaling essentially results in the regression targets having unit variances (Ho et al. 2020). However, the variance of the time score does not equal to the variance of the Stein score. We instead consider
K
1 log Pi (x) = K ==1
8(x, Σ
+ log
(23)
introducing two sources of error, namely the error due to discretizing the integral with K steps and the error due to using the approximate time score 8(x, t). We quantify these errors in the following theorem. We focus on KI. divergence for convenience of the derivations, and remark that the same proof can be used to bound the error between the learned and true density ratios.
X(t) X Eptx..) (0g lo
(20)
for CTSM and CTSM-v. This new weighting, which we call time score normalization, keeps the regressands roughly equal in magnitude. We explicitly compute this novel weighting function in Appendix B its formula depends on a quantity e that is a function of the data distribution's 's mean and variance. A natural choice for c is to compute these statistics from the data, but in our experiments, setting c = often yields better results. In our initial experiments, we found that using the time score normalization was important to achieve stable training. We remark that it is possible to apply time score normalization Eq. t0 CTSM-v as well: upon assuming each dimensionality having equal scales, one can calculate the variances of the objective in each indi- vidual dimension and employ the same weighting scheme.
Theorem 4 (General error bound) Denote by P1 and P1 the densities obtained from Eq. 2 and Eq. 23 using the true and approximate time scores, s(x, t) == Oe log Pe Pc(x) and $(x, t) respectively. Assume that the correct time score evolves smoothly with time, specifically i + s(x, 1) is L(x)- Lipschitz. Denote as follows the discretied distribu- tion pt( k Ei=1 St, (t). The error between the two distributions p1 and P1 is bounded as
KL(PI, P1) S 2K² (x) (Lx))
(24)
$(x, ))
The first term quantifies a discretization error of the integral: it is null when using discretization steps K + oo, or when using paths whose time-evolution / + p(x, I) is smooth, even stationary L(x) + 0 for any point x E E where the density is evaluated. Comparing the constants L(x) of different probability paths is left for future work.
For the specific case of the VP path Eq. 3 the time score normalization can be defined as
ag)² X(()= 2a (at)² + (a()² al
(21)
The second term in Eq. 24 quantifies the estimation error of the time score, collected over the times t; where it is evaluated. While such an estimation error is assumed to be constant in related works (De Bortoli et al., 2022), we specify it for both CTSM and CTSM-v in our next result. Proposition 5 (Error bound for CTSM and CTSM-v) Now consider a parametric model for the time score, se(x, t). Denote by 9 * the parameter for the actual time score O log pt(x), obtained by minimizing the loss from Eq. X Denote by @ the parameter obtained from minimizing that same loss when the expectation is approximated using a finite sample (x,, 2,,44)se[,,N) Then, the expected error over all estimates pi, obtained by integrating the estimated score sg(x, t) over time, is
Importance sampling While time score normalization yields stable training in general, we empirically observe that it may not always yield the best results. Specifically, when the variance of the time score is large, for instance, when a + 1, time score normalization results in heavy down weighting. In certain cases it is beneficial to employ a weighting scheme that is approximately uniform over different values of
Inspired by diffusion models literature (Song et al. 2021a), we employ importance sampling. Specifically, samples of are drawn from another distribution p(t).
A(t)
((0)= Exx(x))
(E dx 22) sx(, t)) p(t)
(22)
1
= X(t) and A(t)= P 1. Further details on the employed importance sampling scheme can be found in Section C.I
with the goal being that, ideally, #())
X(e)
Ep, [L 11))9 (KI((1,)))11
2K² Bp. (x) [x())]
(25)
2 N
<00 .A,)) +
N
5

---

Density Ratio Estimation with Conditional Probability Paths
Note that the expectation of the KL is taken over all esti- mates pi. The error function el-) is specified in Appendix D specifically Eq. [07 with the matrices for CTSM specified in Eq. 114 and the matrices for CTSM-v specified in Eq. 127
Mean-Squared Error of the density ratio. As a basic
measure of estimation error, we approximate the following quantity Ea(z))l log po (z)) -og Pi (z)]l² using Monte-Carke po
The distribution ((xr) is chosen to be the mixture 2P0 + P1
as in the implementation of Choi et al. (2022).
Again, note that the final error decreases with the sample size N and discretization steps K. Moreover, the estimation error of the time score depends on three design choices: the parameterization of the model 0 + 5g(x, f), the chosen probability path (x z), and the weighting function A(t). Interestingly, there is an edge case that is independemt of the parameterization of the score (and therefore of the choice of neural network architecture) where the error is zero. That is when the conditional and marginal scores are equal for CTSM, 0 log Pt (x 2) = 0, log (x), and when the vec- torized version of that statement O, px x z) = [x|(x [o, log Pr (=*1 x z)] holds true for CTSM-v. Choosing paths that approximately verify these condition could reduce the estimation error and would be interesting future work.
likelioodoo of the target distribution. As second measure of success, we approximate the following quantity (x) [log Pr (x) using Monte-Carlo. We report the result in bits per dimension (BPD), obtained by taking the negative log- likelihood, and then dividing by D log 2 where D is the dimensionality of the data.
We note that the metric of log-likelihood should be inter- preted with caution. While commonly reported in related literature (Gao et al.. 2019 Rhodes et al., 2020 Choi et al., 2022 Du et al., (023). that same literature acknowledges that it is specifically designed to measure the likelihood of a normalized model. A model obtained through density-ratio estimation is only normalized in the limit of infinite samples and perfect optimization, meaning it may remain unnor- malized in practice. In such cases, BPD becomes invalid because unnormalized models introduce an additive con- stant that distorts the BPD value. Some literature attempts to address this by re-normalizing the learned model using estimates of the log normalizing constant (Gao et al. 2019
6. Experiments
To benchmark the accuracy of our CTSM objectives, we closely follow the experimental setup of Rhodes et al.] (2020)
2023 How- ever, our experiments show these estimates can be unreliable and may even worsen the unnormalization. For example, the Annealed Importance Sampling estimator (Neal 1998) produces highly variable log normalizing constants (e.g.. ranging between [-1100, 650) depending on the step size in the sampling method). Similarly, the Reverse Annealed Importance Sampling Estimator (Burda et al. 2015) can be numerically unstable for realistic distributions, such as mixtures (Du et al. (023).
Rhodes et al. 2020 Choi et al.) 2022 Du et al.
and Choi et al.] (2022) and also provide further experi- ments. Our code is available at /tt.ttts:://mm /ttpt/uu///ttb..b.p/ com/
kenxtt/dre-probppate
We mainly compare with the TSM objective (Choi et al.] 2022). as it was shown to outperform baseline methods like
NCE Gutmann & Hyvärimen 2012) and TRE (Rhodes et al.
2020 Unless otherwise specified, we use the same score network, VP path, and experimental setup as in [Choi et al. (2022). In these experiments, the TSM estimator obtained using Stein score normalization as in Choi et al. (2022). while our CTSM estimators are always obtained using time score normalization; both weighting functions were defined in Section 4 In fact, we consider time score normalization an integral part of the CTSM method instead of an optional add-on, and thus do not evaluate its effect separately. Details on experiments are specified in Appendix F
6.2. Model Accuracy in Synthetic Distributions with High Diserepancies
We consider synthetic data where two distributions have high discrepancies; this type of problem is considered in previous works (Choi et al., 2022) as it highlights the chal- lenge of the density-chasm problem (Rhodes et al. 2020). For fair comparison, we use the same model architecture, the same interpolation scheme and train for the same number of steps while tuning the learning rates for each scenario.
Overall, these experiments show that vectorized CTSM achieves competitive or better performance to TSM but is orders of magnitude faster, especially in higher dimen- sions. We note the importance of our vectorized CTSM, as in preliminary experiments, the non-vectorized CTSM is essentially not trainable on MNIST.
Gaussians
Consider two distant Gaussians,
po = Nx: (0.....0] T, I), Pi (x) = (xx:; [4...,,4]]] (T.))
(26)
6.1. Evaluation Metrics
(27)
We follow the metrics established by prior work on density ratio estimation (Rhodes et al. 2020 Choi et al. 2022). )
with varying dimensionality. Their density ratio is modeled by a fully-connected neural network ending with a linear
6

---

Density Ratio Estimation with Conditional Probability Paths
layer. Results are reported in Figure 2 We observe that our CTSM methods consistently improve upon 'SM in terms of accuracy for the same number of iterations of the optimization algorithm. Moreover, a single iteration of the optimization algorithm is more than two times faster for our methods than for TSM: CTSM and CTSM-v take around 5ms per iteration, against around 15ms for TSM !
and 0.8 on the off-diagonal, thus making the ground truth MI function of dimensionality. Their density ratio defines the mutual information between two random variables, restricted to even indices and a restricted to odd indices, as explained in Rhodes et al. (2020) Appendix D). Also fol-
lowing Rhodes et al. (2020) Choi et al. (2022), we directly parameterize a quantity related to the covariance; further details can be found in Appendix FF4
Estimating the mutual information is difficult task in high dimensions. Yet, as noted by Choi et al. (0022). TSM can efficiently do so. As shown in Figure 3 (right panel), all methods - TSM, CTSM and CTSM-v - can estimate the mutual information accurately after a sufficiently large number of optimization steps. However, CTSM-v orders of magnitude faster to converge in terms of optimization step. What is more, each optimization step is consistently faster for CTSM and CTSM-v than TSM, and this effect is exacerbated in higher dimensions, as seen in Figure S (left panel). Overall, when running these methods with a fixed compute budget, CTSM-v outperforms both CTSM and TSM, as seen in 3 panel).
10
15
G meniiionalty
75M CTEN
CTSK
Figure 2: For estimating the density ratio between two Gaus- sians, CTSM-v outperforms other methods as the dimen- sionality increases. Full and shaded lines are respectively the means and standard deviations over runs.
Gaussian mixtures Consider two bi-modal Gaussian mix- tures, centered at vectors of entries 2 and -2,
1
ko
ko
Po 2 N22
2 0 1)
N(2
+
2 o
(28)
Time per Iten (ms)
80
Mutual Intormation
2
75
Error (log10)
1
ho
1 + 2
ko
60
1
P1 V
2
2 0 r2)) (29)
40
50
20
25
with a= V 44 We choose the distribution in this way, such that & controls the between-mode distance multiple of a, while either side has unit variance in each dimension.
160
200
300
100
Dimensionality
Dimensionality
Iterations (x8000)
TSM
CTSM
CTSM-V
In this experiment specifically, the default VP path Eq. 3 cannot be used because po is not Gaussian. We therefore use another path specified in Appendix [B.2]
Figure 3: Mutual information estimation. Left: Time per iteration. Middle: Estimated and true (in dashed black) Mutual Information for different dimensions, where we directly report the estimates obtained after a few thousand iterations (see Appendix, Table B.. Right: Error between the estimated and true mutual information for dimensionality 320, during the first steps of optimization. Full and shaded lines are respectively the means and standard deviations over 3 runs.
Results are reported in Appendix E We observe that CTSM and CTSM-v are, again, significantly faster to run than TSM, while being able to achieve competitive performances within the same number of iterations.
6.3. Mutual Information Estimation for High-Ding DinehhDDmnniinnll Gaussians
Following Rhodes et al. (2020); Choi et al. 2022). we con- duct an experiment where the goal is to estimate the mutual information between two high dimensional Gaussian distri- butions
6.4. Energy-based Modeling of Images
Similar Rhodes et al. (2020) and Choi et al. (2022) ), we consider Energy- based Modeling (EBM) tasks on MNIST (LeCun al. 2010). Here, we have
(x: Ox; (ι( = (xx; E),
(30)
Po (x) O, pi(x) =
(31)
where Σ is a structured matrix; specifically it is block- diagonal, where each block is 2 x 2 with 1 on the diagonal
where (x) is a distribution over images of digits. These images may be mapped back to an (approximately) normal distribution using a pre-trained normalizing flow (multivari- ate Gaussian normalizing flow).
For this experiment, Choi et al. 2022)ss 's implementation of TSM had bug (see Appendix F.)) thus the results that we report are better than the ones in their paper.
7

---

Based on preliminary experiments, we employ importance sampling to adjust the effective weighting scheme. For the implementation of the TSM loss, we directly use the original code as provided by Choi al. (2022). ). We remark that the exact speed naturally depends on both the score matching Density Ratio Estimation with Conditional Probability Paths
Table I: EBM results on MNIST. Training is done either in the ambient pixel space, or in a latent space obtained using a pre-trained Gaussian normalizing flow. CTSM-v can achieve comparable results as TSM, while being much faster. For BPD, lower is better.
ODE (Song al. 2021b); both require computing the Stein scores V log pr (x). Based on Eq. 2
we relate these Stein
scores to the time scores
V Vlogp((x)= = V J. 0 log + Pog))(
Space
Methods
Approx. BPD
Time per step
(32)
TSM
1.30
347 ms
Latent
CTSM-v
1.26
58
Stein scores are computed by differentiating through the time scores estimated using the U-Net trained directly in the pixel space. While the time scores themselves may be well-estimated, their gradients might not be, potentially leading to inaccurate Stein scores (Liu et al. 2024). In
TSM
unstable
1103 ms
Ambient
CTSM-V
1.03
142 ms
turn, using inaccurate Stein scores in the sampling process can degrade sample quality (Chen al., 2023). Yet, the generated samples in 4 apppar realistic, suggesting that, in practice, the Stein scores are well-estimated
We note that in practice, CTSM could not be used for this task. Hence, we compare CTSM-v with TSM. To model the vectorized time score used in CTSM-v, we use the same, small U-Net architecture as in Choi et al. (022), with one modification: to condition the network on time, we use popular Fourier feature embeddings (Tancik al. 2020 Song et al. 2021b) instead of linear embeddings in Choi et al. (0022). Preliminary experiments showed this led to more stable training and better final performance.
2
3
5
5
4
2
6
4/
5
3
6
3
3
7
5
4
9 1
0 0
5
2 7
4
3
0
6
5
2
6
0
9
5 5 3
1
7
1
6
8
43 G
0
0
8
9
5
9
9
5
6
7
4
8
6 4 5
50
6
6 9 2 4 3
6 7 5
7
3
7
4 8 8 8
6
3 3
0
577
6
8
2
6
5
8
0
8
2
8
7
6
4
0
9
I
8 4
8
0
Figure 4: We report the samples obtained using ambient pixel space CTSM-v. Left: samples generated using an- nealed MCMC. Right: samples generated using probability flow ODE.
algorithm and implementation details, and in our case may also depend on the way that the flow is utilized; for details we refer readers to Section F.5
We observe that, CTSM objective can train models competi- tive to TSM, while being much faster. Annealed Importance Sampling, which has been used by previous works verify the estimated log densities (Rhodes et al.) 2020) Choi al. 2022). appears to be highly unstable for time score match- ing algorithms, with the estimated log constants varying significantly depending on the step size of HMC algorithm.
7. Discussion
Other estimators of time score In this paper, we compare time score estimators based on different learning objectives. An alternative is to use a simple Monte Carlo estimator, replacing the expectation in Eq. 7 with finite samples. Sim- ilarly, Monte Carlo methods can estimate other quantities like the Stein score Scarvelis et al. (2024). though they rarely used in practice. Recent works suggest that estimators obtained by minimizing a learning objective are preferable when the neural network architecture is well-suited to mod- cling the Stein score (Kadkhodaie et al.] 2024 Kamb & Ganguli 2024). A more careful exploration of these estima- tion methods left for future work.
Additionally, unlike previous related work (Rhodes et al. 2020, Choi al. 2022), we were able to use our algorithms successfully model MNIST images directly the ambient pixel space. We employ architecture U-Net that is closer to the one used by Song et al. (2021b). We observe that these ResNet (He et al. 2016) architectures may result in unstable training for TSM, coinciding with the observation of et al. 2022). Interestingly, the model achieves an approximate BPD value 1. 1.03, surpassing the best reported results in Choi et al. (2022) utilizing pre-trained flows.
Connections with generative modeling literature The learning objectives in this paper rely on probability paths that can be explicitly decomposed into mixtures of simpler probability paths. We used such simpler paths to compute the time score in closed form. Related literature has used these simpler paths to compute other quantities in closed form, such as the Stein score Og log (x [2) (Song et al.
Sampling So far, we have used the estimated time scores to compute the target density, but they can also be used to sample from the target. To do so. we run two sam- pling processes, annealed MCMC and the probability flow
8

---

Density Ratio Estimation with Conditional Probability Paths
b02 or the velocity (Lipman at.l.
2023 Liu et al.
2023
Acknowledgements
Albergo & Vanden-Eijnden 2023 Pooladian et al. 2023
Hanlin Yu and Arto Klami were supported by the Research Council of Finland Flagship programme: Finnish Center for Artificial Intelligence FCAI, and by the grants 345811 I and 363317. Aapo Hyvärinen received funding from CIFAR. Omar Chehab and Anna Korba were supported by funding from the French ANR JCJC WOS. The authors wish to acknowledge CSC - IT Center for Science, Finland, for computational resources.
Tong et al. 2024a) which is vector field that transports samples from Po to P1. Despite the similarities, we learn a fundamentally different quantity and our method differs from the previous ones in terms of the weighting function and an additional vectorization technique.
Connections with multi-class classification Recent works have proposed to perform density ratio estimation by learning a multi-class classifier between all intermediate dis-
Impact Statement
tributions, instead of multiple binary classifiers between con-
secutive intermediate distributions (Srivastava et al., 2023] Yair & Michaeli 2023 Yadin et al.) 2024). Multi-class clas-
This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none of which we feel must be specifically highlighted here.
sification seems to empirically improve the estimation of the density ratio, but compared with TSM, it has limitations in high dimensions (Srivastava et al.) 2023). The limiting
case where the intermediate distributions are infinitesimally close is an interesting direction for future work.
References
Albergo, M. S. and Vanden-Eijndent E. Building Normaliz- ing Flows with Stochastic Interpolants. In The Eleventh International Conference on Learning Representations, 2023.
Optimal design choices In this work, we introduce novel estimators of the time score that depend on many design choices. One of them is the choice of probability path. Xu al. (205)) considered using the learned approximate opti- mal transport path, WW & Xie (2025) considered using the learned approximate probability path given by annealing and Kimura& & Bondell (2025) considered an information geometry formulation. Finding optimal probability paths, in the sense that the final error is minimized, is an active area of research, for example applied to estimating normalizing con- stants Chehab al. (2023a).or sampling from challenging distributions (Guo et al. 2025). Another important design choice is the weighting function that has been empirically in- vestigated in related literature (Kingma & Gao [023] Chen 2023). ). A rigorous study of which design choice influences the final performance is left for future work.
Bach, F. Learning Theory from First Principles. MIT Press, 2024.
Bortoli, V. D., Hutchinson, M., Wirnsberger, P., and Doucet,
A. Target Score Matching, 2024. _eprint: 2402.08667.
Burda, Y. Grosse, R., and Salakhutdinov, R. Accurate and conservative estimates of MRF log-likelihood us- ing reverse annealing. In Proceedings of the Eighteenth International Conference on Artificial Intelligence and Statistics, volume 38 of Proceedings of Machine Learn ing Research, pp. 102-110, San Diego, California, USA, 09-12 May 2015. PMLR.
8. Conclusion
Chehab, O., Gramfort, A., and Hyvarinen, A. Optimiz- ing the Noise in Self-Supervised Learning: from Impor- tance Sampling to eprint: 2301.09696.
We propose new method for learning density ratios. We address a number of problems in previous work (Rhodes et al. 2020 Choi et al., 2022) that culminated in the TSM objective. First, TSM is computationally inefficient, sec- ond, the resulting estimator can be inaccurate, and third, the theoretical guarantees are not clear. Inspired by recent advances in diffusion models and flow matching, we pro- pose the CTSM objective and directly address these three limitations. CTSM drastically reduces the running times while improving the estimation accuracy of the density ratio, especially in higher dimensions. Additionally, we develop techniques for increasing the numerical stability through, for example, novel weighting functions. Finally, we provide theoretical guarantees on the resulting estimators.
Estimation, 2023a.
Chehab, O., Hyvarinen, A., and Risteski, A. Provable ben- efits of annealing for estimating normalizing constants: Importance Sampling, Noise-Contrastive Estimation, and beyond. In Advances in Neural Information Process- ing Systems, volume 36, pp. 45945 Curran Asso- ciates, Inc., 2023b.
Chen, S., Chewi, S., Lee, H., Li, Y., Lu, J., and Salim, A. The probability flow ODE is provably fast. In Thirty- seventh Conference on Neural Information Processing Systems, 2023.

---

Density Ratio Estimation with Conditional Probability Paths
Chen, T. On the importance of noise scheduling for diffusion models, 2023.
Gutmann, M. U. and Hyvärinen, A. Noise-contrastive es-
timation of unnormalized statistical models, with appli- cations to natural image statistics. Journal of Machine Learning Research, 13(11):307-33) 2012.
Choi, K., Meng, C.,, Song, Y., and Ermon, S. Density ratio estimation via infinitesimal classification. In Proceed- ings of The 25th International Conference on Artificial Intelligence and Statistics, volume 151 of Proceedings of Machine Learning Research, pp. 2552-2573. PMLR, 28-30 Mar 2022.
He, K., Zhang, X., Ren, S., and Sun, J. Deep Residual Learn- ing for Image Recognition. In 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), PP. 770-778, 2016. doi: 10. N0.1009/CVR.2016.9209
Clevert, D.-A., Unterthiner, T., and Hochreiter, S. Fast and Accurate Deep Network Learning by Exponential Lin- car Units (ELUs). In Bengio, Y. and LeCun, Y. (eds.), 4th International Conference on Learning Representa- tions, ICLR 2016, San Juan, Puerto Rico, May 2-4, 2016, Conference Track Proceedings, 2016.
Ho, J., Jain, A., and Abbeel, P. Denoising Diffusion Prob- abilistic Models. In Advances in Neural Information Processing Systems, volume 33, pp. 6840-6851. Curran Associates, Inc., 2020.
Hyvärinen, A. Estimation of non-normalized statistical models by score matching. Journal of Machine Learning Research, 6(24):695-709, 2005.
De Bortoli, V., Mathieu, E., Hutchinson, M., Thornton, J., Teh, Y. W., and Doucet, A. Riemannian Score- Basee Gen- erative Modelling. In Advances in Neural Information Processing Systems, volume 35, pp. 2406-2422. Curran Associates, Inc., 2022.
Izbicki, R., Lee, A., and Schafer, C. High--Dienssionl Density Ratio Estimation with Extensions to Approxi- mate Likelihood Computation. In Proceedings of the Seventeenth International Conference on Artificial In- telligence and Statistics, volume 33 of Proceedings of Machine Learning Research, pp. 420 429, Reykjavik, Iceland, April 2014. PMLR.
Dormand, J. R. and Prince, J. A family of embedded Runge-Kuue Kutta formulae. Journal of Computational and Applied Mathematics, 6(1):19-26 26, 1980.
Du, Y., Durkan, C., Strudel, R., Tenenbaum, J. B., Dieleman, S., Fergus, R., Sohl-Dickstein, J. N., Doucet, A., and Grathwohl, W. Reduce, reuse, recycle: Compositional generation with energy- diffusion models and meme. In International Conference on Machine Learning, 2023.
Kadkhodaie, Z., Guth, F., Simoncelli, E. P., and Mallat, S. Generalization in diffusion models arises from geometry- adaptive harmonic representations. In The Twelfth Inter- national Conference on Learning Representations, 2024.
Kamb, M. and Ganguli, S. An analytic theory of creativ- ity in convolutional diffusion models, 2024. -eprint: 2412.20292.
Föllmer, H. Random fields and diffusion processes. Ecole d' Ete de probabilities de Saint-Flour XV-XV11,1885-87.
LNM 1362. Springer-Verlag, Berlin, 1988.
Kendall, A., Gal, Y., and Cipolla, R. Multi-task learning using uncertainty to weigh losses for scene geometry and semanties. 2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 7482-7491, 2017.
Gao, R., Nijkamp, E., Kingma, D. P., Xu, Z., Dai, A. M., and Wu, Y. N. Flow contrastive estimation of energy- based models. 2020 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pp- 7515-7525, 2019.
Kimura, M. and Bondell, H. Density Ratio Estimation via Sampling along Generalized Geodesics on Statistical Manifolds. In Proceedings of The 28th International Con- ference on Artificial Intelligence and Statistics, volume 258 of Proceedings of Machine Learning Research, pp. 253-261. PMLR, May 2025.
Gao, Y., Huang, J., and Jiao, a. Y. Gaussian Interpolation Flows. Journal of Machine Learning Research, 25(253): 1-52, 2024.
Goodfellow, I., Pouget-Abadie, J., Mirza, M., Xu, B., Warde-Farley D., Ozair, S., Courville, A., and Bengio, Y. Generative Adversarial Networks. Commun. ACM, 63 (11):139-144, October 2020. Place: New York, NY, USA Publisher: Association for Computing Machinery.
Kingma, D. and Gao, R. Understanding difffusion objectives as the elbo with simple data augmentation. In Oh, A., Naumann, T., Globerson, A., Saenko, K., Hardt, M., and Levine, S. (eds.), Advances in Neural Information Pro- cessing Systems, volume 36, pp. 65484-65516. Curran Associates, Inc., 2023.
Guo, W., Tao, M., and Chen, Y. Provable Benefit of An- nealed Langevin Monte Carlo for Non-log-concave Sam- pling. In The Thirteenth International Conference on Learning Representations, ICLR 2025, Singapore, April 24-28, 2025. OpenReview.net, 2025.
LeCun, Y., Cortes, C., and Burges, C. MNIST hand- written digit database. ATT Labs [Online]. Available: htpp://yammlecun.ccmm cow.exdb//miii........mmmmomam 2, 2010.
10

---

Density Ratio Estimation with Conditional Probability Paths
Lee, H., Pabbaraju, C., Sevekari, A. P., and Risteski, A. Pitfalls of gaussians noise distribution in NCE. In The Eleventh International Conference on Learning Rep- resentations, 2023.
Scarvelis, C., Borde, H. S. d. O., and Solomon, J. Closed- Form Diffusion Models, 2024.
Sinha, A., O' Kelly, M., Tedrake, R., and Duchi, J. C. Neu- ral Bridge Sampling for Evaluating Safety-Crine Critical Au- tonomous Systems. In Advances in Neural Information Processing Systems, volume 33. pp. 6402-6416. Curran Associates, Inc., 2020.
Lipman, Y., Chen, R. T. Q., Ben-Hamu, H., Nickel, M., and Le, M. Flow matching for generative modeling, In The Eleventh International Conference on Learning Repre- sentations, 2023.
Bohl-Dickstein, J., Weiss, E., Maheswaranathan, N., and Ganguli, S. Deep Unsupervised Learning using Nonequi- librium Thermodynamics.
Liu, B., Rosenfeld, E., Ravikumar, P. K., and Risteski, A. Analyzing and improving the optimization landscape of noise-contrastive estimation. In International Conference on Learning Representations, 2022.
Proceedings of the 32nd In- ternational Conference on Machine Learning, volume 37 of Proceedings of Machine Learning Research, pp. 2256- 2265, Lille, France, July 2015. PMLR.
Liu, S., Yu, J., Simons, J., Yi, M., and Beaumont, M. Min- imizing f-divergences by interpolating velocity fields. In Salakhutdinov, R., Kolter, Z.. Heller, K., Weller, A., Oliver, N., Scarlett, J., and Berkenkamp, F. (eds.), Pro- ceedings of the st International Conference on Machine Learning, volume 235 of Proceedings of Machine Learn- ing Research, pp. 32308-32331 PMLR, 21-27 Jul 2024.
Song, J. and Ermon, S. Understanding the Limitations of Variational Mutual Information Estimators. In Interna- tional Conference on Learning Representations, 2020.
Song, Y., Durkan, C., Murray, I., and Ermon, S. Maximum Likelihood Training of soore-Based Diffusion Models. In Advances in Neural Information Processing Systems, vol- ume 34, pp. 1415-1428, Curran Associates, Inc., 2021a.
Liu, X., Gong, C., and liu, Flow Straight and Fast: Learn- ing to Generate and Transfer Data with Rectified Flow. In The Eleventh International Conference on Learning Representations, 2023.
Song, Y., Sohl-Dickstein, J., Kingma, D. P., Kumar, A., Er- mon, S., and Poole, B. Score-based generative modeling through stochastic differential equations. In International Conference on Learning Representations, 2021b.
Meng, C., Yu, L., Song, Y., Song, J., and Ermon, S. Autore- gressive Score Matching. In Larochelle, H.,, Ranzato, M., Hadsell, R., Balcan, M. F., and Lin, H. (eds.), Advances in Neural Information Processing Systems, volume 33, pp. 6673-6683. Curran Associates, Inc., 2020.
Srivastava, A., Han, S., Xu, K., Rhodes, B., and Gutmann, M. U. Estimating the Density Ratio between Distribu- tions with High Discrepancy using Multinomial Logistic Regression. Transactions on Machine Learning Research, 2023. ISSN 2835-8856.
Neal, R. Annealed importance sampling. Statistics and Computing, 11:125-139, 1998.
Sugiyama, M., Suzuki, T., and Kanamori, T. Density ra- tio estimation: A comprehensive review. In Statistical Experiment and Its Related Topics. Research Institute for Mathematical Sciences Kokyuroku, volume 1703, pp. 10-31, 2010. Presented at Research Institute for Math- ematical Sciences Workshop on Statistical Experiment and Its Related Topics, Kyoto, Japan, Mar. 8-10, 2010.
Nichol, A. Q. and Dhariwal, P. Improved denoising dif- fusion probabilistic models. In Proceedings of the 38th International Conference on Machine Learning, volume 139 of Proceedings of Machine Learning Research, pp- 8172. PMLR, 18-24 Jul 2021.
Pooladian, A.-A Ben-Hamu, H., Domingo-Enninn C., Amos, B., Lipman, Y., and Chen, R. T. Q. Multisample flow matching: Straightening flows with minibatch cou- plings. In International Conference on Machine Learning, 2023.
Tancik, M., Srinivasan, P., Mildenhall, B., Fridovich-Kei S. Raghavan, N., Singhal, U., Ramamoorthi, R., Bar- ron, J., and Ng, R. Fourier features let networks learn high frequency functions in low dimensional domains. In Advances in Neural Information Processing Systems, vol- ume 33, pp. 7537-7547. Curran Associates, Inc., 2020.
Rhodes, B., Xu, K., and Gutmann, M. U. Telescoping Density- Ratio Estimation. In Advances in Neural Infor- mation Processing Systems, volume 33, pp. 4905-4916. Curran Associates, Inc., 2020.
Tong, A., Fatras, K., Malkin, N., Huguet, G., Zhang, Y., Rector-Brooks, J., Wolf, G., and Bengio, Y. Improv- ing and generalizing flow-based generative models with minibatch optimal transport. Transactions on Machine Learning Research, 2024a. ISSN 2835-8856. Expert Certification.
Ruder, S. An overview of multi-task learning in deep neural networks. ArXiv, abs/1706.00 2017.

---

Density Ratio Estimation with Conditional Probability Paths
Tong, A. Y., Malkin, N., Fatras, K., Atanackovie, L., Zhang, Y., Huguet, G., Wolf, G., and Bengio, Y. Simulation- free Schrödinger bridges via score and flow matching. In Proceedings of The 27th International Conference on Artificial Intelligence and Statistics, volume 238 of Proceedings of Machine Learning Research, pp. 1279- 1287. PMLR, 02-04 May 2024b.
Yair, O. and Michaeli, T. Thinking fourth dimensionally: Treating time random variable in EBMs, 2023.
van der Vaart, A. Asymptotic Statisties. Asymptotic Statis- tics. Cambridge University Press, 2000.
Vincent, A connection between score matching and de- noising autoencoders. Neural Computation, 23:1661- 1674, 2011.
Virtanen, P. Gommers, R., Oliphant, T. E.,, Haberland, M., Reddy, T., Cournapeau, D., Burovski, E., Peterson, P., Weckesser, W., Bright, J., van der Walt, S. J., Brett, M., Wilson, J., Millman, K. J., Mayorov, N., Nelson, A. R. J., Jones, E., Kern, R., Larson, E., Carey, C. J., Polat, i. Feng, Y., Moore, E. W., VanderPlas, J., Laxalde, D., Perktold, J., Cimrman, R., Henriksen, I., Quintero, E. A. Harris, R., Archibald, A. M., Ribeiro, A. H., Pedregosa, F., van Mulbregt, P., and SciPy 1.0 Contributors. SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python. Nature Methods, 17:261-272, 2020.
Wang, H., Yu, Z., Yue, Y., Anandkumar, A., Liu, A., and Yan, Learning Calibrated Uncertainties for Domain Shift: A Distributionally Robust Learning Approach. In Proceedings of the Thirty-Seconal International Joint Con- ference on Artificial Intelligence, IJCAI-23, pp. 1460- 1469. International Joint Conferences on Artificial Intelli- gence Organization, August 2023.
Williams, D. J., Wang, L., Ying, Q., Liu, S., and Kolar, M. High- Dimensional Differential Parameter Inference in Exponential Family using Time Score Matching. In Proceedings of The 28th International Conference on Artificial Intelligence and Statisties, volume 258 of Pro- ceedings of Muchine Learning Research, pp. 3493-3501. PMLR, May 2025.
Wu, D. and Xie, Y. Annealing Flow Generative Models Towards Sampling High-Dimensional and Multi-Modal Distributions, 2025. _eprint: 2409.20547.
Xu, C., Cheng, X., and Xie, Y. Computing sigh-dimensional optimal transport by flow neural networks. Proceed- ings of The 28th International Conference on Artificial Intelligence and Statistics, volume 258 of Proceedings of Machine Learning Research, pp. 2872-2880. PMLR, May 2025.
Yadin, S., Elata, N., and Michaeli, T. Classification Diffu- sion Models: Revitalizing Density Ratio Estimation. In The eight Annual Conference on Neural Informa- tion Processing Systems, 2024.
12

---

Density Ratio Estimation with Conditional Probability Paths
Appendix
The paper and appendix are organized as follows.
Introduction
2
Background
3
Novel Objectives for Time Score Estimation
3
Basic Method
3.2
Vectorized Variant
3.3
General Framework
Design Choices
Theoretical Guarantees
Experiments
Evaluation Metrics
6.2
Model Accuracy in Synthetic Distributions with High Discrepancies
6.3
Mutual Information Estimation for High-Dimensional Gaussians
6.4
Energy-based Modeling of Images
7
Discussion
8
8
Conclusion
Useful Identities
15
Probability Paths
16
B.
Variance-Preeervinng Probability Path
16
B.2 Schrödinger Bridge Probability Path
17
Weighting Scheme
19
Details on Importance Sampling
19
D Theoretical Results
19
D.1 Proof of Eq.
19
D.2 Proots of Theorems and 3
19
D33 Proof of Theorem
20
D.4 Proof of Proposition)
21
Additional Experimental Results
23
13

---

Density Ratio Estimation with Conditional Probability Paths
Experimental Details
25
F.
Bug of TSM Implementation for Toy Experiments in Choi et al. (2022)
25
F2
Implementation Details
25
H3
Distributions with High Discrepancies
25
Mutual Information Estimation
26
F55
Energy-based Modeling
27
F.S.1
General Methodology
27
F.5.2
Experimental Details
28
F.6
Sampling
28
14

---

Density Ratio Estimation with Conditional Probability Paths
A. Useful Identities
We here organize useful identities that will be used to prove subsequent results in the form of a lemma.
Lemma 6 (Variance of a specific random variable) Consider two independent random variables, e ~ NN(0 I) and æ with mean I and covariance E. Then for scalars o, b E R,
Var[al|e] + be = 2a2 + 2 D2,D,
(33)
where c (Trace (Σ) + // D depends on the first two moments of x and on the dimensionality D.
Proof of Lemma
o |||| follows a x - which has mean D and variance 2D.
E [|e|]"]] Var [||||]]
+
E
||e||]] 2D D².
(34)
[< T x] = E Σ 4,00 = Σ [ej E [as]] = O,
(35)
E [x]]]
Var [2x]] + (TE)
[axa)) 2 Ex + n
(36)
E (e x) 2 E
Σ E [err,eja Σ E [e?a??
(37)
E
[ EE [
Σ
(Mi + mT = -Tr(>)) + ||||11
(38)
Var [cT x]
E (e x)
E
[E]
x))
2 E (e x) 2
-r(EE) + +112012
(39)
E ||ee// T x] E
Σ c
Σ ejoy
EE
Σ 2/
+E Σ & Σ وjo
(40)
(+
Σ E |o
E [xyy] +E
Σ 2
Σ
(41)
Var [alle[] + Te T x] = E [ulle]] + Te T x) >) [alle]]] +b T x))
(42)
= E |aallell* =B(a2|el|*
+ 2abl/el1 cT x+ (eT x)²| - (E [alle|] + be <<bxxx))) x))
(43)
=d +0+ 6² (Tr(E) + 11 - -aDD)² = 2a D + 6² (Tr(E) + [|p|l²))
(44)
=
(45)
15

---

Density Ratio Estimation with Conditional Probability Paths
B. Probability Paths
Definition In this paper, we consider probability paths pe( that are explicitly decomposed as a mixture of simpler probability paths p,(x z), where z indexes the mixture. Formally, this is written as
px(x) pp(z) [ (x z)] Ep(z) [W(x pz(z), I)]
(46)
The conditional paths chosen to be Gaussian (x z) = and &e in Sections B. and B.2
1). We will specify popular choices of (z(
Time score
We have
1
lo z)
log (kg 1)
(x - -x4)
ko (x -4)) + const
(47)
2
2
1
log (k:P)
||x Hell const
(48)
2kit
(kg(
||x - 11l2 8= + const,
(49)
2
2k
0y (xoge((x22)
1 D
Ocki
Ochic
|| -eell ke 2x -Mi) (-0:ui))
(50)
2
2
k
Ochit
Ō,ke
+
|x x-
- 14,11
+ hit
(x - m1) Oe 0c14c. p4.
(51)
2
kt
2k
As such, the time score is
0 log (x)
-Dki
+
1
i0, c(xx2)=
+
is 1x)((
2(x,2)= 1 -(x Mc(2)).
vkr vi((x
(52)
2hg
2k.
In fact, we can formally write the time score without the conditioning variable,
p=(z|x) p(z) exp (-
1 1.111 ||x (z(2))
0g log (xx)
Ep.( x) lo [z)].
(53)
2kg
Vectorized time score
The vectorized version is simply given by
- ke
ks
vec(O, (x
in (x(xx))
+ 2
ex 2)?
(54)
2k
Vkr
which can be obtained by decomposing the time score as the sum of D terms.
Stein score
The Stein score (Kingma & Gao 2023)
(x 1 (x- (x))). Vir
0g log lo
Vkg (x,, z),
(55)
Stein score normalization Observe that for fixed f, is, by definition, sampled from standard normal distribution. As such, the Stein score in Equation Eq. has variance The Stein score normalization in Eq. 19 is therefore given by
A(t) x lt-
(56)
B.I. Pariance-Preserviie Probability Path
Simulating the path This path is simulated by interpolating the random variables (Xo, x1) ~ po 8 Pt.
x = 04 X1 + V of xo
(57)
16

---

Density Ratio Estimation with Conditional Probability Paths
Definition Conditioning
and X1, and choosing a Gaussian reference distribution po(x) Nx; /), yields
(z) a X1: h
a.
(58)
These choices define a popular probability path, sometimes called
as the variance of pr (a:) is constant
for all t E [0, 1I (Sohl-Dickstein et al. 2015 Ho et al., 2020 Song et al. 2021b| Lipman et al. 2023). This path is in fact the default choice in the work most related to ours (Choi et al. 2022). the above, a₁ is positive and increasing,
such that ag = and a1 1. It is sometimes referred to as the noise schedule (Chen, 2023). Popular choices include exponential 0₁ min(1, 1, 2())))) Song al. 2021b) for some fixed T 2 0, or linear at min(1, t) functions (Albergo & Vanden-Lijnnden 2023 Gao al. 2024).
We remark that in diffusion models literature (Song et al. 2021b), Po denotes data and P1 denotes noise. We follow the flow matching convention, and use Po to denote noise and to denote data.
Stein score
The resulting Stein score from Eq. 55 is
1
0x logp(xx/22 12)=
Es (x.
(59)
VI aj
Stein score normalization
We have
A(1) x -ag-
(60)
Time score
The resulting time score from Eq.
52 is
Oe lo 2) D D_ 0,
0,ay
||x -c, X1 x1|12 3
(x- (xx al xi
(61)
(1 at)
1 az
a,a D
0/
||l|
+
T d6xi
(62)
Of
7
oj
V1
Vectorized time score
The vectorized version of the time score can be obtained as
0 (x)2)) D
(x -ax x1)² + 1 oz
(x -0 x1) of
(63)
of
(1-a
(1 (1-o)
0:0/
a:a/
1
2 +
E =d,x1.
(64)
1z
2
07
V1 o
Time score normalization
We have
20 (at)² + (a1)² (1 at)
Var (arp logg:op//x)))
(65)
(1 - (1-a})* a?)²
where c (Trace (E)= + || depends on the mean μa, covariance Σ and dimensionality D x.
To compute the variance of the time score, observe that the first term is deterministic and therefore does not participate in
the computation of the variance. To obtain the variance of the two remaining terms, we apply Lemma 6 with a = a(t)a'(( (t) and b
V1-a
Interestingly, the variance can explode Var [d, log (xxx)21 [)] + oo near the target distribution att) + 1.
B.2. Schrödinger Bridge Probability Path
Simulating the path This path is simulated by interpolating the random variables (Xo, xi) ~ n (x0, x1), generated from a coupling æ of the marginals Po and p1, and adding Gaussian noise e ~ N(00, IO between the endpoints,
x xx +11 t) X to / 1(1 t)
(66)
17

---

Density Ratio Estimation with Conditional Probability Paths
Definition Conditioning and (x1, Xo), yields
[=(2) (1 X +Xx1:
h h= -x= t)o?.
(67)
These choices define another path of distributions the literature. Typically, the coupling from which z drawn is either the product distribution Po @ P1 or a coupling that satisfies optimal transport. In the latter case, the ensuing path is known Schrödinger bridge (Föllmer 1988 Tong et al. 2024b). In practice, the optimal transport coupling can be approximated using limited samples from both po and pi (Pooladian al., [2023] Tong et al., 2024a). For simplicity, we use the product distribution. Note that for this path, Po need not be a Gaussian.
When using independent couplings with Po and pi having equal variance var, arguably the most natural choice of a is to set o = 2var.. In this case, the variance is preserved along the path. In order to see that, observe that under this setting the variance of x, given by
Cvar (1 v) + 20(1 l)var var.
However, empirically one may achieve better results with other choices of o.
Stein score
The resulting Stein score from Eq. 55 is
1 8x logg(x 2) = /vt(11-9 -t) cx z).
(68)
Stein score normalization
The Stein score normalization is given by
X(t) x t(1 t)o?.
(69)
Time score
The resulting time score from Eq. 52 is
1 1 2t
1 - 2t
a log (x 12)
+
2 t))² 0² ||1- (1- x)x xs tx11
(70)
2 t(1 t)
+
(x (1 - 1) X0 -txi)² T (x) - -x0)
(71)
t(1 t)o²
1 1 1 2t
1 - 2t
1 6T [xxx1= - -xo).
D
+
|c|1 +
(72)
2 t(1 t)
2t(1 - t)
Vt(1 t)o
Vectorized time score
The vectorized time score is given by
vee(a, (x 2)) = - 1-
+ [t - t))² 0² (x- - (1 111-)) xp - 1xx1)²
(73)
2 1(1 - 1)
+
(x - (1 -1)) t) X0 -txi) (xl -x0)
(74)
1(1 - 1)02
1 D 1 - 2t
+ 1 2 +
1
V1(1 - 1)0 E (X1 10-x0 Xu)
(75)
1(1 1)
21(1 1)
Time score normalization To compute the variance, treat we observe that, similar to VP path, it can be written as a
random variable with mean μ and covariance Σ,
1 2/ 2[ and
Vi(1+))
We have
1 4t 4t2 2ct 2ct² D. 213
Varp, log (x z)]
(76)
t)2
Note that t approaches 0 or 1, the variance may be infinite.
18

---

Density Ratio Estimation with Conditional Probability Paths
C. Weighting Scheme
C.1. Details on Importance Sampling
We consider the simple VP path, given by x = xx +VI Xo, where X is standard Gaussian, and X is a distribution 1++2 with = 1. One divided by the time score normalization is given by (1-12)2. Treating this as unnormalized probability density defined between 0 and I, one can derive that the normalization constant is given by Z I-1. and the CDF is given by v(1) Z 1-p7. We can calculate the inverse CDF as
+ V + 42 2yZ
2
(77)
VI+ 4y +1
As such, we can draw samples between 0 and l using the inverse CDF transform. Re-normalize 1 to lie between 0 and 1 - yields the final samples.
In practice, we choose 1 0.9 and employ this heuristic scheme for EBM experiments though we are using different variant of the VP path.
D. Theoretical Results
D.1. Proof of Eq.
Proof of Eq. The derivations are similar to denoising score matching (Vincent 2011 Bortoli et al. 2024
wish relate time score Ô, po and the conditional time score 0, log z).
We have
(x(x)= - | 22p(z)dz,
(78)
(xp((x)= f (dpp(((12))(z))) / o,
(79)
therefore
(x) px(x) = / 0 log log (2) Pi(x z)p(z) (x(x)
dz = / logpi(x
8 logp:(x))
(80)
D.2. Proofs of Theorems 1 and
We note that Theorems 1 and
special cases of Theorem 3
For Theorem [(x, ( 2) 0 (x 2). in which case g(x, () (x(() (0, (x = 0 log Dog.,((), i.e. the time score itself.
For Theorem
2 f(x, t|2) = vec(O, log (x z)), in which case g(x, t) = Epp(a|x) [vec px 2))] is clear that
Σ (x)() [ār log = xx1=)]] = (x)()
Σ a. log (x0) x² ,2) = Ep((z/x) (0x log 2)] = 0. log (xx),
(81)
Le. the sum of g(x, gives the time score.
We prove Theorem in what follows.
Proof of Theorem
The derivations are similar to Lipman et al. 2023
Tong et al. 2024a). First, we compute the gradients
19

---

Density Ratio Estimation with Conditional Probability Paths
of both cost functions, Jg and Jr.
(0) =
(t)lg((x 8) - (x +)][²] VoBp(t)
(82)
((t) (IIs(x, t) 2 (8(x, (), so 1)) + ||s0(x 0)
(83)
(oBp(t))..
A(t) ((t) (2 ()) f)) + [1so(xx))1))
(84)
VoJr(0) = V (x
A(t)]i((x t 2) so +)][²]
(85)
Vo
((t) (IIf(xx
(f(x, f| ( so (x, t)) + (x,x
(86)
(x
(10) f)) + (x, [x1)))
(87)
We then proceed to show that the two terms coincide:
E (lx (x. t)1l (x))))/ E)1??
(88)
g(x,t) I (x z)p(z)
f(x. t| z)d
(89)
(x(x)
Bpi(x) (g( (x(x 1)) Bepcco (/ Pi(xx f(x, 1(x11122)
(90)
(x(
/ m(x122)pl)
f(xx:122)d2.00(xx))) (x(x)
(91)
(x(x)
/ ( pu piixx//neee dx
(92)
= / / ((Cx)(1)s((x3)))xxxxxxxx0ee
(93)
D.3. Proof of Theorem 4
Proof of Theorem 1 We have
KL(pI,D1)² = (Epi(x) [logpI (x) logEi (x)))²
(94)
< E (x0)
[loggi(x) - log Pi (x))
(95)
K
27
= Ep (x)
s(x, t)dtt
1
Σ
8(x, ti)
(96)
K
1=1
(1/² s(x, t)dt
K
K
1 K
3(x, ti) )
27
= (x)
1
Σ
s(x,
+
1
Σ
s(x,
Σ
(97)
K
K
K 1=1
1=1
1=1
< Epp(N (x)
2 [ s(x, t)dt
1
Σ
K
s s(x,
2
+
1
K
s(x, tx) -
1 K
8(x,
ts)
27
(98)
K
K
K 1=1
i=1
1=1
[(x)
1
< E (pp(xx) (x)
2
+2
(s(x, ti) - -s(xx
(99)
2K
2K2 F px, [[xx]]] + 2F
(x)))x() [(s(x,)) 1) - 6(x; 1))
(100)
where we used Jensen's inequality and bound the discretization error of a Riemannian integral using the left rectangular sum.
20

---

Density Ratio Estimation with Conditional Probability Paths
D.4. Proof of Proposition 5
Proof of Proposition
Denote by 2(), de lo z) the conditional score and by 1x(x, The population and empirical losses defined from Eq.
X() (8(x,2,1) 80 (x 1))²
(oxx1))²
8 are respectively
N Σ lo(xx 2, 4))
(0) p a, [)],
(CTSM(O)
(101)
where the empirical loss uses i...d samples (x,, 21, keI.NN the following, we suppose that the model is well-specified. which means that there exists 0* that parameterizes the true score.
Error formulas Theorem 5.23),
First, we compute the error in the parameters. Using Bach 2024
Section 4.7) and van der Vaart 20000
- 0-) ~ ~N, -0000000000)
(102)
where G(0*) and (0') are matrices that will be later specified.
Then, obtain the error in the scores, using the delta method
( (x, so- (x, ~ (0, Vx 0) (I1 Vosaxx, ()lo).
(103)
From there, we compute the squared error in the scores. We now specify the remainder term in the asymptotic + oo analysis: is in oNN) and justified under the standard technical conditions van der Vaart (2000) Th. 5.23). We write it in expectation with respect to the law of 0,
1
E p(() [x)((x))))))) s0 (x,
x.) + +(N
(104)
where
elx,,t, t,,, X, (a t)
(105)
And then in expectation with respect to the law of tx.
E pi (eteesexc00 sg (x, = -e x.p) + +
(106)
where
e(0", A.p) (()) (() baanaa Vosoo Ol-- Voso(x,
(107)
Specifying the matrices The following matrices were used above: we now recall their definition, using the same notation
as in Bach 2024 Section 4.7).
G(O*) (0*)) Eeleoa (10=))
(o)oo 2, t)lo-
(108)
(109)
Case of CTSM
We specify
Volo(x, 2) 22(() (6(x,2,0) so (x, ()) V (oso(x,1). ():
(110)
(2,,(x,,,t) 2A(t). Vese (xs t)_ 2A(t) (s(x, 2, s0(x, 8)) (zs0(x,t)
(111)
and evaluate them at 0", To simplify notations, write w/ (x2 2, t) s(x, 80- (x. t) O, log p. (x 2) a log Px(x).
volo(x2,t))o -2((t)x(xx,zt) Voso(x t)lo=
(112)
Volo 2X(e) t) Vosa (x, O)6 ((())((x,z()) (x)0(xx1))o--
(113)
21

---

Density Ratio Estimation with Conditional Probability Paths
Finally, this yields
G(Ø")= p0p0)). A(t))((x,, Vose t)le- Vese(x, 8)lo-) [A(t) Voso(x, f) Voso t)lo- - A(t) mxxx, (g00 t)lo.]
(114)
((0") = 2()
(115)
A sufficient condition to make the error null in Eq. 107 is to have z,t) 0.
Case of CTSM-v
The derivations are largely the same. We have
2x(x,,,t) X(t) [e]]]] 2.t)) vec(se(x, t))11 a(t) Σ (s)(x.2,t)) -o(xx t),)²
(116)
where vec(s(x, 2, f));
= a Pe 2) indicates the i-th component of the vector vec(s 2,,, ())
[0, Pe(' xx* z)) D1
All that remains to specify the error are the matrices G and H. We have
Volo(x, -22(t) Σ 2,(x,,,t): -56x, t) (xss(xtt);
(117)
t) = =2((t) Σ Vasa (x,, Voso e) - 2A(t) Σ (s(x, 2,t;; -8((x, t):) V2s0(x, t)i.
(118)
We now wish to evaluate these at 0" To simplify notations, we now denote by w(x,2, t); = s(x, -0- (x, t);
d log Pe (x')x<*,+ z) [x)(x|>)) [ log (x x .z]]- Now we can write
Volo(x,2,)) = -2A(t) Σ w(x, z, Voso t);lo=
(119)
2,,
2,t) = -2((t) Σ
Vxs0(x, t) Vxso t)elo- - 2A(t) Σ w(x, z, t), V8s0 t)lor.
(120)
As result, we have
27
G(0*) 41E (x1)
X(t)
2) t);lo-
(121)
(0))) 21E (x))
X(t) Σ
Vese(x,
Vosexxx X(t) Σ w 2, (3.s
(122)
A sufficient condition to make the error null in Eq. 107 is to have w(x, 2, t); 0 for all i.
22

---

Density Ratio Estimation with Conditional Probability Paths
Table 2: Results on Gaussians with D being 2, 5 or 10. D is dimensionality, MSE is MSE to ground truth reported in the form of [mean, std], T is average time per step in ms. Unif indicates uniform weighting, Stein indicates Stein score normalization and Time indicates time score normalization, with Time indicating using the real c and Time indicating using c 1.
==2
D
= 10
Algo
MSE
MSE
T
MSE
T
TSM+Unif
[0.21, 0.036]
11.1
[2.982, 1.738]
12.5
[12.478, 6.026]
12.1
TSM+Stein
[0.253, 0.142]
13.4
[2.408, 1.205]
15.4
[7.343, 1.378]
14.0
CTSM+Time
[0.158, 0.049]
3,9
[1.37, 0.821]
6.3
[16.285, 11.047]
5.3
CTSM+Time
[0.078, 0.017]
4.5
[0.987, 0.28]
6.0
[10.032, 5.476]
4.8
CTSM-v+Time 0
[0.175, 0.045]
8.3
[0.86, 0.199]
5.2
[4.331, 0.727]
5.1
CTSM-v+Time 1
[0.104, 0.014]
4.0
[0.814, 0.219]
5.0
[1.616, 0.203]
4.9
Table 3: Results on Gaussians with D being 15 or 20. D is dimensionality, MSE is MSE to ground truth reported in the form of [mean, std], T is average time per step in ms. Unif indicates uniform weighting, Stein indicates Stein score normalization and Time indicates time score normalization, with Time 0 indicating using the real c and Time indicating using c =
D 15
D = 20
Algo
MSE
T
MSE
T
TSM+Unif
[74.932, 60.02]
13.8
[335.45, 83.226]
13.0
TSM+Stein
[91.328, 48.905]
14.3
[329.779, 156.634]
12.9
CTSM+Time 0
[36.922, 20.238]
6.2
[125.234, 30.715)
3.9
CTSM+Time 1
[61.902, 19,891]
5.5
[50.756, 12.708]
4.7
CTSM-v+Time
[16.529, 3.101]
5,4
[41.945, 13.973]
5,0
CTSM-v+Time
18888, 1.921]
4.8
[43.861, 17.132]
5.9
E. Additional Experimental Results
Distributions with high discrepancies We report the results of the algorithms under different settings and different weighting schemes. For TSM we additionally report the results under uniform weighting, i.e. X(t) 1.
Gaussians report the main results in Table 2 and Table CTSM-v is consistently among the fastest and the best, The plot in the main paper is generated using TSM with Stein score normalization, CTSM with time score normalization and 1 and CTSM with time score normalization and 1.
We additionally report the results of using time score normalization for TSM in Table 4 We did not observe decisive improvements, and remark that CTSM-v yields better results with the same weighting scheme.
Table 4: Additional results on Gaussians. D is dimensionality, MSE is MSE to ground truth reported in the form of [mean, std], T is average time per step in ms. Unif indicates uniform weighting, Stein indicates Stein score normalization and Time indicates time score normalization, with Time 0 indicating using the real and Time indicating using
TSM+Time 0
TSM+Time
MSE
MSE
T
2
[0.217. 0.063]
11.7
[0.451, 0.206]
11.6
5
[3.764, 2.10
13.1
[5.088, 4.481]
12.0
10
[13.647, 2.953]
13.9
[30.195 12.414]
12.3
15
[96.588, 53,982]
14.5
[99.062, 34.036]
31.8
20
(218.046, 70.411]
14.2
[135.942, 53.202]
13.9
23

---

Density Ratio Estimation with Conditional Probability Paths
Table 5: Results on GMMs with o = 1.0. k determines the distance between two GMM components, MSE is MSE to ground truth reported in the form of [mean, std], is average time per step in ms. Unif indicates uniform weighting. Stein indicates Stein score normalization and Time indicates time score normalization, with Time 0 indicating using the real and Time indicating using
k=0.5
k=1.0
k=2.0
Algo
MSE
T
MSE
T
MSE
TSM+Unif
[173.473, 52.466]
28.9
[276.345, 97.042]
12.6
[14643.815, 13997.560
61.8
TSM+Stein
2232,488, 133.647]
14.6
[459.645, 260.768]
17.6
[3427.258, 3545.452]
12.0
CTSM+Time O
[880,47, 172.594]
4.1
[480.847, 151.097]
4.5
[646,945, 210.44]
4.7
CTSM+Time 1
(923.082, 131.758]
4.6
(460.546, 186.5]
4.2
[547.603, 200.504]
4.8
Timm-v+Time
[173.804, 108.326]
4.8
[211.046, 69.472]
4.0
[319.981, 100.91]
7.4
CTSM-v+Time
[221.519, 98.112]
5.8
[181.082, 68.879]
4.7
1266.486, 150.877]
4.3
Table 6: Results on GMMs with 220.. determines the distance between two GMM components, MSE is MSE to ground truth reported in the form of [mean, std], T is average time per step in ms. Unif indicates uniform weighting, Stein indicates Stein score normalization and Time indicates time score normalization, with Time 0 indicating using the real and Time indicating using = 1.
k=0.5
k=1.0
k=2.0
Algo
MSE
T
MSE
T
MSE
T
TSM+Unif
1106..18, 550.442]
33.3
[1293.421, 2
12.5
[6614.483, 1169.068]
13.5
TtM+Stein
1460.023, 502.921]
39.0
[1564.266, 360.361]
36.0
[5180.453, 1786.018]
12.7
CTSM+Time O
[1934.401, 269.515]
4.5
[1872.342, 467.047]
4.6
[5961.52, 683.578)
4.6
CTSM+Time 1
22113.975, 403.59]
8.0
[2238.261 123.69]
4.6
[6017.094, 344.537]
4.0
CTSM-v+Time 0
[745.15, 158.202]
4.6
[1558.495 379.161 179.161]
4.5
[5009.627, 1943.244)
8.5
CTSM-v+Time
[762.231, 288.029)
5.3
[1762.826, 431.333]
4.8
19226.993, 861.191]
9.2
Gaussian mixtures We report the main results on Gaussian mixtures in Table 5 We set a in the Schrödinger bridge probability path to 1.0 due strong empirical results while enabling direct comparisons between TSM and CTSM(-v).
We additionally report the results with /2.0 in Table and the results with 0.0 in Table We observe that, setting V220 results in worse performances for all methods. For TSM under uniform weighting, one can consider using = 0.0, in which case the performance improves, though CTSM-v under o 1.0 remains competitive.
Table 7: Results on GMMs with a = 0.0. k determines the distance between two GMM components, MSE is MSE to ground truth reported in the form of [mean, std], average time per step in ms. Unif indicates uniform weighting. Stein indicates Stein score normalization and Time indicates time score normalization, with Time 0 indicating using the real and Time indicating using
k=0.5
k=1.0
k=2.0
Algo
MSE
MSE
T
MSE
T
UnM+
[148, 688, 97.058]
14.6
[70.908, 5.85]
12.9
[898.01 016, 847.255]
49.1
24

---

Density Ratio Estimation with Conditional Probability Paths
F. Experimental Details
F.I. Bug of TSM Implementation for Toy Experiments in Choi et al. (2022)
We observed a bug for the TSM implementation of the code Choi et al.
2022).
Recall that the TSM objective is given by
L0s = pox) (x. 0)] 2 (x) (s,((, (x, 1)]+
(123)
E Ep((.x) 120 () + (x(()sx((x))))) () + A(1)so(x, 6)²].
However, Choi et al. (0022 implemented
(fs (f) = 21E (s 0)] pI (x)) (x) ( (x, 1)]+
(124)
E
t) + (x,) t) + (xt) t)
i.c. the scaling in front of (xx) t) is incorrect. We remark that this bug only applies when attempting to train purely based on TSM objective on toy experiments.
E2. Implementation Details
Our implementation of TSM is largely based on the code provided by Choi et al. (2022). However, especially for other than the EBM experiments, we improve their code in several ways. Apart from bug fixes, we use analytical expressions for the weighting quantities.
For both TSM and CTSM, following Choi et al. (2022), we add small number e to the time during training and inference. We follow the convention that is added when the probability path results in approximately degenerate distribution that time. For the toy experiments, we set e le 5, while for EBM experiments set le 4 during training and le 5 during inference.
For experiments apart from EBM, for each task we employ a fixed validation set of size 10000 and select the learning rates based on results on the sets. After a certain number of steps, an evaluation step is performed, and the model is evaluated based on both the validation set and a test set, consisting of 10000 samples dynamically generated based on the data generation process. The best test set results are obtained by selecting the steps corresponding to the best validation set results.
Following Choi al. 2022
the density ratios are evaluated using the initial value problem ODE solver implemented in where we use the default RK45 integrator (Dormand & Prince 1980) with rtol 1e 6 and
SciPy [Virtanen al. 2020
atol= = le 1e-6. 6.
E3. Distributions with High Discrepancies
The experimental setup is similar to Choi al. (0022). We use score model simple MLP with structure [D + 1, 256, 256, 256.Nootoa and ELU activation (Clevert et al. of the data and Noutput D for CTSM-v and otherwise. Note that the input shape is 1) 1, as the time / is concatenated to the input. All models are trained for 20000 iterations. After each 1000 iterations, the model is evaluated. For each scenario, the best learning rate is selected based on the best val set performances of single run. Afterwards two runs under the same learning rate but different random seeds are run, and the final results on the test set is reported.
2016) based on Choii et al. (2022), where D is the dimensionality
Gaussians Following Choi et al. 2022), we employ the variance- probability path, with a, =
The learning rate is tuned between [5e 4, le 3, 2e 3, 5e 3, 1e 2]. Following Choi et al. (022). the MSEs are evaluated using samples from both and P1-
2e 3, 5e 3, 1e 2, 2e 2, 5e 2]. There is one case where the selected learning rate for each algorithm the smallest, and we manually verify that using Irs 5e 5 or 2e 5 does not result in improved results. Following Choi et al. (2022). the MSEs are evaluated using samples from both Po and p1-
Gaussian mixtures The learning rate is tuned between [le- 4, - 5. -, 1e 3. 3,
25

---

Density Ratio Estimation with Conditional Probability Paths
The components are isotropic, with the covariance given by o² I. We use k specify the distance between the means of the two components as multiple of the standard deviation a.
We know that the mean of a GMM is simply given by the mean of the means of each component, while the covariance of a GMM with two components of equal weights is given by the following formula
1 ス
Σ + (mn p2) - -22)
Σ -
(125)
Consider the case where #1 11--6 42 ko. One has that, in order for the GMM have variance equal in each dimension, 4 The means of the two components given by μ - ko and μ tšōko, respectively.
In principle, using var = 2 for SB path results in preserved variance along the path. However, empirically we observe that it is beneficial to use smaller variance, e.g. var- = 1.
E4. Mutual Information Estimation
The probability path is given by
Px z) = (x|xxxxxxx (1- (1-
() 1).
(126)
The derivations for the objective of TSM objective can be found in Choi et al. (002). Here we derive the training objective for the CTSM-v objective.
Using similar settings and notations in Choi al. (2022). parameterize single matrix S, defined below.
Denote the covariance matrix of pl as E. Use S to denote E I.
Recall that the true time score is given by the posterior expectation of à log (x z). We have
1 T Σ +const.
p(z) = log (( 0, Σ) =
(127)
2
1 T T-22"6 12 +oonst.
px z) = log.N(( x | z, (1 - 1) = 1 112
(128)
2
The posterior distribution (z x) can be solved closed-form, which a Gaussian distribution, with covariance Σ = Σ + 1-12 I) and mean 1
I-1² Σx. x. Similar to Choi et al. (2022), the above quantities can be expressed in terms of
the inverse of I I+t2 ( (E- -) (1 - (2) [
f I-
Σ)
we have
12
(1
E It 1 1- f2 Σ x))
Σ
1)
(129)
1
22
-1
Σ
Σ = (1- (I+
( 1)) -
(130)
The expectation of θ log (x z). which by definition is also the value of Ô log Pt (x), can also be obtained in closed-form. The expectation of the individual entries of O log (x z) are also given in closed-form.
[8 (xgg(((|22))
t
t
[x (x-]x))²],
+ i 12 [[x -ixi) X11;
1
(131)
1 12
(1- 12)
21-
t(1 - t?) - (1|2
+ Exa)
+||xx|11 2 (t2 + 1)xx; F4:
(pc x)) [8, log Pt (x|2)];
(132)
- 12)2
where j and Σ are the mean and covariance of the posterior distribution as discussed above. As such, perhaps unsurprisingly. CTSM-v does not induce much computational overhead above TSM.
For CTSM objective, the model is trained to match the time score, while for CTSM-v objective, the model is trained to match the entire vec (0, log Pe (x z)).
The hyperparameters are inspired by Choi et al. (022) and listed in Table g For all methods, the learning rates are tuned
between le 4, le 3 and le
2.
26

---

Density Ratio Estimation with Conditional Probability Paths
Table 8: Hyperparameters for MI experiment. After every eval freq steps, an evaluation is performed, with the first result after the first eval freq steps.
D
n iters
eval freq
batch size
40
20001
2000
512
80
50001
5000
512
160
200001
5000
512
320
400001
8000
256
F.5. Energy-based Modeling
F.5.1. GENERAL METHODOLOGY
We employ the same variance-preservingg probability path as used in Choi et al. (2022) which in turn comes from diffusion
models literature (Ho et al. 2020 Song et al. 2021b).
For reproducing TSM results, we use a batch size of 500 and use polynomial interpolation with buffer size 100, matching the reported hyperparameters in Choi al. (2022). Following Choi et al. (2022). we tune the step size of TSM between [2c- 4,56 4, le - 3]. For CTSM-v, largely reuse the hyperparameters, while tuning the step size between [5e 4, le 3, 2e - 3].
For CTSM-v objective, we parameterize the model to output the time score normalized by the approximate variance. Specifically, for a given f, we calculate Var (0, log (x z)) where e is assumed to be 1, and the score network trained to it log
predict Sulä, log x)} this ensures that the regression target is zero mean and having reasonable variances across t.
In previous works (Rhodes et al. 2020 Choi al. 2022). different normalizing flows are fitted to the data, and DRE is carried out making use of the flows.
The flows can naturally be utilized in different ways. Denote the latent space of the flow as u, and the ambient space of the flow as x. Choi et al. 2022) consider the following scheme:
1. An SDE is defined on u space, interpolating between Gaussian and the empirical distribution induced by final samples on u space obtained by transforming the data points from x space,
2. Intermediate samples on space are transformed into space using the flow, inducing a time varying distribution on space,
3. The score network takes as input x and I, and is trained to predict the time score.
Note that flow a bijection. Consider time-varying density (x)... For any I, we use the same bijective transformation T obtain the pair of and We have
3 log p (x) 0, log (p) (detJr 8, (o) + O log [d)tJr = O log
(133)
As such, the time score is invariant across bijections.
With CTSM, inspired by previous approaches, we also consider a probability path in u space. One needs the time score of the conditional distribution, which needs to be computed in space. One can in principle train the score network either by feeding in coordinates of points in the u space or the corresponding coordinates in x space, where the conditional target vector field is computed space.
1. An probability path is defined on space, interpolating between Gaussian and the empirical distribution induced by samples,
2. The score network takes input either x or along with and learns the time score.
Note that it is correct to feed in the score network either x or u, when the model takes as input x, one can interpret that the normalizing flows is a part of the score network, i.e. (o t) = (f (x),t), t), where is the normalizing flows that
27

---

Density Ratio Estimation with Conditional Probability Paths
fixed and does need to be learned and s is the score network that we parameterize. As such, the correctness is guaranteed by standard CTSM / CTSM-v identities. We empirically observe that directly feeding in x coordinates leads to better BPD estimates.
We remark that both TSM and CTSM need to map between u and x coordinates using the normalizing flows. However, while CTSM only need to map both u to x and x to u exactly once, TSM requires an extra u to x map due to needed by the boundary condition.
F.5.2. EXPERIMENTAL DETAILS
In terms of EBM with Gaussian flows, we observe that, possibly due the parameterization, models trained using CTSM-v may require a larger number of integration steps compared with TSM when evaluating the density ratio using ODE integrator with specific error tolerances as described in Section F22 on MNIST test set with batch size 1000, TSM requires on average 489.2 evaluations, while CTSM-v requires on average 830.6 evaluations. However, we remark that it is unclear what the true time scores are like.
Especially with TSM, the BPD estimate on the validation set can vary greatly. general, we report the final results using the checkpoint that resulted in the best validation set result, unless the BPD estimate falls below in which case we consider the training unstable.
Previous works (Rhodes et al., 2020 Choi et al., 2022 experimented with performing DRE utilizing copula flows and RQ-NSF flows. We observe that CTSM-v suffers from unstable optimization, with the BPD dropping at first but shoots back to large value afterwards. We hypothesize that the parameterization does not agree with the inductive bias of the employed score network.
When training CTSM-v in the ambient space, we employ as score network a more advanced U-Net largely based on Song et al.] (202Tb) and Choi et al.] (2022), which, among others, employs residual blocks. As noted by Choi et al. (2022). residual blocks result in unstable training for TSM. However, CTSM-v works well with this version of the U-Net. We also experimented with training using TSM with the U-Net as used by Choi et al. (2022) in the ambient space, but found the BPD estimates on the val set to be generally larger than 5 and did not explore it further.
With Gaussian flows, all experiments were run using one NVIDIA V100 GPU each. With ambient space, the models were trained and evaluated using one NIVDIA A A10 GPU each, while the running times were obtained based on 10000 steps using one NVIDIA V100 GPU each. We observed that the training dynamics of the models may vary across the employed GPUs, even with the other settings kept the same.
F.6. Sampling
We employ annealed MCMC draw samples from the learned score network. We draw total of 100 samples. For each sample, we construct 1000 intermediate distributions, where each intermediate distribution is targeted using single HMC step. The intermediate distributions are constructed by linearly interpolating between 0 and and setting
(x() = log (x) S Ô (x)d7.
(134)
After which, we run another 100 steps of HMC to further refine the samples.
Each step of HMC contains 10 leapfrog steps. When using Gaussian flows, we observe a correlation between sample quality and the estimated log constant, where the sample quality is good when the estimated log constant is close to O. Based on the observation, we tune the step size of HMC on a grid in the form of [le n, 2.5e n, 5e n, 7.5e n).
For Gaussian flows, using TSM and CTSM-v are shown in Figure 5
annealed MCMC with batch size of 100. The first 64 samples drawn from models trained
For pixel space CTSM-v. we consider both annealed MCMC and the Probability Flow (PF) ODE (Song et al. 2021b). For annealed MCMC, we use a batch size of 50 and run 2 individual batches. For PF ODE, we use a batch size of 64. The first 64 samples were reported in the main paper.
28

---

Density Ratio Estimation with Conditional Probability Paths
9
7
7
6
4
2
2
8
4
4
7
6
L 2
4
1
9
3
8 4 1
3
3
3
2
4
3
8
8
8
2
12
8
6
5
8
5
4
8
3
5
#
7
4
1
5 8 / 4
4
7
7
Q
4
4
9 5
6
3
2
9
9
6
6 4
2
:
9
7
6
8 9
E
3
2
5
6
3
/
0 4
3
3
q
3
0
1
J
3
0 7
3
!
6
6
6
4
2
9
Y
0
4
9
6
7 5 3
Figure 5: Left: samples drawn from model trained with TSM, Gaussian flows; right: samples drawn from model trained with CTSM-v, Gaussian flows.
29


Reviews:
[{"review_id": "DMYEh7unNz", "reviewer": "Reviewer_MEax", "summary": "This paper proposes the conditional time score matching (CTSM) objective, which is a variant of the time score matching (TSM) objective proposed in the recent density ratio estimation literature.\nThe CTSM objective provides a principled way to detour the computational drawback of TSM, which requires two automatic differentiation steps. The idea is to introduce a conditioning variable z, and the mathematical trick is essentially same as that of denoising score matching. The authors provide a vector variant that can be useful in practice. \nThey also analyze the quality of the estimated density $\\hat{p}_1(x)$, where the integration of the time score is approximated by a discretization, assuming that $p_0(x)$ is given.\nThe experiments support that CTSM can perform as well as TSM with less computation.\nOverall, it is a well-written paper with well-executed analyses and experiments.\n\n## update after rebuttal\n\nI thank the authors' response. I will keep my positive score.\nFor the final version, it'd be helpful if the authors further comment on the choice of KL divergence for the analysis and its limitation.", "questions": "", "limitations": "", "rating": 5, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for the very positive assessment and for spotting the typo in Eq. 45 which is now corrected. Regarding our theoretical guarantees, we chose the KL divergence out of convenience for the derivations in Appendix D.3. A similar analysis could indeed be used to bound the expected value of the square of the difference in the log density ratios. To do so, we would start from Eq. 81 and replace the log densities with log density ratios.", "claims_and_evidence": "The theoretical claims are solid. Experimental validations are also thorough showing the benefit of CTSM compared to TSM.", "methods_and_evaluation_criteria": "The proposed method is a very natural yet effective modification of TSM leading to computational efficiency.\nThe evaluation criteria in experiments seems also adequate.", "theoretical_claims": "I checked the proofs of the statements, but only skimming over the proof of Proposition 5.", "experimental_designs_or_analyses": "The experiments are solid enough to demonstrate the benefit.", "relation_to_broader_scientific_literature": "Density ratio estimation in general is a key task in machine learning.\nRecent years density ratio estimation based on infinitesimal classification has received attention as a solution to the so-called \"density chasm\" problem. The paper's idea is not a rocket science, but it provides a very thorough analysis on the benefit of conditioning both in theory and practice.", "essential_references_not_discussed": "References seem adequate.", "other_strengths_and_weaknesses": "It was a pleasant read overall and I do not have any specific comment on weakness.", "other_comments_or_suggestions": "N/A", "questions_for_authors": "- In theoretical guarantees, is there a particular reason for the KL divergence considered in the analysis? I am just curious if this is a convenient choice for the ease of analysis, or something else. Can this relationship be translated to, for example, the expected value of the square of the difference in the log density ratios?\n- Is the last term a typo in Eq. (45)?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "W8wCrwzxF3", "reviewer": "Reviewer_h72Z", "summary": "The authors tackle the problem of estimating the ratio of two probability densities, improving upon speed and accuracy of prior work. They also establish a theorem on a guarantee of the error in the estimated density ratio. The method applies the \"marginalization trick\" (a la flow matching) to make the learning objective tractable. Essentially a latent variable is introduced to the stochastic interpolant between the test and target densities, and object modified to match the expectation of score matching marginalizing out thelatent. The distributions are chosen in a clever way similar to previous literature so that the objective is tractable (so called Conditional Time Score Matching objective). The authors also introduce a variant where a multivariate test and target distributions are broken up into n terms in an autoregressive fashion and the terms matched separately. A novel contribution is the design of a weighting function, time score normalization, to stabilize training. This could become a standard statistical method.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your positive assessment. We agree that our learning objective (CTSM) is similar to the learning objective in flow matching (Eq 5, [1]). However, our method is not a straightforward application of flow matching given that, fundamentally, we learn a different quantity. Flow matching learns a velocity field while we learn a time score.\n\nFurthermore, our vectorized learning objective (CTSM-v) is a novel contribution of our work that is essential for high-dimensional experiments (see Fig 2). In fact, our CTSM-v objective has no obvious counterpart in the flow matching literature, to the best of our knowledge. Moreover, our time score matching weighting scheme is tailored for learning the time score and is different from weighting schemes used in flow matching (Eq 5, [1], i.e. a uniform weighting).  \n\nWe are happy to include these clarifications in the main text. \n\n[1] Lipman et al., Flow Matching for Generative Modeling, ICLR 2023", "claims_and_evidence": "Claims in the paper are supported by ample ablation studies and other experiments (with uncertainty estimates).", "methods_and_evaluation_criteria": "Proposed evaluation makes sense for the problem at hand and are quite comprehensive.", "theoretical_claims": "I didn't review the proof for Theorem 4 and Proposition 5, which are in the supplementary materials.", "experimental_designs_or_analyses": "Experimental design is sound.", "relation_to_broader_scientific_literature": "Estimation of density ratios is a fundamental problem in generative modeling with many applications, as the paper describes.", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "N/A", "other_comments_or_suggestions": "N/A", "questions_for_authors": "I think the paper would benefit from some discussion on how this work isn't just a straightforward application of the ideas behind flow matching. Is this something you would consider?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "gT9Uiwx1XK", "reviewer": "Reviewer_B2JP", "summary": "This paper introduces a new method for density ratio estimation called conditional time score matching (CTSM). CTSM estimates the time score along a probability path connecting two densities instead of directly estimating the ratio of two densities. By conditioning on additional variables, the authors propose an easy-to-estimate objective and a faster variant. They further provide the theoretical guarantee of error bounds and then show strong empirical results, especially in high-dimensional settings.\n\n## update after rebuttal:\nI'm maintaining my score.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for the very positive assessment. Below we respond to the comment on the MNIST experiment.\n\nWe indeed showed results only for pre-trained Gaussian NF, since it was the most computationally stable. We have some experimental results also for Copula and RQ-NSF utilizing essentially the same score network as in [1] but found the optimization to be unstable. \n\nMoreover, after the submission we also experimented more with directly solving the same problem in the ambient space itself using a different score network, instead of using any pre-trained flow and its latent space. Whereas previous methods reported that using a pre-trained flow was important to obtain good results (see [1, 2]), our method works extremely well in the ambient space scenario, reaching a BPD of 1.03, outperforming any of the previous results utilizing a latent space. We also checked that the learnt time score in ambient space was capable of generating reasonable samples using two sampling methods: annealed MCMC and the Probability Flow ODE from [3]. We will also add this result.\n\n[1] Choi et al., Density Ratio Estimation via Infinitesimal Classification, AISTATS 2022\n\n[2] Rhodes et al., Telescoping Density-Ratio Estimation, NeurIPS 2020\n\n[3] Song et al., Score-Based Generative Modeling through Stochastic Differential Equations, ICLR 2021", "claims_and_evidence": "Yes, the claims are supported by clear theoretical analysis and empirical results.", "methods_and_evaluation_criteria": "The method and chosen benchmarks clearly evaluating the improvements over existing methods. The evaluation criteria follows prior work of Rhodes 2020 and Choi 2022 and they make sense for the problem.", "theoretical_claims": "I review the key theoretical results and proofs (mainly section 5) in the paper. The proofs are correct and rigorous.", "experimental_designs_or_analyses": "The experimental design in this paper is sound and consistent with prior works. However, in Table 1, the evaluation is limited with only a pretrained Gaussian normalizing flow. The paper could benefit from additional experiments from prior work. Specifically, the paper could include experiments using Copula interpolation and RQ-NSF interpolation following [Choi 2022] Table 1. \n\n\nChoi 2022. Density Ratio Estimation via Infinitesimal Classification.", "relation_to_broader_scientific_literature": "This paper contributes to the broader research on density ratio estimation, which has numerous application in machine learning.", "essential_references_not_discussed": "The current set of references is robust.", "other_strengths_and_weaknesses": "See \"Experimental Designs Or Analyses\" for additional experiments.", "other_comments_or_suggestions": "Not applicable.", "questions_for_authors": "Not applicable.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: This paper proposes the conditional time score matching (CTSM) objective, which is a variant of the time score matching (TSM) objective proposed in the recent density ratio estimation literature. The CTSM objective provides a principled way to detour the computational drawback of TSM, which requires two automatic differentiation steps. The idea is to introduce a conditioning variable and apply the "marginalization trick" (a la flow matching) to make the learning objective tractable. 

All reviewers are unanimously positive about this paper: the density ratio estimation problem is important, the methodology is clever and sound, and the paper is well-written with with well-executed analyses and experiments. Based on the reviews and my own reading, I happily recommend acceptance.
Decision: Accept (poster)