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Steering Diffusion Models to Rare Events with Sequential Monte Carlo

Aavash Subedi, Tim Reichelt, Christopher Williams, Philip Stier, Yee Whye Teh, Saifuddin Syed

arXiv:2610.08652v1stat.MLcs.LG

TL;DR

Estimating a rare-event probability with Monte Carlo becomes increasingly costly as the event grows rarer. DireSMC guides weighted diffusion samples with sequential Monte Carlo and a smooth relaxation of the event set. It reports net speed-ups over crude Monte Carlo across rare-event tests.

  • Problem

    Crude Monte Carlo needs N ∝1/p0[E] samples for stable estimation, making rare-event probabilities computationally difficult to estimate.

    Source: §1 · p. 1

  • Method

    DireSMC guides weighted samples toward rare events with sequential Monte Carlo and an analytical relaxation of the event set.

    Source: §1 · p. 2

  • Results

    9 to 1413× net speed-ups over crude Monte Carlo, improving with rarity, are reported for synthetic examples and a score-based climate emulator.

    Source: §5 · p. 10

  • Takeaways & Limitations

    The method provides rare-event samples and probability estimates that can support failure-mode characterization, planning, and resource allocation.

    Source: §1 · pp. 1–2

    Estimates target the discretized, approximate law sampled, so they carry reducible discretization bias and errors from an approximate score.

    Source: §5 · p. 10

Abstract

from arXiv · show

Diffusion models are increasingly used as surrogates for expensive simulators in weather prediction, molecular dynamics, and materials design. In these models, computing the probability $p_0[E]$ of an event $E$ is difficult, especially when the event of interest is rare. A stable estimate using Monte Carlo becomes computationally intractable, requiring a growing sample size $\propto\!1/p_0[E]$ to compensate for an increasing rarity. In this paper, we present Diffusion Importance Sampling of Rare Events or DireSMC, a sequential Monte Carlo scheme that guides a population of weighted samples towards the rare event, giving access not only to samples but also to a calibrated estimate of its probability. We set up our guidance using an analytical relaxation of the event set, allowing the method to easily extend to a wide range of user-defined rare events. We validate our method on a toy problem with analytical solutions and on a score-based climate emulator, where we obtain accurate rare-event probabilities on a range of rarities from $10^{-3}$ to $10^{-5}$, achieving net speed-ups of $9\times$ to $1413\times$ over Monte Carlo.

1 INTRODUCTION

Rare-event probabilities matter because they inform downstream planning, yet crude Monte Carlo requires increasingly many samples as events become rarer. DireSMC combines diffusion guidance with sequential Monte Carlo to produce event samples and probability estimates.

1 INTRODUCTION

  • Monte Carlo requires N ∝1/p0[E] samples to stabilize rare-event probability estimates, making it prohibitively expensive as event rarity increases.Rare-event samples and calibrated probabilities are both useful for characterizing failure modes and informing planning and resource allocation.Source: §1 · p. 1
  • A hard indicator reward recovers the desired conditional rare-event distribution, but its gradient cannot steer particles toward the event.More generally, approximate rewards may steer toward a distribution different from the intended tilted target.Source: §1 · pp. 1–2
  • Earlier diffusion-SMC and rare-state methods have high compute costs in rare-event settings, motivating the paper’s probability-estimation approach.Figure 1 depicts guided particles accumulating importance weights that support resampling and an estimate of p0[E].Source: §1 · p. 2, Figure 1 · p. 2
  • DireSMC uses sequential Monte Carlo with tilted marginals to estimate rare-event probabilities and produce samples, with variance governed by steering quality rather than event rarity.For threshold-exceedance events, it uses a smooth surrogate reward whose look-ahead guidance is closed-form under a Tweedie posterior approximation.Source: §1 · p. 2

2 BACKGROUND

The paper treats a discretized reverse diffusion chain as the sampling process and seeks the probability of an event under its endpoint law. Rare events make the standard Monte Carlo estimator increasingly costly.

2 BACKGROUND

  • A diffusion model reverses a noising path from data to a Gaussian reference, using a learned score to evolve samples back toward data.The paper uses a discretized schedule and numerical transition kernels because the reverse-time SDE cannot be simulated directly.Source: §2 · p. 2
  • The endpoint law p0 is the law of the discretized chain; the method assumes access to transition simulation and the score, but not generally to the intractable densities.As discretization is refined, the chain endpoint approaches the data distribution at a rate determined by the integrator’s weak-order convergence.Source: §2 · p. 3
  • The crude Monte Carlo estimator has relative variance (1 −p0[E])/(Np0[E]), so fixed relative error requires N ∝1/p0[E] runs for rare events.The target is the probability of the event under the endpoint law p0.Source: §2 · p. 3

3 RARE-EVENT SAMPLING AND PROBABILITY ESTIMATION

The section constructs rare-event estimates by tilting diffusion marginals and applying sequential Monte Carlo to a guided particle population. Reward design then targets low variance while retaining a smooth gradient for steering.

3.1 STEERING THE DIFFUSION MODELS TOWARDS RARE EVENTS

  • Particles propagate through guided reverse-diffusion proposals, receive incremental weights, and may be resampled and rejuvenated with an invariant MCMC kernel.The sampler tracks a sequence of tilted marginals, with the reward set to zero at the noise endpoint.Source: §3.1 · pp. 3–4

3 RARE-EVENT SAMPLING AND PROBABILITY ESTIMATION

  • DireSMC’s weighted particle system yields both an estimate of p0[E] and weighted samples from the conditional event distribution.The paper presents the construction for discretized kernels and states that it also holds for the continuous-time diffusion construction.Source: §3 · p. 3, §3.1 · p. 4

3.1 STEERING THE DIFFUSION MODELS TOWARDS RARE EVENTS

  • The probability estimator is unbiased for any reward choice, while reward quality affects variance; its normalizing constant is accumulated along denoising.The conditional sample estimate converges almost surely as particle count grows; under an exact score and a zero-probability event boundary, the probability estimate converges to the data-law probability as discretization is refined.Source: §3.1 · pp. 4–5

3.2 REWARD SELECTION

  • The ideal Doob-transformed trajectory law is the base chain conditioned on the event, and its transitions minimize Pearson χ2 divergence among candidate guided transitions.The minimum is bounded by χ2(π0∥p0) and is zero when the initial and terminal states are independent under the base chain.Source: §3.2 · p. 5
  • For threshold-exceedance events, a smooth reward relaxes the indicator so its gradient can guide particles outside the event toward it.A Gaussian-CDF reward is adopted; for a linear observable its look-ahead integral is closed-form under the Gaussian posterior approximation, and guidance is tempered as that look-ahead becomes reliable.Source: §3.2 · p. 6

3.3 RELATED WORK

  • Prior diffusion-guidance methods do not compute probabilities, while the cited molecular approach combines a sequence of events rather than estimating p0[E] directly.The paper contrasts its single-denoising-pass weight accumulation and resampling with a climate-emulator importance-sampling approach.Source: §3.3 · p. 6

4 EXPERIMENTS

Experiments evaluate DireSMC on a correlated Gaussian mixture and a pretrained climate emulator, testing probability estimation and sample quality across rare-event settings. The reported results include lower variance than crude Monte Carlo and improved performance over a competing estimator.

4.1 CORRELATED GAUSSIAN MIXTURE MODEL (GMM)

  • At compute-normalized particle budgets, DireSMC variants have lower variance across rarities, and Manshausen et al. (2026) show larger discretization bias.At equal particle counts, DireSMC with resampling and MADM also has lower CV despite the comparator requiring 3.5× as many function evaluations.Source: Figure 2 · p. 7, §4.1 · pp. 7–8
  • DireSMC recovers the reference exceedance probability with analytical and learned scores, while its coefficient of variation stays flat across rarities for fixed N.The GMM experiment varies analytic event mass from 6.9×10^-3 to 1.4×10^-5.Source: §4.1 · p. 7

4.2 GENERATING HEATWAVES WITH PRE-TRAINED CLIMATE MODEL EMULATORS

  • At least an order-of-magnitude variance reduction over standard Monte Carlo is reported for PNW extreme-temperature events, including a threshold where only 2/16 Monte Carlo runs produced a sample.The figure shows the mean July temperature field and highest-weight threshold-crossing guided samples at different exceedance thresholds.Source: Figure 3 · p. 8, §4.2 · p. 9

4.1 CORRELATED GAUSSIAN MIXTURE MODEL (GMM)

  • Discretization bias decreases approximately as O(1/K), increases with rarity at fixed K, and is lower for the tested SDE integrators than for DDIM.The MADM kernel also introduces an O(1/K) leading-order error and assumes a conservative score; the method extends without modification to interval and multi-constraint events.Source: §4.1 · p. 8

4.2 GENERATING HEATWAVES WITH PRE-TRAINED CLIMATE MODEL EMULATORS

  • The hot-and-dry event estimates recover the joint probability and dependence ratio with positive net speed-up over crude Monte Carlo; dependence strengthens as the event becomes rarer.The reported dependence ratio changes from d = 31.4 to 1.4 × 10^3.Source: Table 1 · p. 9, §4.2 · p. 9
  • From ∼10^-6 at ΔT = 0 to near-certainty at ΔT = 4 K, DireSMC tracks the event probability across six orders of magnitude.Figure 4 compares the estimates with unguided Monte Carlo at equal compute; points average 16 runs and bars show ±1 standard deviation.Source: Figure 4 · p. 9, §4.2 · p. 9

5 CONCLUSION

The conclusion presents DireSMC as steering diffusion generation toward user-defined rare events with a smooth relaxation and accumulated importance weights. The appendix describes the tilted-chain identities and proves estimator properties under the stated setup.

5 CONCLUSION

  • DireSMC steers generation toward user-defined rare events using a smooth relaxation of the event set, accumulating importance weights to estimate event probability.The conclusion reports net speed-ups of 9 to 1413× over crude Monte Carlo on synthetic examples and a score-based climate emulator.Source: §5 · p. 10

A.1 NOTATION AND TWO IDENTITIES

  • The appendix invokes the tower identity and a telescoping identity under which incremental reward ratios collapse along a trajectory because rK = 0.Source: §A.1 · p. 20

A.2 STRUCTURE OF THE TILTED CHAIN

  • The tilted chain is described as a Markov chain with step-k marginals π⋆k and twisted transition kernels.Source: §A.2 · p. 20

A.3 PROOF OF THEOREM 1

  • The proof establishes unbiasedness of the probability estimator for every K, N, reward sequence, and resampling schedule, with sample convergence and a limit to pdata[E] under the stated assumptions.Source: §A.3 · p. 21

A.4 PROOF OF THEOREM 2

The appendices support the rare-event estimation approach by explaining its reward and guidance constructions, diffusion and SMC machinery, and the role of annealing. They also describe how the rare-event tilt complicates normalizer estimation, motivating splitting or annealing.

A.4 PROOF OF THEOREM 2

  • With indicator-proportional reward, the tilted chain equals the event-conditioned chain and the bound is tight; the CDF reward’s normalizer converges to p0[E] as δ decreases.The zero-variance Doob-transform case is circular because its normalizer is the quantity being estimated; the boundary condition rK = 0 incurs the bound in Eq. (33).Source: §A.4 · p. 22

A.5 THE OPTIMAL REWARD RECURSION

  • The Doob reward minimizes the variance of each incremental weight for any proposal, and iterating its variance-minimizing recursion recovers Doob rewards.The reward-dependent variance term vanishes only when the reward equals the optimal reward up to an additive constant.Source: §A.5 · p. 22

A.6 PROOF OF LEMMA 1

  • For Gaussian kernels, reward guidance shifts the mean by c_k∇x r_k(x_k), and the exact weight expression depends on the standardized shift; the integrator enters through c_k and v_k.The lemma does not cover multi-stage solvers, deterministic solvers, or anisotropic noise, for which its single-score Gaussian-kernel assumptions fail.Source: §A.6 · pp. 22–23

B DIFFUSION MODEL DETAILS

  • Score networks are trained by denoising score matching, while VP-SDE details use linear or cosine schedules and Tweedie/ΠGDM estimates supply posterior mean and covariance surrogates (Song et al., 2021b).The ΠGDM covariance proxy is exact for a single-Gaussian data distribution and used as a tractable surrogate otherwise.Source: B DIFFUSION MODEL DETAILS · pp. 23–24

C SEQUENTIAL METHODS

  • AIS and SMC temper rewards, while multilevel splitting uses nested rare-event sets; these methods transport particles along a sequence, with SMC and splitting resampling to focus on higher-weight trajectories (Del Moral et al., 2006; Neal, 2001; Glasserman et al., 1999; Kahn & Harris, 1951).The approaches differ in how they construct their sequences, with splitting using hard indicators in place of smoothed distributions.Source: C SEQUENTIAL METHODS · p. 24

C.1 ESTIMATING THE NORMALISATION CONSTANT

  • Small tempering steps keep weights near one, whereas a large step can collapse ESS; splitting instead estimates each level’s conditional reach probability by its surviving fraction.Both methods estimate the normalizer through a product of incremental ratios along overlapping bridge distributions rather than a single rare-event ratio.Source: §C.1 · p. 25

C.2 GENERIC SMC ALGORITHM

  • SMC resamples below an ESS threshold and rejuvenates particles; its unnormalized integral estimator is unbiased, whereas its normalized estimator is finite-sample biased but converges almost surely as N grows.Source: §C.2 · p. 26

C.3 WHY ANNEALING BECOMES NECESSARY

  • Even exact samples from the tilted target yield a poorly behaved endpoint estimate of the normalizer, motivating annealing by splitting or accumulated importance weights.The rare-event indicator shields the numerator from regions that the tilted distribution approximates poorly.Source: §C.3 · p. 26
  • The tilted distribution π0 largely avoids the bulk where the denominator integrand is largest, lowering effective sample size and producing a high-variance estimate of Z0 even with perfect π0 samples.The numerator remains bounded because it is evaluated where q samples effectively; the denominator requires bulk samples that q avoids.Source: §C.3 · p. 27
  • Breaking the estimate into transition importance distributions leads to splitting or annealing (Del Moral et al., 2006; Morio et al., 2014; Neal, 2001).Source: §C.3 · p. 27

C.3.1 EMPIRICAL EVIDENCE ON THE 2D GMM

The appendix examines probability estimation on the 2D GMM and develops splitting estimators, path weights, and guidance approximations. It also analyzes factorized guidance for multiple constraints and its effect on estimator variance.

C.3.1 EMPIRICAL EVIDENCE ON THE 2D GMM

  • With a known denominator, a small number of direct π0 samples can estimate the rare-event probability; without it, estimation fails, motivating annealing.Source: Table 2 · p. 27, §C.3.1 · p. 27

C.4 SPLITTING ESTIMATOR

  • A shared lower-level tilt cancels its normalizer but leaves a sparse numerator; two threshold-specific tilts avoid that ESS collapse but require estimating their normalizer ratio with sufficient overlap.Source: §C.4 · p. 28

D WEIGHT COMPUTATION

  • Girsanov’s theorem gives an exact change-of-measure weight between path measures; its control cost prices drift perturbation, with a martingale correction for favorable noise.Source: D WEIGHT COMPUTATION · p. 29

D.1 ζk FOR COMMON INTEGRATORS

  • As discretization increases or Δℓk shrinks, the integrator variants recover the Euler–Maruyama ζ²k/vk form.Source: §D.1 · p. 31, Table 3 · p. 31

E THE GUIDANCE GRADIENT VIA LOCAL LINEARISATION

  • A Gaussian denoising-posterior approximation and local constraint linearization turn the look-ahead reward into a closed-form Gaussian CDF, with posterior uncertainty inflating effective observation variance.Source: E THE GUIDANCE GRADIENT VIA LOCAL… · pp. 31–32

E.1 FACTORISED GUIDANCE FOR MULTIPLE CONSTRAINTS

  • Diagonal factorization drops off-diagonal look-ahead covariance and changes the proposal, but leaves the endpoint target unchanged, so the approximation contributes only to estimator variance.Source: §E.1 · p. 33

F THE CORRELATED 2D GMM BENCHMARK

The correlated 2D GMM benchmark provides closed-form quantities for evaluating rare-event estimation and guidance. Experiments also examine learned-score tail bias and identify score approximation and discretization as sources of bias relative to the true event probability.

F THE CORRELATED 2D GMM BENCHMARK

G.2 SOURCES OF BIAS

  • Bias relative to the true event probability arises from learned-score approximation and from the sampled discretized law differing from the continuous law; the path-weighted estimator remains unbiased for its sampled law.Source: §G.2 · p. 37

G.2.1 TAIL BIAS OF THE LEARNED SCORE

  • Within 7% across three decades of rarity, the two tail-bias estimates agree; both show widening bias as rarity increases and monotonically degrading score estimates.Source: Table 5 · p. 37, §G.2.1 · p. 37

G.2.2 THE DISCRETISED LAW: GRID AND INTEGRATOR

The section compares discretization bias across integrators and grid placements, finding that SEEDS-1 generally reduces bias at lower step counts while both integrators become comparable at higher counts. Grid placement also affects variance diagnostics, and finer grids bring the discrete chain closer to the continuous dynamics.

G.2.2 THE DISCRETISED LAW: GRID AND INTEGRATOR

  • For K ≥30, the uniform-t fitted 1/K coefficients are +7.27 for EM and −4.41 for SEEDS-1, with largest residuals of 1.6 and 2.4 standard errors, respectively.The fit excludes K = 10–20 as outside the asymptotic regime; on uniform-ℓ, the SEEDS-1 coefficient changes sign between K = 30 and 50.Source: §G.2.2 · pp. 38–39
  • On uniform-t grids SEEDS-1 has lower bias at every K, while uniform-ℓ reverses the ranking at K = 10 and 15; the authors hypothesize that coarse ℓ-grids undersample the denoising middle.The integrators converge to each other by K = 200.Source: §G.2.2 · p. 39
  • At K ≥20, SEEDS-1 has lower discretization bias, the integrators converge by K = 200, and both are within 0.2% of the continuous limit at K = 1000 on the uniform-ℓ grid.The K = 1000 configuration is the one deployed in the larger climate experiments.Source: Table 6 · p. 38, §G.2.2 · p. 39
  • At K = 100, EM’s ESSE is 0.713 on uniform-ℓ versus 0.600 on uniform-t, while its PSIS k̂ is 0.24 versus 0.40.Across the uniform-t EM arm, CV falls from 0.338 to 0.009 and ESSE rises from 0.032 to 0.943 as K increases.Source: Table 6 · p. 38, §G.2.2 · p. 39
  • With exact Doob guidance, finer grids bring the discrete chain closer to continuous dynamics, shrinking discretization bias and moving the proposal toward the variance-minimizing Doob proposal.With approximate guidance, the passage says Doob h-transform approximation error can dominate discretization bias.Source: §G.2.2 · p. 39

G.3 THE THRESHOLD-EXCEEDANCE EXPERIMENT

The deployed threshold-exceedance experiment evaluates conditional samples, resampling and ESS behavior, and scaling with particle budget. The reported outcome is recovery of the correct probability estimates and shape statistics.

G.3 THE THRESHOLD-EXCEEDANCE EXPERIMENT

  • The deployed threshold-exceedance experiment recovers the correct probability estimates and shape statistics, as summarized in Table 7.The conditional clouds, no-resampling arm, whole-path ESS diagnostics, and particle-budget scaling are examined.Source: §G.3 · p. 39
  • The subsection covers conditional sample clouds, a no-resampling arm, whole-path ESS diagnostics, and estimator scaling with particle budget N.These are presented as components of the Section 4.1 threshold-exceedance experiment.Source: §G.3 · p. 39
  • Unless otherwise stated, experiments use K = 1000, the SEEDS-1 integrator, and a uniform-ℓ grid with the same guidance.Source: §G.3 · p. 39

G.3.1 CONDITIONAL-SAMPLE COVERAGE: CLOUDS FOR LEARNED AND ANALYTICAL

The conditional-sample plots show guided samples visually indistinguishable from exact conditional samples. The ESS analysis tracks a flat untilted phase followed by decay after guidance begins, with similar profiles for learned and analytical scores.

G.3.1 CONDITIONAL-SAMPLE COVERAGE: CLOUDS FOR LEARNED AND ANALYTICAL

  • The guided sample clouds are visually indistinguishable from exact conditional samples, qualitatively supporting conditional-distribution coverage.The plots use N = 2048 samples with the deployed configuration; Table 7 provides KS and EMD ratios.Source: §G.3.1 · p. 39, Table 7 · p. 41

G.3.2 ESS ALONG THE TRAJECTORY

  • Without resampling, terminal ESS asymptotically measures the gap between the implemented guidance and exact Doob guidance.The terminal ESS approaches 1/(1 + χ²(P⋆∥Q0:K)).Source: §G.3.2 · p. 39
  • Figure 7 indicates that SDE integrators require fewer discretization steps than the ODE integrator to reach bias below 10% across rarities.It compares Euler-Maruyama, SEEDS-1, and DDIM using unconditional base-model samples.Source: Figure 7 · p. 40
  • Figure 8 reports O(1/K) bias for analytical scores with Doob transitions and 1000 particles, while Figure 9 measures convergence to an unconditional-MC reference under deployed learned guidance.Figure 9 uses 10,000 particles and no rejuvenation.Source: Figure 8 · p. 40, Figure 9 · p. 41
  • For five thresholds, ESS/N stays at 1 until t ≈0.9, when guidance turns on, then decays monotonically for both scores; the learned and analytical profiles remain close.The passage attributes the profile to guidance geometry rather than the score itself.Source: §G.3.2 · pp. 40–41

G.3.3 SELECTION AND REJUVENATION

Across the experiments, resampling and rejuvenation affect weight quality and variance, while correlated guidance improves the rarest compound-event estimate. Results also show boundaries of these benefits, including approximate rejuvenation and the cost of added computation.

G.3.3 SELECTION AND REJUVENATION

  • Rejuvenation reduced CV across all tested GMM rarities, while resampling alone could adversely prune lineages.Across the climate-emulator ablation, resampling lowered CV in eight of nine cells; rejuvenation's CV effects were mixed and within sampling error.Source: §G.3.3 · p. 42, Table 8 · p. 43, §H.2.1 · p. 51
  • For learned scores, MADM may fail to preserve πk when the score is non-conservative, so rejuvenation is turned off as non-conservativity rises.The score network becomes increasingly non-conservative as denoising time approaches t = 0.Source: §G.3.3 · pp. 43–44
  • At τ = 10, ungated rejuvenation had KS 1.48/1.59 and EMD 2.10, versus 1.23/1.07/0.98 when gated at t ≥ 0.30.The authors adopted the gate ℓt ≤ −0.21; at τ = 7 and 8, gated and ungated results coincided within noise.Source: Table 9 · p. 44, §G.3.3 · p. 45
  • MADM targets continuous-time marginals rather than the discretised γ0 law, introducing discretisation bias that grows with M.The reported bias decreases as K increases; the two integrators have different error profiles.Source: §G.3.3 · p. 45, Figure 13 · p. 45

G.4 BOX AND MULTIPLE BOX EVENTS

  • For rare learned-score box events (p⋆ ≤ 2 × 10^-4), rejuvenation lowered CV by 1.5–2.5×; on easier events (p⋆ ≥ 10^-2), it was neutral or slightly worse.The caption identifies Table 10 as the box-event results for the learned score.Source: §G.4 · pp. 45, 47, Table 10 · p. 47

H.2.1 SELECTION AND REJUVENATION

  • Without resampling, median ˆk exceeded 0.5 in seven of nine climate-emulator cells, indicating heavy-tailed weights and slow estimator convergence.The authors note the accompanying CV is optimistic.Source: §H.2.1 · p. 51, Table 16 · p. 54

H.2.4 FACTORISED VERSUS MULTIVARIATE GUIDANCE

  • At 10^-4, the joint reward CV was 0.080 versus 0.310 for the factorised reward (variance ratio 14.8, p < 0.001).The corresponding ˆd/d⋆ values were 1.06 and 1.12, respectively.Source: §H.2.4 · p. 55, Table 18 · p. 55

H.2.5 CONDITIONAL FIELDS FOR COMPOUND EVENTS

  • Guiding on temperature alone reduced humidity across the region in the compound-event fields, and the schedule ablation found lower CV with rejuvenation than without.The no-rejuvenation variants converged on λt ≡ 1, whereas the parameter schedules were selected heuristically and not independently tuned for other settings.Source: Table 11 · p. 48, §H.2.5 · p. 55, Figure 18 · p. 56

H.3 EXPERIMENT 2: A CHANGING CLIMATE

The warming sweep compares DireSMC estimates with large Monte Carlo references where those references are available, and also compares its precision with compute-matched Monte Carlo. At lower warming levels, validation against a large reference was not feasible.

H.3 EXPERIMENT 2: A CHANGING CLIMATE

  • At ΔT ≥ 1.5 K, guided estimates matched all four large-Monte-Carlo reference points for the frozen τ = 5.81 K event.The table reports 16 seeds per point; no large Monte Carlo reference was available below ΔT = 1.5 K.Source: Table 19 · p. 57, §H.3 · p. 57
  • At ΔT = 4.0 K, the frozen-threshold event had estimated probability p0[E] = 0.95; validation at ΔT ≤ 1.2 K was not possible under the available compute.A 5% relative-error Monte Carlo estimate was estimated to require about 10^8/10^7/10^6 unconditional draws at ΔT = 0.0/1.0/1.2 K, respectively.Source: §H.3 · p. 57

H.4 COMPUTATIONAL COST

  • The path-weighted sampler's net speed-up generally outweighed the added cost of guided steps, and it was described as feasible where crude Monte Carlo produced no event samples.The comparison is at matched CV and includes each arm's measured wall-clock cost.Source: §H.4 · pp. 58–59, Table 21 · p. 59

H.5 HYPERPARAMETER ABLATION

The schedule ablation varies guidance start time and tempering shape on a marginal p⋆ = 10^-3 problem. The authors selected the reported parameters heuristically and reused them across experiments.

H.5 HYPERPARAMETER ABLATION

  • The schedules were selected heuristically on marginal hot and dry p⋆ = 10^-3 problems, without independent tuning for other settings.The same adopted hyperparameters were used across experiments.Source: §H.5 · p. 59
  • The ablation varied the guidance start point ton and the exponent p in λ(t) = 1 − (t/ton)^p.The caption describes a one-knob schedule ablation at p⋆ = 10^-3 using the same 16 seeds in every row.Source: §H.5 · p. 59, Table 22 · p. 60
  • Collapsed runs were included: one hot run at λmax = 2.0 and two hot and two dry runs at δhi = 0.25 SD.Source: Table 22 · p. 60

I METROPOLIS-ADJUSTED LANGEVIN CORRECTIONS

The section presents a Langevin proposal and a Metropolis-adjusted correction for rejuvenation, then contrasts DireSMC's path weights with a PF-ODE estimator. Both estimators have finite-discretisation considerations, but their weighting constructions and assumptions differ.

I METROPOLIS-ADJUSTED LANGEVIN CORRECTIONS

  • ULA proposes moves using the guided score but accepts every proposal, leaving πk invariant only in the ρ → 0 limit.MADM adds a Metropolis-Hastings correction to address this bias.Source: I METROPOLIS-ADJUSTED LANGEVIN… · p. 60
  • MADM estimates the intractable ptk ratio by a score line integral, which is valid when the score is a gradient field but only approximated by quadrature here.The Simpson-rule quadrature has O(ρ^5/2) truncation error, so the adjusted kernel is not exactly πk-invariant.Source: I METROPOLIS-ADJUSTED LANGEVIN… · pp. 60–61
  • The line-integral correction adds score evaluations, while the endpoint scores and rewards are shared with the proposal and acceptance test.Source: I METROPOLIS-ADJUSTED LANGEVIN… · p. 61

J.1 THE PF-ODE ESTIMATOR

  • DireSMC accumulates a Girsanov path ratio during the guided reverse-SDE run; the compared PF-ODE method uses a ratio of forward-evaluated endpoint densities.The path ratio depends on the whole trajectory, not only its endpoint.Source: J RELATED WORKS EXTENDED · p. 61, §J.1 · p. 62
  • At finite K, the PF-ODE weight is exact only under stated density-matching conditions that the authors say do not hold for their explicit Euler integrator.In non-analytical settings Dmap is unknown, so the PF-ODE approach assumes p↼ = p0; DireSMC instead requires its K-step chain law to approach the model's.Source: §J.1 · pp. 62–63

J.1.1 EMPIRICAL RESULTS

In the 2D GMM threshold-exceedance test at τ=10, DireSMC is compared with Manshausen et al. under discretization and compute-matched settings. The reported comparisons favor DireSMC in finite-K bias and CV, with higher crossing fractions under adaptive resampling.

J.1.1 EMPIRICAL RESULTS

  • In practice, the Hutchinson trace estimator adds variance that shrinks with more probes, at the cost of greater computational overhead.The bias-isolation comparison instead supplies the true divergence.Source: §J.1.1 · p. 63
  • At K=50/100/200/500, DireSMC’s crossing fractions were 86.5%/80.4%/81.3%/79.4% versus 10%, with lower seed-level standard error and lower finite-K bias.The bias comparison is against the analytical ground truth.Source: §J.1.1 · p. 63, Table 23 · p. 63
  • For τ=10, DireSMC uses about 2032 function calls per particle versus 7000 for the odds-ratio estimator when P=1.The call count includes score, reward, and reward-gradient evaluations; rejuvenation accounts for 32 calls.Source: §J.1.1 · p. 64

J.1.2 DETAILED COMPARISON

  • At K=1000 and τ=10 with learned scores, DireSMC with rejuvenation outperforms Manshausen et al. in CV in the particle-count comparison.The comparison covers CV of the estimated probability over 24 seeds.Source: §J.1.2 · p. 64, Table 25 · p. 64
  • The bias-versus-K evaluation uses analytical scores, Doob transition kernels, no resampling, and N=1000 to isolate errors introduced in Eq. 124.The reported statistic is mean estimated probability relative to the model’s threshold-exceedance probability, with standard error over 24 seeds.Source: §J.1.2 · p. 64, Table 27 · p. 65
  • At equal compute across rarities, all DireSMC variants outperform Manshausen et al. in CV, with the comparator using 286 particles versus DireSMC’s 1000.The comparison uses learned scores and K=1000.Source: §J.1.2 · p. 64, Table 26 · p. 65
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