Simulation-Based Inference

For a given data distribution $(X_t)_{t \in \mathbb{N}} \sim Q$ i.i.d., we investigate the hypothesis testing problem: $H_0: Q = P_0$ vs. $H_1: Q = P_1$, for two different model probability distributions $P_0$ and $P_1$. In contrast to the standard setting, where analytic densities $p_0$ and $p_1$ are given, here, we consider the density-free setting, where we only have access to i.i.d. simulations $(Z^0_t)_{t \in \mathbb{N}} \sim P_0$ and $(Z^1_t)_{t \in \mathbb{N}} \sim P_1$. For this simulation-based hypothesis testing setting, we construct an e-test martingale, resulting in a sequential test with anytime-valid type-I error guarantees, approximate growth optimality, geometrically decaying type-II error bounds, and asymptotic power one. Most ingredients used in our constructions are variants of well known concepts. The value of this paper lies in the compact presentation of an effective, anytime-valid solution for the density-free simulation-based sequential hypothesis testing case.

Simulation-based inference (SBI) has become a powerful approach to Bayesian inference in complex scientific models whose likelihoods are difficult or impossible to evaluate. Amortized SBI learns reusable inference models from simulated data, enabling rapid posterior inference for new observations, and modern generative models have made these models increasingly expressive. However, this reuse is limited to the prior distribution chosen during training, whereas scientific analyses often need revised priors as knowledge accumulates or alternative assumptions are tested. We introduce Spectra, a test-time adaptation method for diffusion-based SBI. Spectra uses an exact score-transport identity to obtain the adapted score from a frozen diffusion model in closed form for structured prior changes, without additional simulation or training. Across six SBI benchmarks, Spectra achieves accurate adaptation under strong prior shifts at low online sampling cost. This enables pretrained SBI models to incorporate updated prior information at test time.

We present a new perspective on the problem of likelihood approximation within the framework of simulation-based inference that promotes scalable and controllable simulation routines for large-scale data analysis, allows efficient parameter space exploration or smooth interpolation in high-dimensions and, thus, supports valid statistical treatments of hypothesis testings as well as uncertainty quantification. In particular, we consider a chain of $n$-aggregated normalizing flows for likelihood approximation scheme, where a set of upfront replicated observation datasets from the forward complex simulation model pass through the first set of bijective transformations, and then subsequently pass to the other sets of bijective transformations. Here, we assume that, for any $k \in \{1,\,2, \ldots, n\}$, the parameters corresponding to the first $k$ sets of bijective transformations are estimated sequentially, in some sense of optimality, for constructing flexible probability distributions, regardless of the remaining $(n-k)$ sets of bijective transformations. Moreover, our objects of interest are to highlight two complementary mathematical arguments that leverage an informatics-theoretic formalization, based-on empirical likelihood estimators under moment restrictions, and a sequential decision-making paradigm, with mixing distributions, for updating and aggregating the estimated parameters of the overall normalizing flows. As a by-product, the framework provides a reliable surrogate model, conditioned on the model parameters defining the forward computational simulation, that allows samples generation, with statistical powers, and facilitates computationally tractable scheme in the Bayesian paradigm for inference, hypothesis testings and uncertainty quantification.

Casting Bayesian inference as a neural network optimization problem targeting an amortized posterior is attractive, as it extends to otherwise intractable statistical models and offers near instantaneous inference for new datasets after prepaying the training cost. Although theory guarantees faithfulness under ideal convergence, practical amortized inference still requires iterating over architectures and optimization choices and ultimately ``satisficing'' under finite simulation, compute, and time budgets. Even the best-performing solution may thus retain avoidable representation gaps that typically require problem-specific fixes. Here, we propose a generic alternative which improves training dynamics with auxiliary guidance losses applied to internal representations. Specifically, we show how such guidance leads to faster convergence when training data is abundant and to better performance when it is scarce. We formalize representation gaps as getting stuck in a local optimum at the information bottleneck between the parts of the network tasked with feature learning and those tasked with conditional distribution learning, and offer a generic diagnostic to separate summary failures from inference failures. Finally, we demonstrate that auxiliary supervision improves convergence speed and accuracy on a range of challenging real-world inference problems.

Models of quantum systems faithfully map system parameters to observations, but the inverse problem of parameter inference from measurement data presents a fundamental challenge: computationally intractable likelihoods due to an exponentially large Hilbert space. Here, we introduce simulation-based quantum system inference, a unified, likelihood-free framework that learns parameter posteriors directly from classical simulation data. The central idea is to pair polynomial-cost classical simulators, such as Pauli propagation and tensor networks, with normalizing flows or other neural density estimators for accurate, reusable inference. A single model, trained once, maps any new measurement record to its posterior in one forward pass---turning per-experiment inference into a fixed, up-front cost. We numerically demonstrate the framework's versatility across Pauli noise learning, quantum error mitigation, quantum state tomography, and Hamiltonian learning, with examples involving 81-qubit shallow circuits and 735-parameter inference. In each case, the approach yields accurate estimates of identifiable parameters, while posterior uncertainty provides additional diagnostics of non-identifiability and indicates where further characterization is needed. Our framework reduces data-acquisition requirements in quantum experiments and accelerates parameter inference, providing a practical route to characterizing and improving large-scale quantum systems.

Validation of generative social simulators often stops at face validity: emergent network structure is compared descriptively, without quantified parameter uncertainty or an adequacy check. We present an adequacy-aware calibration protocol that couples amortized posterior estimation with a synthetic identifiability assessment, a matched-sample-size adequacy check (prior-predictive reachability plus per-statistic posterior-predictive localization), a diagnosis-guided repair, and a statistic-held-out audit. We demonstrate it on a real second-hand luxury resale market with four channel-by-residency cells, each a bipartite buyer-brand network, using a forward model built from persona profiles elicited once, offline, by a language model. The behavioural parameters are recoverable in all four cells, though calibration is approximate and overconfident for one parameter. The observed summary falls outside the simulator's reachability reference in every cell, with the mean purchased tier as the pervasive discrepancy. The repair meets the value-block criterion in two of four cells but does not restore adequacy, and the held-out audit surfaces a buyer-breadth-dispersion miss no earlier diagnostic detected. A profile-source ablation finds the language-model profiles beat a flat rule baseline in all four cells, yet within-category brand relabelling causes no consistent degradation, so the profiles are a partially validated input whose value rests on structure, not brand identity. Making no causal claim, we conclude that an independent-aggregation account, without agent interaction or a buyer-breadth mechanism, cannot jointly reproduce the market's purchased-tier level, head-brand concentration, community structure and buyer-breadth heterogeneity.

We present a simple decomposition of a neural-network-based normalizing flow that naturally uncovers a pivotal statistic (or something close) in the presence of nuisance parameters, based only on a sample generator from the distribution of interest. We show that the statistic is near-pivotal in the sense of minimum average KL-divergence of its $p$-values versus uniform and we argue that it can be expected to have good power when the dimension of the statistic equals the dimension of the parameter. It is able to incorporate prior knowledge about group invariances such as translation and scale. It can discover the one-sample $t$-test almost exactly, outperforms the Welch test in terms of worst-case size over a constrained variance-ratio range and achieves good calibration on partial biserial correlations, while showing higher power (and being much faster) on small-to-moderate samples than profile likelihood-ratio techniques.

We perform field-level likelihood-free inference of the matter density parameter $Ω_m$ from simulated galaxy catalogs using machine learning models with differing inductive biases. Using hydrodynamic simulations from CAMELS, we examine how observable choice and architecture govern cosmological information extraction. We consider galaxy positions and line-of-sight peculiar velocities, separately and jointly, and compare permutation-invariant Deep Sets, implemented with either multilayer perceptrons (MLPs) or Kolmogorov-Arnold Networks (KANs), to graph neural networks (GNNs), which explicitly encode spatial relations. We test in-distribution and out-of-distribution (OOD) performance across simulations with different subgrid galaxy-formation prescriptions. Deep Sets infer $Ω_m$ from velocities alone with mean relative errors of approximately $18\%$ in-distribution and $\sim25\%$ OOD, with KANs and MLPs achieving comparable performance. In contrast, the same set-based approach does not yield useful $σ_8$ predictions in either in-distribution or cross-suite tests. Adding positions does not improve Deep Sets, while GNNs infer $Ω_m$ with mean relative errors of about $10\%$ in-distribution and $10$--$17\%$ OOD. These results indicate that peculiar velocities provide the dominant source of $Ω_m$ information for set-based models in this setting, while spatial information is most effectively used by architectures that explicitly encode galaxy-galaxy relations. Because the velocity inputs are exact simulated peculiar velocities, applications to survey data will require validation under realistic velocity-measurement noise, selection effects, and survey geometry.

Simulation-based science often requires a distribution over simulator parameters whose push-forward reproduces a set of real observations: this is the source distribution estimation (SDE) problem. Existing methods fit the source against a likelihood surrogate trained once from a fixed proposal prior. Their objective is therefore stated only in terms of the surrogate instead of the true simulator, which may fail for inaccurate areas in parameter space where the surrogate was never trained. We instead solve SDE by expectation maximization: an E-step trains an amortized posterior on fresh simulations from the current source estimate, and an M-step refits the source to the average of that posterior over the observed data. We give two parameterizations, (1) separate source and posterior flows and (2) a single shared conditional flow. We evaluate our method on three benchmark tasks under both broad and misspecified initial priors. Both improve on existing fixed surrogate approaches and on iterated variants of each, most clearly on Lotka--Volterra, where no baseline falls below 0.96 data-space C2ST while our methods reach 0.64-0.68 in three of four initial-prior settings.

Computational models of blood clotting improve understanding of thrombus formation, but their clinical application remains limited because many model inputs are difficult to measure and patient-specific data are often sparse. We present a computational framework based on latent neural differential equations that infers unknown model parameters from sparse measurements and forecasts thrombosis progression. We demonstrate the framework using data generated from a multiphysics blood-clotting model in which clot growth is governed by the coagulation cascade and diffusion. Four known biochemical inputs (fibrinogen and factors IX, VIII, and V), together with sparse early clot-size observations, are used to infer the tissue-factor parameter and predict subsequent clot growth. We compare seven probabilistic methods: stochastic neural ordinary differential equations (SNODE), stochastic neural functional differential equations (SNFDE), a latent neural-process baseline, a monotone probabilistic deep ensemble, empirical trajectory retrieval, PCA-ridge Gaussian posterior, and Gompertz-curve retrieval. SNODE achieved the best performance in inferring the unknown input and forecasting future clot-growth trajectories. SNFDE performed similarly and consistently outperformed the other non-differential models. Prediction accuracy improved as more observations became available, whereas longer forecasting horizons increased uncertainty and decreased accuracy. Latent neural differential equations thus effectively combine parameter inference and clot-growth forecasting from sparse measurements, providing a promising foundation for personalized thrombosis modeling.

With AI systems gaining more access to individuals' information, it is important to protect privacy when reporting statistical answers. Equally important is to privatize the reporting of uncertainty in such answers. To this end, we adopt a Bayesian likelihood-free framework and make simulation from the posterior private. In particular, we propose a new private instantiation of the Bayesian bootstrap using a blocking strategy. Rather than assigning idiosyncratic random weights to each individual, we randomly group individuals and assign a single weight to each group. By concealing individuals' contributions within a group, we fortify differential privacy gates. We harness amortized inference that decouples private learning from posterior sampling. A push-forward map from observation weights to posterior samples is learned privately by adding calibrated noise during training. Subsequent posterior draws require no additional privacy and computation budget. We call the resulting method the Private Generative Bayesian Bootstrap (PGBB). We establish a differential privacy guarantee, analyze convergence to the non-private blocked-bootstrap target, and quantify the discrepancy between the ordinary and blocked Bayesian-bootstrap posteriors. In addition, we derive data-free tuning of the block Dirichlet concentration parameter that restores posterior dispersion asymptotically. We also show a single fit of PGBB can support a family of loss-based decision rules simultaneously without additional privacy cost. In simulations and in applications to U.S. Census returns to schooling and U.S. natality birthweight quantiles, PGBB gives competitive private uncertainty quantification and improves over private Bayesian alternatives that require a specified data-generating model in common settings.

Understanding and controlling complex dynamical systems often requires executing thousands of numerical simulations across vast parametric landscapes, which is time-consuming. Machine learning surrogates significantly accelerate simulation by predicting state trajectories across different initializations and parameter values. However, surrogate models are specialized to one simulation model. Modifying the underlying differential equations - e.g., adding a physiological state or altering an epidemiological contact network - renders trained models obsolete and forces computationally expensive retraining from scratch. We introduce HyperODE, a surrogate capable of operating across an entire class of approximately mass-conserving compartmental models without retraining. By mapping the structure of ordinary differential equations (ODEs) into directed hypergraphs, HyperODE decouples the functional form of system interactions from the neural network architecture. HyperODE takes a compartmental model in the form of an ODE with an arbitrary parameter distribution defined through quantiles and transforms it into a hypergraph. It outputs the distribution of the trajectories for all the states in the original ODE in the form of quantiles. We then use this surrogate to build an encoder that takes a noisy trajectory and outputs a distribution over the parameters of the original ODE, thus calibrating the model in a single pass. On families and system sizes never seen in training, HyperODE produces calibrated quantile bands in a single forward pass, with weighted-interval score and coverage on par with specialized surrogates for each structure. For inverse inference, HyperODE produces calibration from noisy state trajectories in a few milliseconds with a single shared encoder, competitive with existing methods. HyperODE extends zero-shot to ODEs that break mass conservation and to external forcing.

In this work, we present a simulation-based parameter estimation framework for a model defined by a computational simulation of a physical system. We specifically outline an estimation framework consisting of two closely-integrated steps that facilitate an overall end-to-end parameter estimation scheme. The first step involves utilizing an embedded normalizing flow which is used to transform the unknown complex distribution of the residual information into a simple base distribution corresponding to the transformed residual information. In the second step, an empirical-likelihood estimator, under moment restrictions, is utilized for imposing an indirect constrain on the base distribution, where such an instantiated task reasonably allows us to treat the transformed residual information as random variables arising from discretely distribution population with each transformed data point as a single-cell from a set of finite-cell contingencies. Moreover, we use first-order gradient methods for updating the estimated parameter values of the model defined by the computational simulation and the corresponding parametrized embedded normalizing flow, that call for all gradient-related information by leveraging implicitly differentiations of the empirical-likelihood function, which is constructed from the implied empirical probabilities under moment restrictions. Here, it is worth mentioning that the problem formulation presented in this work, which highlights an information-theoretic interpretation, allows to present a computational framework for algorithmic implementations. Finally, as a-by-product, the inverse of the parametrized embedded normalizing flow, w.r.t. the estimated parameter values, serves as a surrogate model for the computational simulation model, which provides useful information for quantifying model discrepancies and sensitivity analysis.

Empirical Bayes (EB) performs simultaneous inference across many related latent variables. Classical EB assumes that the likelihood p(x | z) is tractable. In many scientific applications, however, the likelihood is available only through a simulator. This paper develops EB for such implicit likelihoods. We introduce simulation-based empirical Bayes (SBEB), which connects nonparametric EB to simulation-based inference (SBI). SBEB computes EB estimates without an explicit density by using the observed data, simulator samples, and an amortized inference network. SBEB iteratively refines the fitted EB prior toward the population prior. With several scientific simulators and real-world data, we demonstrate that SBEB improves accuracy over SBI with a fixed prior.

Simulation-based inference (SBI) with machine learning is an increasingly important tool for solving inverse problems in science and engineering, including parameter inference and the inversion of detector effects. We provide an overview of the Bayesian and frequentist statistical frameworks, describe how machine-learning-based SBI methods, such as neural posterior estimation and neural likelihood estimation, can be used for parameter estimation within these frameworks, and show that the same methods can also be applied to Empirical Bayes or unfolding tasks. We also discuss how to validate inference results and the limitations of SBI with machine learning.

Neural simulation-based inference enables parameter estimation for complex models, but typically requires the user to specify a simulator encoding a fixed model structure. We present a framework for joint model selection and parameter estimation that combines large language models for program synthesis with neural simulation-based inference. Given a natural language description of the system and data under investigation, an LLM proposes candidate simulator programs which are iteratively refined via feedback-driven mutation and evaluated using neural density estimation. The approach enables simulation-based inference over a pool of models, not just parameters within a fixed model. On benchmarks spanning deterministic dynamics, stochastic epidemic models, and dark matter substructure inference from gravitational-lensing images, the method identifies plausible model families from open-ended prompts, with accuracy that reflects the information content of the data and identifiability of candidate models.

Contact-rich manipulation requires pose estimates that are often more accurate than what depth-only sensing provides. Existing methods, relying on vision and contact, employ costly offline training procedures that need to be retrained for new environments and geometries. We propose BayesContact, a Simulation-Based Inference framework for visuo-tactile pose estimation in peg-in-hole insertion. BayesContact maintains a particle belief over object pose and fuses depth observations with force/torque-derived contact evidence. We employ simulation based forward models to approximate these observation likelihoods. For each pose hypothesis, a renderer predicts depth measurements and a physics simulator predicts contact outcomes under guarded probing actions; both are scored against real observations to update the belief. The resulting multimodal belief also enables information-gain-based probing for active disambiguation. Across simulated geometries and real-robot experiments, BayesContact improves pose observability and insertion success over vision-only inference by 30%

The prevailing approach to inferring weak gravitational lensing shear from images involves detecting galaxies, estimating their ellipticities, and calibrating these estimates to correct for image noise, selection bias, and model misspecification. Characterizing the statistical model and assumptions underlying this pipeline is challenging, which makes it difficult to propagate uncertainty through its various stages. As an alternative, we propose to infer shear using neural posterior estimation (NPE), a type of simulation-based inference. We train a deep neural network to map a simulated multiband image to a variational distribution over the underlying shear field, thereby folding galaxy detection, deblending, measurement, and calibration into a single implicit inference step. Once trained, the network accounts for all features present in the simulated images, including potential sources of bias. In experiments on simulated constant-shear images with increasingly complex observational effects, NPE produces accurate and well-calibrated posterior approximations for both shear components in the presence of blended galaxies, spatially varying point spread functions, stars, and detector artifacts. These results demonstrate that NPE can be a viable shear estimation method in settings where all anticipated features and artifacts can be simulated, a requirement that will become increasingly feasible as simulation fidelity improves in the coming decades.

Simulation-Based Inference (SBI) is critical for scientific discovery, with generative models offering a promising path toward efficient inference. However, existing methods struggle with effective multimodal modeling. They often rely on brute-force fusion strategies that ignore the structural disparities between parameters and observations, thus limiting estimation fidelity. In this work, we introduce FUSE (Feynman-Kac steered mUlti-modal flow matching for efficient Simulation-based posterior Estimation). Unlike prior work, FUSE employs a dual-track architecture that preserves the distinct features of multimodal inputs while facilitating dynamic interaction. Additionally, we propose an FK-steered sampling strategy that leverages intermediate observation likelihoods to guide the generative trajectories, effectively improving the sample quality during inference. Our approach outperforms state-of-the-art baselines on standard SBI benchmarks, producing posteriors that closely match ground-truth MCMC. Furthermore, in a real-world exoplanet orbital estimation task, FUSE successfully resolves complex parameter degeneracies that challenge existing methods, highlighting its potential to accelerate complex scientific discoveries in astrophysics and beyond.

Reinforcement Learning (RL) is a sequential decision-making framework in which an agent learns optimal policies through interaction with an environment by maximizing cumulative rewards. Among RL methods, Bayesian Reinforcement Learning (BRL) addresses common practical challenges related to data scarcity by leveraging prior knowledge about the environment and sequential belief updates. However, most BRL approaches require an explicit likelihood function, which is frequently inaccessible or intractable in real-world settings. We propose Likelihood-Free Iterated Batch Importance Sampling (LF-IBIS), a novel algorithm for BRL that updates the agent's beliefs online as new interactions become available. By combining Approximate Bayesian Computation with Iterated Batch Importance Sampling, LF-IBIS enables full Bayesian inference in settings where the environment dynamics are not described by an explicit or tractable likelihood. The method yields approximate posterior distributions over both environment parameters and optimal policies, providing a quantification of policy uncertainty useful for a Bayesian treatment of the exploration-exploitation trade-off. We test the method on a simulation study in response-adaptive randomization in clinical trials, where closed-form posteriors enable validation. Additional experiments address settings where the posterior has no closed form and illustrate online policy updating based on the posterior distribution of the optimal policy.

We introduce a neural network-based framework for learning time series estimators through a process we term decision-theoretic pretraining. Analysts specify a generative world, a distribution over data-generating processes, and a target decision objective. A neural network trained on stratified simulations from this world approximates the corresponding optimal decision rule, yielding a neural estimator that provides forecasts, parameter estimates, predictive intervals, or model-selection for zero-shot inference on previously unseen time series. The joint specification of the generative world and objective enables the estimators to directly approximate process-level, finite-sample properties: near-optimal risk, bias control, minimax performance, and uniform calibration. Our experiments demonstrate that these neural estimators can outperform traditional baselines such as maximum likelihood estimation and model selection via AICc, for the same model structural model classes. Furthermore, even when trained purely on simulations of structural models, they achieve competitive or state-of-the-art forecasting accuracy on major real-world benchmarks, compared with statistical, neural or large pre-trained models. We illustrate the framework by addressing two longstanding challenges: finite-sample bias and miscalibration in AR(p) models, and the forecast combination puzzle. These applications highlight the approach's main advantage: its ability to approximate solutions to analytically intractable or computationally prohibitive time series problems, including complex structural equations or optimality criteria. Ultimately, by enabling explicit control over decision-theoretic trade-offs, the framework equips analysts with highly efficient estimation tools tailored to their specific analytical needs.

Mechanistic epidemiological models are widely used to support infectious disease forecasting and public-health decision making. Bayesian calibration of such models is commonly performed using Markov chain Monte Carlo (MCMC), which can become computationally expensive for high-dimensional nonlinear systems and repeated near-real-time analyses. Here, we investigate simulation-based inference (SBI) using neural posterior estimation as a scalable alternative for Bayesian calibration of a mechanistic SECIR epidemiological model using COVID-19 intensive care unit (ICU) occupancy data from Germany during 2020. We compared SBI and MCMC across multiple epidemic phases using both 31-day inference windows and a substantially more challenging 201-day reconstruction problem involving multiple transmission change points. Posterior agreement was evaluated quantitatively using Wasserstein distances and Kullback-Leibler divergences together with posterior predictive checks. Across the 31-day windows, SBI recovered posterior distributions in strong agreement with MCMC while accurately reproducing observed ICU trajectories. In the 201-day setting, SBI preserved the dominant posterior structure despite increased uncertainty. SBI, by combining CPU and GPU resources, substantially reduced computational runtime compared with MCMC, which was restricted to running on CPUs. Whereas MCMC required approximately 1000 seconds for the 31-day inference problems, SBI achieved comparable posterior and predictive performance in approximately 60-70 seconds on a single GPU. For the 201-day inference problem, SBI required an average of 157 seconds, while the MCMC runs took over 19,000 seconds. Our results demonstrate that SBI provides a rapid and computationally efficient framework for Bayesian calibration of mechanistic epidemiological models, supporting repeated near-real-time inference and rapid outbreak analysis.

Modeling chaotic systems is crucial yet challenging. Inverse problems in chaotic dynamics, namely inferring initial conditions from final states, remain largely unsolved because of ill-posedness, non-uniqueness, instability, and potentially chaotic time-reverse dynamics. We address this open problem with Bidirectional Conditional Flow Matching (Bi-CFM), which learns bidirectional mappings between distributions of initial and final states to capture the stochasticity of chaotic evolution and mitigate exponential error accumulation over time. Furthermore, for systems with conservation laws, we extend it to Conservation-constrained Bi-CFM (CBi-CFM). Across the classic Lorenz, Circuit, and high-dimensional Lorenz 96 systems, Bi-CFM improves five distribution-level metrics over baselines while achieving a speedup of more than two orders of magnitude. In the three-body planet-planet scattering problem in planetary dynamics, CBi-CFM better respects conservation laws, with conservation errors comparable to those of the ground truth. Finally, on real observations of globular clusters, collisional million-body systems shaped by $\sim 10^{10}$ years (10 Gyr) of evolution, our method represents an advance in accuracy, establishing a scalable route to solving inverse problems of long-timescale real-world chaotic dynamics.

When two or more parameters or labels produce similar data, they are degenerate, or hard to distinguish. Degeneracies render both label prediction and inverse problems difficult, since both machine learning algorithms and probabilistic samplers rely on the distinguishability of data and its gradients with respect to parameters. However, identifying degeneracies in physical models or real-world datasets can be elucidating about the choice of model or the underlying process that produces the data. We present the degeneracy distillery, a method that (1) detects and (2) resolves degenerate parameter combinations (a) automatically and (b) symbolically, from parameter-data (or parameter-simulation) pairs alone, through estimation and flattening of the Fisher information matrix. By exploring the information geometry of the likelihood, we characterize degeneracies as an intrinsic property of the physical model, requiring no realised data observation. We demonstrate our approach on a range of synthetic and real-world problems, discovering symbolic coordinate transformations that identify the combinations of parameters of a model which yield independent effects on the data. The resulting coordinates flatten the Fisher information in expectation globally, in contrast to posterior-based methods that flatten only at a single point, and substantially reduce the simulation budget required for downstream neural posterior estimation. In test cases we require up to $10\times$ fewer simulations for posterior estimation at matched validation calibration whilst simultaneously gaining physical insight on the system.

We perform a realistic KiDS-Legacy mock analysis with field-level neural compression and simulation-based inference using just 60 $N$-body simulations. The weak lensing shear field encodes substantially more cosmological information than standard two-point summary statistics such as the power spectrum. Field-level inference can fully exploit this information, but physical realism at the field-level requires very high-fidelity simulations. This poses a major challenge for simulation-based inference (SBI): accurate empirical density modelling and deep-learning-based neural compression require tens of thousands of training samples, but achieving physical realism at the field level makes each simulation extremely costly. We demonstrate that multifidelity SBI can alleviate this tension by substantially reducing the number of high-fidelity simulations needed for accurate cosmological inference. We pre-train neural inference models on realistic KiDS-Legacy-like shear mocks using fast log-normal \texttt{GLASS} simulations and fine-tune them on a small set of high-fidelity $N$-body simulations. We show that $60$ high-fidelity simulations are sufficient to obtain informative and well-calibrated cosmological posteriors, enabling at least an order-of-magnitude reduction in simulation cost for accurate field-level inference in a realistic setting.

Simulation-based inference (SBI) of latent parameters is often hindered by simulator misspecification, the mismatch between simulated and real-world observations caused by inherent modeling simplifications. RoPE, the recent state-of-the-art for robust SBI, addresses this through optimal transport between learned representations of real and simulated observations, but requires ground-truth parameter calibration pairs that are typically unavailable in the very settings where SBI is needed. What practitioners do have is unstructured side-information such as regime labels, instruction text, and policy bulletins. We propose Misspecification-Aware Simulation-Based Inference (MA-SBI), a calibration-free framework that turns this side-channel into a posterior correction. A learned corrector maps side-channel text to an observation-space shift applied before any pre-trained amortized posterior, requiring no retraining and no parameter ground-truth. Our main theorem bounds achievable bias reduction by the mutual information between misspecification and side-channel, with a non-vacuous constant that extends to all sub-Gaussian noise via Donsker-Varadhan. On hide-the-calibration benchmarks, MA-SBI with text alone matches the oracle posterior across 10 seeds and two backbones (TOST equivalence), while RoPE given more data does not. The two approaches are complementary: where misspecification is structural and recoverable from parameter pairs, RoPE dominates, as the theory predicts. A stochastic variant improves posterior-predictive log-likelihood on real COVID and OxCGRT epidemiological data, and correctly leaves the posterior unchanged on a well-specified cognitive-science corpus.

Accurate posterior estimation is central to scientific inference, as uncertainties determine what can be reliably learned from observational data. While Markov chain Monte Carlo methods provide asymptotic convergence guarantees, they are computationally demanding in high-dimensional settings. Neural network-based generative models for entire discretized 3D fields enable fast amortized inference but often lack convergence guarantees and principled accuracy assessment. Using Hamiltonian Monte Carlo to obtain reference posterior samples, we conduct a controlled field-level evaluation of an implicit generative model (Stochastic Interpolants) and an explicit likelihood-based model (GLOW normalizing flows). This comparison, unavailable in typical applications, enables the detection of posterior geometry failures that standard metrics cannot capture. As a case study, we consider the cosmological inverse problem of inferring cosmic initial conditions from present-day large-scale structure. To match the precision of modern cosmological data, this problem increasingly relies on complex, non-linear, and non-differentiable simulators, which are incompatible with gradient-based inference frameworks. Generative models offer a route to address these challenges, provided their inferred posteriors are reliable. In this work, we show that matching posterior means, marginal distributions, or achieving high cross-correlation does not imply correct uncertainty structure, as revealed by posterior variance fields and sample-based evaluations. Through this work, we aim to raise awareness of the challenges of uncertainty estimation in high-dimensional field-level settings, highlighting the importance of careful design and validation of neural generative approaches for scientific applications.

Accurate calibration of particle-based simulators is crucial for robotic earthwork simulation, but analytical calibration is challenging due to this task's highly nonlinear particle dynamics and the black-box nature of conventional simulators. Although simulation-based inference (SBI) can estimate posterior distributions over simulation parameters solely from forward simulations, applying SBI directly to high-fidelity (HF) particle simulators is often computationally prohibitive. Low-fidelity (LF) simulators with coarser particles can reduce this cost, but changes in particle size and particle count shift the parameter values needed to reproduce the same observation, producing biased LF posteriors. We propose Bridged SBI, which leverages a biased but informative LF posterior to guide HF inference. This method first uses inexpensive LF simulations to identify a coarse high-density parameter region, and then it learns a local residual bridge to transport LF posterior samples toward HF-consistent regions by correcting the LF--HF discrepancy. We analyze how sequential multi-fidelity SBI (Naive-MF) can suffer from LF-induced posterior miscoverage when it directly relies on the LF posterior without discrepancy correction. We then show that Bridged SBI is designed to alleviate this issue by explicitly modeling the LF--HF discrepancy through residual correction. Experiments on both sim-to-sim particle-parameter calibration and real-to-sim calibration with real soil observation show that Bridged SBI produces more accurate and reliable HF posteriors than HF-only SBI or the Naive-MF baseline, especially under limited HF simulation costs.

Multi-arm bandit algorithms are increasingly used in online platforms, clinical trials, and social science experiments, but valid statistical inference on their performance remains an open challenge. After deploying bandits, a natural question is whether one can construct a confidence interval for its mean reward and assess whether it reliably outperforms a baseline policy. The total reward achieved in any single bandit deployment is random, and deploying a bandit twice on the same population typically yields different reward trajectories due to stochastic rewards. Standard statistical inference methods cannot be used because bandit algorithms introduce complex dependencies in the collected data, which violate the i.i.d. assumption underlying many classical approaches. Moreover, existing inference methods for adaptively collected data only apply to estimands that do not depend on the data-collection algorithm (such as the mean reward under a fixed action). We propose Bandit Simulation for Inference (BSI), a framework that fits a simulator of the bandit environment from observed data--either on-policy or off-policy--and uses it to estimate the mean reward under any evaluation policy, including adaptive blackbox algorithms. BSI formally propagates uncertainty in the estimated simulator parameters into the confidence interval construction. Furthermore, for BSI to be valid, it requires only weak exploration assumptions on the behavior policy and avoids importance weighting. We prove that BSI yields asymptotically valid confidence intervals, and demonstrate empirically that it maintains nominal coverage in settings where standard off-policy evaluation methods fail.

We are witnessing a surge in observations of the cosmic dawn (CD) and epoch of reionisation (EoR), driving an increasing demand for fast and robust theoretical interpretation frameworks. In response, machine learning (ML), and emulation in particular, has emerged as a powerful approach to accelerate and enhance inference pipelines. In this work, we present 21cmEMUv3, an emulator trained on 21cmFASTv3 simulations that model both atomically and molecularly cooling galaxies. 21cmEMUv3 is conditioned on $σ_8$ and ten astrophysical parameters to produce seven summary observables: (i) the cylindrical 21cm power spectrum (PS), emulated for the first time at such high resolution and accuracy across a wide redshift range of $z \sim$ 6--30; (ii) the spherically-averaged 21cm PS; (iii) the mean neutral fraction of the intergalactic medium (IGM); (iv) the mean 21cm spin temperature; (v) the global 21cm signal; (vi) the ultraviolet (UV) luminosity functions (LFs); and (vii) the Thomson scattering optical depth. Notably, the cylindrical 21cm PS is emulated via score-based diffusion, while the remaining six summaries are emulated via long-short term memory (LSTM) networks, all achieving sub-percent median accuracy. We use the emulator to reinterpret current 21cm PS upper limits from HERA, for the first time using state-of-the-art hydrodynamical simulations to inform priors on star formation inside molecularly cooling galaxies. We find that our inferred soft-band X-ray luminosity per unit star formation rate is consistent with extrapolations of high-mass X-ray binaries to the low-metallicity regimes expected in the first galaxies, excluding values below $10^{39.2}$ erg s$^{-1}M^{-1}_\odot \rm{yr}$ at $95\%$ confidence. Finally, we produce forecasts for the detection of the cosmic 21cm PS with the Square Kilometre Array for different array configurations. The 21cmEMU package is publicly available.