Estimating a distribution given access to its unnormalized density is pivotal in Bayesian inference, where the posterior is generally known only up to an unknown normalizing constant. Variational inference and Markov chain Monte Carlo methods are the predominant tools for this task; however, both are often challenging to apply reliably, particularly when the posterior has complex geometry. Here, we introduce Soft Contrastive Variational Inference (SoftCVI), which allows a family of variational objectives to be derived through a contrastive estimation framework. The approach parameterizes a classifier in terms of a variational distribution, reframing the inference task as a contrastive estimation problem aiming to identify a single true posterior sample among a set of samples. Despite this framing, we do not require positive or negative samples, but rather learn by sampling the variational distribution and computing ground truth soft classification labels from the unnormalized posterior itself. The objectives have zero variance gradient when the variational approximation is exact, without the need for specialized gradient estimators. We empirically investigate the performance on a variety of Bayesian inference tasks, using both simple (e.g. normal) and expressive (normalizing flow) variational distributions. We find that SoftCVI can be used to form objectives which are stable to train and mass-covering, frequently outperforming inference with other variational approaches.

We study the problem of likelihood maximization when the likelihood function is intractable but model simulations are readily available. We propose a sequential, gradient-based optimization method that directly models the Fisher score based on a local score matching technique which uses simulations from a localized region around each parameter iterate. By employing a linear parameterization for the surrogate score model, our technique admits a closed-form, least-squares solution. This approach yields a fast, flexible, and efficient approximation to the Fisher score, effectively smoothing the likelihood objective and mitigating the challenges posed by complex likelihood landscapes. We provide theoretical guarantees for our score estimator, including bounds on the bias introduced by the smoothing. Empirical results on a range of synthetic and real-world problems demonstrate the superior performance of our method compared to existing benchmarks.

Many machine learning problems can be seen as approximating a *target* distribution using a *particle* distribution by minimizing their statistical discrepancy. Wasserstein Gradient Flow can move particles along a path that minimizes the $f$-divergence between the target and particle distributions. To move particles, we need to calculate the corresponding velocity fields derived from a density ratio function between these two distributions. Previous works estimated such density ratio functions and then differentiated the estimated ratios. These approaches may suffer from overfitting, leading to a less accurate estimate of the velocity fields. Inspired by non-parametric curve fitting, we directly estimate these velocity fields using interpolation techniques. We prove that our estimators are consistent under mild conditions. We validate their effectiveness using novel applications on domain adaptation and missing data imputation. The code for reproducing our results can be found at https://github.com/anewgithubname/gradest2.

We introduce Sequential Neural Posterior Score Estimation (SNPSE), a score-based method for Bayesian inference in simulator-based models. Our method, inspired by the remarkable success of score-based methods in generative modelling, leverages conditional score-based diffusion models to generate samples from the posterior distribution of interest. The model is trained using an objective function which directly estimates the score of the posterior. We embed the model into a sequential training procedure, which guides simulations using the current approximation of the posterior at the observation of interest, thereby reducing the simulation cost. We also introduce several alternative sequential approaches, and discuss their relative merits. We then validate our method, as well as its amortised, non-sequential, variant on several numerical examples, demonstrating comparable or superior performance to existing state-of-the-art methods such as Sequential Neural Posterior Estimation (SNPE).

Computer simulations have proven a valuable tool for understanding complex phenomena across the sciences. However, the utility of simulators for modelling and forecasting purposes is often restricted by low data quality, as well as practical limits to model fidelity. In order to circumvent these difficulties, we argue that modellers must treat simulators as idealistic representations of the true data generating process, and consequently should thoughtfully consider the risk of model misspecification. In this work we revisit neural posterior estimation (NPE), a class of algorithms that enable black-box parameter inference in simulation models, and consider the implication of a simulation-to-reality gap. While recent works have demonstrated reliable performance of these methods, the analyses have been performed using synthetic data generated by the simulator model itself, and have therefore only addressed the well-specified case. In this paper, we find that the presence of misspecification, in contrast, leads to unreliable inference when NPE is used naïvely. As a remedy we argue that principled scientific inquiry with simulators should incorporate a model criticism component, to facilitate interpretable identification of misspecification and a robust inference component, to fit ‘wrong but useful’ models. We propose robust neural posterior estimation (RNPE), an extension of NPE to simultaneously achieve both these aims, through explicitly modelling the discrepancies between simulations and the observed data. We assess the approach on a range of artificially misspecified examples, and find RNPE performs well across the tasks, whereas naïvely using NPE leads to misleading and erratic posteriors.