Many probabilistic models of interest in scientific computing and machine learning have expensive, black-box likelihoods that prevent the application of standard techniques for Bayesian inference, such as MCMC, which would require access to the gradient or a large number of likelihood evaluations. We introduce here a novel sample-efficient inference framework, Variational Bayesian Monte Carlo (VBMC). VBMC combines variational inference with Gaussian-process based, active-sampling Bayesian quadrature, using the latter to efficiently approximate the intractable integral in the variational objective. Our method produces both a nonparametric approximation of the posterior distribution and an approximate lower bound of the model evidence, useful for model selection. We demonstrate VBMC both on several synthetic likelihoods and on a neuronal model with data from real neurons. Across all tested problems and dimensions (up to D = 10), VBMC performs consistently well in reconstructing the posterior and the model evidence with a limited budget of likelihood evaluations, unlike other methods that work only in very low dimensions. Our framework shows great promise as a novel tool for posterior and model inference with expensive, black-box likelihoods.

Many probabilistic models of interest in scientific computing and machine learning have expensive, black-box likelihoods that prevent the application of standard techniques for Bayesian inference, such as MCMC, which would require access to the gradient or a large number of likelihood evaluations. We introduce here a novel sample-efficient inference framework, Variational Bayesian Monte Carlo (VBMC). VBMC combines variational inference with Gaussian-process based, active-sampling Bayesian quadrature, using the latter to efficiently approximate the intractable integral in the variational objective. Our method produces both a nonparametric approximation of the posterior distribution and an approximate lower bound of the model evidence, useful for model selection. We demonstrate VBMC both on several synthetic likelihoods and on a neuronal model with data from real neurons. Across all tested problems and dimensions (up to D = 10), VBMC performs consistently well in reconstructing the posterior and the model evidence with a limited budget of likelihood evaluations, unlike other methods that work only in very low dimensions. Our framework shows great promise as a novel tool for posterior and model inference with expensive, black-box likelihoods.

We introduce here a novel sample-efficient inference framework, V ariational Bayesian Monte Carlo (VBMC). VBMC combines variational inference with Gaussian-process based, active-sampling Bayesian quadrature, using the latter to efficiently approximate the intractable integral in the variational objective.

Neural Information Processing Systems http://nips.cc/

We thank the reviewers very much for their time and valuable feedback. Our MCMC method significantly outperforms the other MCMC methods. Monte Carlo (PMC) [35] which is an iterated importance sampling method with connections to SMC. SA-MCMC uses a "global" proposal distribution like IMH but unlike many MCMC methods. We will add the references and discuss future directions in our revision.

Now, since as mentioned above the KL divergence is a non-negative quantity, from Eq. S2 we have

Transformer-based models for amortized probabilistic inference, such as neural processes, prior-fitted networks, and tabular foundation models, excel at single-pass marginal prediction. However, many real-world applications, from signal interpolation to multi-column tabular predictions, require coherent joint distributions that capture dependencies between predictions. While purely autoregressive architectures efficiently generate such distributions, they sacrifice the flexible set-conditioning that makes these models powerful for meta-learning. Conversely, the standard approach to obtain joint distributions from set-based models requires expensive re-encoding of the entire augmented conditioning set at each autoregressive step. We introduce a causal autoregressive buffer that preserves the advantages of both paradigms. Our approach decouples context encoding from updating the conditioning set. The model processes the context once and caches it. A dynamic buffer then captures target dependencies: as targets are incorporated, they enter the buffer and attend to both the cached context and previously buffered targets. This enables efficient batched autoregressive generation and one-pass joint log-likelihood evaluation. A unified training strategy allows seamless integration of set-based and autoregressive modes at minimal additional cost. Across synthetic functions, EEG signals, cognitive models, and tabular data, our method matches predictive accuracy of strong baselines while delivering up to 20 times faster joint sampling. Our approach combines the efficiency of autoregressive generative models with the representational power of set-based conditioning, making joint prediction practical for transformer-based probabilistic models.

Now, since as mentioned above the KL divergence is a non-negative quantity, from Eq. S2 we have