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Variational Bayesian Monte Carlo (VBMC) - v1.0


  • New paper accepted (Sep/25/2020) The "Variational Bayesian Monte Carlo with Noisy Likelihoods" paper [2] has been accepted at NeurIPS 2020! This is the second VBMC paper at NeurIPS. The revised version of the manuscript is available on arXiv.
  • Major update (Jun/16/2020) VBMC v1.0 has been released! (see tweeprint) This update includes full support for noisy log-likelihood evaluations, a linear transformation of the inference space to better represent the variational posterior, and a number of tweaks to the algorithm's settings for improved performance. See the new preprint for more information [2].
  • The original VBMC paper has been published at NeurIPS 2018 [1], and an exploration of various VBMC features has been published in PMLR [3]. VBMC has been presented at seminars in Oxford (UK), Bristol (UK), NYU (NY) and online conferences. See here for slides, and here for a recorded talk (at CDS@NYU) with implausible automated captions.

What is it

What if there was a model-fitting method similar to Bayesian optimization (e.g., BADS), which, instead of returning just the optimal parameter vector, would also return its uncertainty (even better, the full posterior distribution of the parameters), and maybe even a metric than can be used for Bayesian model comparison?

VBMC is an approximate inference method designed to fit and evaluate computational models with a limited budget of potentially noisy likelihood evaluations (e.g., for computationally expensive models). Specifically, VBMC simultaneously computes:

  • an approximate posterior distribution of the model parameters;
  • an approximation — technically, an approximate lower bound — of the log model evidence (also known as log marginal likelihood or log Bayes factor), a metric used for Bayesian model selection.

Extensive benchmarks on both artificial test problems and a large number of real model-fitting problems from computational and cognitive neuroscience show that VBMC generally — and often vastly — outperforms alternative methods for sample-efficient Bayesian inference [1,2].

VBMC runs with virtually no tuning and it is very easy to set up for your problem (especially if you are already familiar with BADS, our model-fitting algorithm based on Bayesian optimization).

Should I use VBMC?

VBMC is effective when:

  • the model log-likelihood function is a black-box (e.g., the gradient is unavailable);
  • the likelihood is at least moderately expensive to compute (say, half a second or more per evaluation);
  • the model has up to D = 10 continuous parameters (maybe a few more, but no more than D = 20);
  • the target posterior distribution is continuous and reasonably smooth (see here);
  • optionally, log-likelihood evaluations may be noisy (e.g., estimated via simulation).

Conversely, if your model can be written analytically, you should exploit the powerful machinery of probabilistic programming frameworks such as Stan or PyMC3.


Download the latest version of VBMC as a ZIP file.

  • To install VBMC, clone or unpack the zipped repository where you want it and run the script install.m.
    • This will add the VBMC base folder to the MATLAB search path.
  • To see if everything works, run vbmc('test').

Quick start

The VBMC interface is similar to that of MATLAB optimizers. The basic usage is:


with input parameters:

  • FUN, a function handle to the (unnormalized) log posterior distribution of your model (that is, log prior plus log likelihood of a dataset and model, for a given input parameter vector);
  • X0, the starting point of the inference (a row vector);
  • LB and UB, hard lower and upper bounds for the parameters;
  • PLB and PUB, plausible lower and upper bounds, that is a box that ideally brackets a region of high posterior density.

The output parameters are:

  • VP, a struct with the variational posterior approximating the true posterior;
  • ELBO, the (estimated) lower bound on the log model evidence;
  • ELBO_SD, the standard deviation of the estimate of the ELBO (not the error between the ELBO and the true log model evidence, which is generally unknown).

The variational posterior vp can be manipulated with functions such as vbmc_moments (compute posterior mean and covariance), vbmc_pdf (evaluates the posterior density), vbmc_rnd (draw random samples), vbmc_kldiv (Kullback-Leibler divergence between two posteriors), vbmc_mtv (marginal total variation distance between two posteriors); see also this question.

Next steps

  • For a tutorial with many extensive usage examples, see vbmc_examples.m. You can also type help vbmc to display the documentation.

  • For practical recommendations, such as how to set LB and UB and the plausible bounds, and any other question, check out the FAQ on the VBMC wiki.

  • If you want to run VBMC on a noisy or stochastic log-likelihood, see below.

For BADS users

If you already use Bayesian Adaptive Direct Search (BADS) to fit your models, setting up VBMC on your problem should be particularly simple; see here.

How does it work

VBMC combines two machine learning techniques in a novel way:

  • variational inference, a method to perform approximate Bayesian inference;
  • Bayesian quadrature, a technique to estimate the value of expensive integrals.

VBMC iteratively builds an approximation of the true, expensive target posterior via a Gaussian process (GP), and it matches a variational distribution — an expressive mixture of Gaussians — to the GP.

This matching process entails optimization of the evidence lower bound (ELBO), that is a lower bound on the log marginal likelihood (LML), also known as log model evidence. Crucially, we estimate the ELBO via Bayesian quadrature, which is fast and does not require further evaluation of the true target posterior.

In each iteration, VBMC uses active sampling to select which points to evaluate next in order to explore the posterior landscape and reduce uncertainty in the approximation.

VBMC demo

In the figure above, we show an example VBMC run on a "banana" function. The left panel shows the ground truth for the target posterior density. In the middle panel we show VBMC at work (contour plots of the variational posterior) across iterations. Red crosses are the centers of the mixture of Gaussians used as variational posterior, whereas dots are sampled points in the training set (black: previously sampled points, blue: points sampled in the current iteration). The right panel shows a plot of the estimated ELBO vs. the true log marginal likelihood (LML).

In the figure below, we show another example VBMC run on a "lumpy" distribution.

Another VBMC demo

See the VBMC paper for more details [1].

VBMC with noisy likelihoods

VBMC v1.0 (June 2020) introduced support for noisy models [2]. To run VBMC on a noisy problem, first you need to ensure that your target function fun returns:

  • as first output, the noisy value of the log-posterior (where the noise usually comes from a stochastic evaluation of the log-likelihood);
  • as second output, an estimate of the noise in the returned log-posterior value (expressed as standard deviation, SD).

Noisy evaluations of the log-likelihood often arise from simulation-based models, for which a direct expression of the (log) likelihood is not available. We recommend Inverse Binomial Sampling (IBS) as a method that conveniently computes both an unbiased estimate of the log-likelihood and an estimate of its variability entirely through simulation — however VBMC is compatible with any estimation technique.

Once you have set up fun as above, run VBMC by specifying that the target function is noisy

OPTIONS.SpecifyTargetNoise = true;

For more information, see the VBMC FAQ and Example 5 in the VBMC tutorial.

In the figure below, we show the difference in performance between the original VBMC (old) and VBMC v1.0 (new) when dealing with noisy target evaluations.

VBMC2020 demo


The VBMC toolbox is under active development. The toolbox has been extensively tested in several benchmarks and published papers, but as with any approximate inference technique you need to double-check your results. See the FAQ for more information on diagnostics.

If you have trouble doing something with VBMC, spot bugs or strange behavior, or you simply have some questions, please contact me at, putting 'VBMC' in the subject of the email.

VBMC for other programming languages

VBMC is currently available only for MATLAB. A Python version is being planned, ideally before the great filter.

If you are interested in porting VBMC to Python or another language (R, Julia), please get in touch at (putting 'VBMC' in the subject of the email); I'd be willing to help. However, before contacting me for this reason, please have a good look at the codebase here on GitHub, and at the papers [1,2]. VBMC is a fairly complex piece of software, so be aware that porting it will require considerable effort and programming/computing skills.


  1. Acerbi, L. (2018). Variational Bayesian Monte Carlo. In Advances in Neural Information Processing Systems 31: 8222-8232. (paper + supplement on arXiv, NeurIPS Proceedings)
  2. Acerbi, L. (2020). Variational Bayesian Monte Carlo with Noisy Likelihoods. To appear in Advances in Neural Information Processing Systems 33. arXiv preprint arXiv:2006.08655 (preprint on arXiv).

Please cite both references if you use VBMC in your work (the 2018 paper introduced the framework, and the 2020 paper includes a number of major improvements, including but not limited to support for noisy likelihoods). You can cite VBMC in your work with something along the lines of

We estimated approximate posterior distibutions and approximate lower bounds to the model evidence of our models using Variational Bayesian Monte Carlo (VBMC; Acerbi, 2018, 2020). VBMC combines variational inference and active-sampling Bayesian quadrature to perform approximate Bayesian inference in a sample-efficient manner.

Besides formal citations, you can demonstrate your appreciation for VBMC in the following ways:

  • Star the VBMC repository on GitHub;
  • Follow me on Twitter for updates about VBMC and other projects I am involved with;
  • Tell me about your model-fitting problem and your experience with VBMC (positive or negative) at (putting 'VBMC' in the subject of the email).

You may also want to check out Bayesian Adaptive Direct Search (BADS), our method for fast Bayesian optimization.

Additional references

  1. Acerbi, L. (2019). An Exploration of Acquisition and Mean Functions in Variational Bayesian Monte Carlo. In Proc. Machine Learning Research 96: 1-10. 1st Symposium on Advances in Approximate Bayesian Inference, Montréal, Canada. (paper in PMLR)


VBMC is released under the terms of the GNU General Public License v3.0.


Variational Bayesian Monte Carlo (VBMC) algorithm for posterior and model inference in MATLAB





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