Markov chain Monte Carlo (MCMC) is an inference mechanism that approximates a target probability distribution by a sequence of states (a.k.a.

We demonstrate for the first time that ill-conditioned, non-smooth, constrained distributions in very high dimension, upwards of 100,000, can be sampled efficiently \emph{in practice}. Our algorithm incorporates constraints into the Riemannian version of Hamiltonian Monte Carlo and maintains sparsity. This allows us to achieve a mixing rate independent of smoothness and condition numbers. On benchmark data sets in systems biology and linear programming, our algorithm outperforms existing packages by orders of magnitude. In particular, we achieve a 1,000-fold speed-up for sampling from the largest published human metabolic network (RECON3D). Our package has been incorporated into a popular Bioinformatics library.

Hamiltonian Monte Carlo (HMC) has emerged as a powerful Markov Chain Monte Carlo (MCMC) method to sample from complex continuous distributions. However, a fundamental limitation of HMC is that it can not be applied to distributions with mixed discrete and continuous variables. In this paper, we propose mixed HMC (M-HMC) as a general framework to address this limitation. M-HMC is a novel family of MCMC algorithms that evolves the discrete and continuous variables in tandem, allowing more frequent updates of discrete variables while maintaining HMC's ability to suppress random-walk behavior. We establish M-HMC's theoretical properties, and present an efficient implementation with Laplace momentum that introduces minimal overhead compared to existing HMC methods. The superior performances of M-HMC over existing methods are demonstrated with numerical experiments on Gaussian mixture models (GMMs), variable selection in Bayesian logistic regression (BLR), and correlated topic models (CTMs).

Gaussian latent variable models are a key class of Bayesian hierarchical models with applications in many fields. Performing Bayesian inference on such models can be challenging as Markov chain Monte Carlo algorithms struggle with the geometry of the resulting posterior distribution and can be prohibitively slow. An alternative is to use a Laplace approximation to marginalize out the latent Gaussian variables and then integrate out the remaining hyperparameters using dynamic Hamiltonian Monte Carlo, a gradient-based Markov chain Monte Carlo sampler. To implement this scheme efficiently, we derive a novel adjoint method that propagates the minimal information needed to construct the gradient of the approximate marginal likelihood. This strategy yields a scalable differentiation method that is orders of magnitude faster than state of the art differentiation techniques when the hyperparameters are high dimensional. We prototype the method in the probabilistic programming framework Stan and test the utility of the embedded Laplace approximation on several models, including one where the dimension of the hyperparameter is 6,000. Depending on the cases, the benefits can include an alleviation of the geometric pathologies that frustrate Hamiltonian Monte Carlo and a dramatic speed-up.

Markov Chain Monte Carlo (MCMC) is a flexible approach to approximate sampling from intractable probability distributions, with a rich theoretical foundation and comprising a wealth of exemplar algorithms. While the qualitative correctness of MCMC algorithms is often easy to ensure, their practical efficiency is contingent on the `target' distribution being reasonably well-behaved. In this work, we concern ourself with the scenario in which this good behaviour is called into question, reviewing an emerging line of work on `robust' MCMC algorithms which can perform acceptably even in the face of certain pathologies. We focus on two particular pathologies which, while simple, can already have dramatic effects on standard `local' algorithms. The first is roughness, whereby the target distribution varies so rapidly that the numerical stability of the algorithm is tenuous. The second is flatness, whereby the landscape of the target distribution is instead so barren and uninformative that one becomes lost in uninteresting parts of the state space. In each case, we formulate the pathology in concrete terms, review a range of proposed algorithmic remedies to the pathology, and outline promising directions for future research.

Here, we propose to use neural networks as a powerful and scalable parametric model, while staying as close to a truly Bayesian treatment as possible.

HMC include being sensitive to parameter tuning and being restricted to continuous distributions.

Hamiltonian system built on the distribution of interest.