26 research outputs found
Gradient-free Hamiltonian Monte Carlo with Efficient Kernel Exponential Families
We propose Kernel Hamiltonian Monte Carlo (KMC), a gradient-free adaptive
MCMC algorithm based on Hamiltonian Monte Carlo (HMC). On target densities
where classical HMC is not an option due to intractable gradients, KMC
adaptively learns the target's gradient structure by fitting an exponential
family model in a Reproducing Kernel Hilbert Space. Computational costs are
reduced by two novel efficient approximations to this gradient. While being
asymptotically exact, KMC mimics HMC in terms of sampling efficiency, and
offers substantial mixing improvements over state-of-the-art gradient free
samplers. We support our claims with experimental studies on both toy and
real-world applications, including Approximate Bayesian Computation and
exact-approximate MCMC.Comment: 20 pages, 7 figure
Geometric ergodicity of the Random Walk Metropolis with position-dependent proposal covariance
We consider a Metropolis-Hastings method with proposal kernel
, where is the current state. After discussing
specific cases from the literature, we analyse the ergodicity properties of the
resulting Markov chains. In one dimension we find that suitable choice of
can change the ergodicity properties compared to the Random Walk
Metropolis case , either for the better or worse. In
higher dimensions we use a specific example to show that judicious choice of
can produce a chain which will converge at a geometric rate to its
limiting distribution when probability concentrates on an ever narrower ridge
as grows, something which is not true for the Random Walk Metropolis.Comment: 15 pages + appendices, 4 figure
Kernel Sequential Monte Carlo
We propose kernel sequential Monte Carlo (KSMC), a framework for sampling from static target densities. KSMC is a family of sequential Monte Carlo algorithms that are based on building emulator models of the current particle system in a reproducing kernel Hilbert space. We here focus on modelling nonlinear covariance structure and gradients of the target. The emulator's geometry is adaptively updated and subsequently used to inform local proposals. Unlike in adaptive Markov chain Monte Carlo, continuous adaptation does not compromise convergence of the sampler. KSMC combines the strengths of sequental Monte Carlo and kernel methods: superior performance for multimodal targets and the ability to estimate model evidence as compared to Markov chain Monte Carlo, and the emulator's ability to represent targets that exhibit high degrees of nonlinearity. As KSMC does not require access to target gradients, it is particularly applicable on targets whose gradients are unknown or prohibitively expensive. We describe necessary tuning details and demonstrate the benefits of the the proposed methodology on a series of challenging synthetic and real-world examples
K2-ABC: Approximate Bayesian Computation with Kernel Embeddings
Complicated generative models often result in a situation where computing the
likelihood of observed data is intractable, while simulating from the
conditional density given a parameter value is relatively easy. Approximate
Bayesian Computation (ABC) is a paradigm that enables simulation-based
posterior inference in such cases by measuring the similarity between simulated
and observed data in terms of a chosen set of summary statistics. However,
there is no general rule to construct sufficient summary statistics for complex
models. Insufficient summary statistics will "leak" information, which leads to
ABC algorithms yielding samples from an incorrect (partial) posterior. In this
paper, we propose a fully nonparametric ABC paradigm which circumvents the need
for manually selecting summary statistics. Our approach, K2-ABC, uses maximum
mean discrepancy (MMD) as a dissimilarity measure between the distributions
over observed and simulated data. MMD is easily estimated as the squared
difference between their empirical kernel embeddings. Experiments on a
simulated scenario and a real-world biological problem illustrate the
effectiveness of the proposed algorithm
Kernel methods for Monte Carlo
This thesis investigates the use of reproducing kernel Hilbert spaces (RKHS) in the context of Monte Carlo algorithms. The work proceeds in three main themes. Adaptive Monte Carlo proposals: We introduce and study two adaptive Markov chain Monte Carlo (MCMC) algorithms to sample from target distributions with non-linear support and intractable gradients. Our algorithms, generalisations of random walk Metropolis and Hamiltonian Monte Carlo, adaptively learn local covariance and gradient structure respectively, by modelling past samples in an RKHS. We further show how to embed these methods into the sequential Monte Carlo framework. Efficient and principled score estimation: We propose methods for fitting an RKHS exponential family model that work by fitting the gradient of the log density, the score, thus avoiding the need to compute a normalization constant. While the problem is of general interest, here we focus on its embedding into the adaptive MCMC context from above. We improve the computational efficiency of an earlier solution with two novel fast approximation schemes without guarantees, and a low-rank, Nyström-like solution. The latter retains the consistency and convergence rates of the exact solution, at lower computational cost. Goodness-of-fit testing: We propose a non-parametric statistical test for goodness-of-fit. The measure is a divergence constructed via Stein's method using functions from an RKHS. We derive a statistical test, both for i.i.d. and non-i.i.d. samples, and apply the test to quantifying convergence of approximate MCMC methods, statistical model criticism, and evaluating accuracy in non-parametric score estimation