## Bayesian Analysis

- Bayesian Anal. (2017), 27 pages.

### Designing Simple and Efficient Markov Chain Monte Carlo Proposal Kernels

Yuttapong Thawornwattana, Daniel Dalquen, and Ziheng Yang

#### Abstract

We discuss a few principles to guide the design of efficient Metropolis–Hastings proposals for well-behaved target distributions without deeply divided modes. We illustrate them by developing and evaluating novel proposal kernels using a variety of target distributions. Here, efficiency is measured by the variance ratio relative to the independent sampler. The first principle is to introduce negative correlation in the MCMC sample or to reduce positive correlation: *to propose something new, propose something different*. This explains why single-moded proposals such as the Gaussian random-walk is poorer than the uniform random walk, which is in turn poorer than the bimodal proposals that avoid values very close to the current value. We evaluate three new bimodal proposals called Box, Airplane and StrawHat, and find that they have similar performance to the earlier Bactrian kernels, suggesting that the general shape of the proposal matters, but not the specific distributional form. We propose the “Mirror” kernel, which generates new values around the mirror image of the current value on the other side of the target distribution (effectively the “opposite” of the current value). This introduces negative correlations, leading in many cases to efficiency of $>100\mathrm{\%}$. The second principle, applicable to multidimensional targets, is that a sequence of well-designed one-dimensional proposals can be more efficient than a single $d$-dimensional proposal. Thirdly, we suggest that variable transformation be explored as a general strategy for designing efficient MCMC kernels. We apply these principles to a high-dimensional Gaussian target with strong correlations, a logistic regression problem and a molecular clock dating problem to illustrate their practical utility.

#### Article information

**Source**

Bayesian Anal. (2017), 27 pages.

**Dates**

First available in Project Euclid: 10 November 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.ba/1510282998

**Digital Object Identifier**

doi:10.1214/17-BA1084

**Keywords**

asymptotic variance bimodal kernel Metropolis–Hastings algorithm Mirror kernel molecular clock dating variable transformation

**Rights**

Creative Commons Attribution 4.0 International License.

#### Citation

Thawornwattana, Yuttapong; Dalquen, Daniel; Yang, Ziheng. Designing Simple and Efficient Markov Chain Monte Carlo Proposal Kernels. Bayesian Anal., advance publication, 10 November 2017. doi:10.1214/17-BA1084. https://projecteuclid.org/euclid.ba/1510282998

#### Supplemental materials

- Supplementary Material of “Designing Simple and Efficient Markov Chain Monte Carlo Proposal Kernels”. (I) Efficiency curves for Box, Airplane and StrawHat kernels for a range of a values. (II) Two-dimensional Gaussian target example. (III) MCMC algorithms for the phylogenetic problem. (IV) Effect of μ∗ on efficiency.Digital Object Identifier: doi:10.1214/17-BA1084SUPP