I have come across when surfing the web, that I think have use in Metropolis Light Transport(MLT) Implementations

**A general construction for parallelizing Metropolis−Hastings**

This is a paper for augmenting metropolis sampling with multiple proposals in a way that,

as opposed to earlier approaches such as multiple try metropolis and prefetching,

does not discard all but one of the proposals in one iteration.

**Adaptive Markov Chain Monte Carlo**

Adaptive Markov Chain Monte Carlo(AMCMC) methods adapt the parameters(usually those of the proposal distribution)

of the sampler to achieve an ideal acceptance ratio.

**Exploring an Adaptive Metropolis Algorithm**

This is a paper on adapting the step size on a logarithmic scale,

which allows it to find a step size that fulfills the target acceptance ratio much faster than adapting the step size directly.

**ROBUST ADAPTIVE METROPOLIS ALGORITHM WITH**

COERCED ACCEPTANCE RATE

COERCED ACCEPTANCE RATE

This paper introduces a new

robust adaptive Metropolis algorithm estimating the shape of the target distribution and simultaneously coercing the acceptance rate.

**Kernel Adaptive Metropolis-Hastings**

A Kernel Adaptive Metropolis-Hastings algorithm is introduced, for the purpose of sampling

from a target distribution with strongly nonlinear support. The algorithm embeds the trajectory of the Markov chain into a reproducing kernel Hilbert space (RKHS), such that the feature space covariance of the samples informs

the choice of proposal. The procedure is computationally efficient and straightforward to implement, since the RKHS moves can be integrated out analytically: our proposal distribution in the original space is a normal distribution

whose mean and covariance depend on where the current sample lies in the support of the target distribution, and adapts to its local covariance structure.

Link to example code

**Adaptive Metropolis Algorithm Using Variational**

Bayesian Adaptive Kalman Filter

Bayesian Adaptive Kalman Filter

A new adaptive MCMC algorithm, called the variational Bayesian

adaptive Metropolis (VBAM) algorithm, is developed. The VBAM algorithm

updates the proposal covariance matrix using the variational Bayesian adaptive

Kalman filter (VB-AKF). A strong law of large numbers for the VBAM algorithm

is proven. The empirical convergence results for three simulated examples and for

two real data examples are also provided.

**DRAM: Efficient adaptive MCMC**

We propose to combine two quite powerful ideas

that have recently appeared in the Markov chain Monte Carlo

literature: adaptive Metropolis samplers and delayed rejec-

tion. The ergodicity of the resulting non-Markovian sampler

is proved, and the efficiency of the combination is demon-

strated with various examples. We present situations where

the combination outperforms the original methods: adap-

tation clearly enhances efficiency of the delayed rejection

algorithm in cases where good proposal distributions are

not available. Similarly, delayed rejection provides a sys-

tematic remedy when the adaptation process has a slow

start.

web page

**Learn From Thy Neighbor: Parallel-Chain and**

Regional Adaptive MCMC

Regional Adaptive MCMC

Starting with the seminal paper of Haario, Saksman and Tamminen (Haario

et al. (2001)), a substantial amount of work has been done to validate adaptive

Markov chain Monte Carlo algorithms. In this paper we focus on two practi-

cal aspects of adaptive Metropolis samplers. First, we draw attention to the

deficient performance of standard adaptation when the target distribution is

multi-modal. We propose a parallel chain adaptation strategy that incorpo-

rates multiple Markov chains which are run in parallel. Second, we note that

the current adaptive MCMC paradigm implicitly assumes that the adaptation is

uniformly efficient on all regions of the state space. However, in many practical

instances, different “optimal” kernels are needed in different regions of the state

space. We propose here a regional adaptation algorithm in which we account

for possible errors made in defining the adaptation regions.

Slides

**Delayed Rejection**

**Delayed rejection schemes for efficient Markov-Chain Monte-Carlo**

sampling of multimodal distributions

sampling of multimodal distributions

A number of problems in a variety of fields are characterised by target dis-

tributions with a multimodal structure in which the presence of several isolated

local maxima dramatically reduces the efficiency of Markov chain Monte Carlo

sampling algorithms. Several solutions, such as simulated tempering or the use

of parallel chains, have been proposed to facilitate the exploration of the relevant

parameter space. They provide effective strategies in the cases in which the di-

mension of the parameter space is small and/or the computational costs are not

a limiting factor. These approaches fail however in the case of high-dimensional

spaces where the multimodal structure is induced by degeneracies between re-

gions of the parameter space. In this paper we present a fully Markovian way

to efficiently sample this kind of distribution based on the general Delayed Re-

jection scheme with an arbitrary number of steps, and provide details for an

efficient numerical implementation of the algorithm.

**Markov Chain Quasi Monte Carlo**

Markov Chain Quasi Monte Carlo(MCQMC) methods reformulate Markov Chain Monte Carlo methods

to allow for the use of low discrepancy sequences instead of a pseudo-random number generator(PRNG).

**Discrepancy bounds for uniformly ergodic**

Markov chain quasi-Monte Carlo

Markov chain quasi-Monte Carlo

In [Chen, D., Owen, Ann. Stat., 39, 673–701, 2011] Markov chain

Monte Carlo (MCMC) was studied under the assumption that the

driver sequence is a deterministic sequence rather than independent

U (0, 1) random variables. Therein it was shown that as long as the

driver sequence is completely uniformly distributed, the Markov chain

consistently samples the target distribution. The present work extends

these results by providing bounds on the convergence rate of the dis-

crepancy between the empirical distribution of the Markov chain and

the target distribution, under the assumption that the Markov chain

is uniformly ergodic.

In a general setting we show the existence of driver sequences for

which the discrepancy of the Markov chain from the target distribution

with respect to certain test sets converges with (almost) the usual

Monte Carlo rate of n^−1/2 .

**CONSISTENCY AND CONVERGENCE RATE OF MARKOV**

CHAIN QUASI MONTE CARLO WITH EXAMPLES

CHAIN QUASI MONTE CARLO WITH EXAMPLES

Markov Chain Monte Carlo methods have been widely used in various science areas

for generation of samples from distributions that are difficult to simulate directly.

The random numbers driving Markov Chain Monte Carlo algorithms are modeled

as independent U[0, 1) random variables. By constructing a Markov Chain, we are

able to sample from its stationary distribution if irreducibility and Harris recurrence

conditions are satisfied. The class of distributions that could be simulated are largely

broadened by using Markov Chain Monte Carlo.

Quasi-Monte Carlo, on the other hand, aims to improve the accuracy of estimation

of an integral over the unit cube [0, 1)k . By using more carefully balanced inputs,

under some smoothness conditions the estimation error can be shown to converge at a

speed of O(log^k n/n), while the plain Monte Carlo method can only give a convergence

rate of Op(1/√n).

The improvement is significant when n is large.

In other words, Markov Chain Monte Carlo is creating a larger world, and quasi-

Monte Carlo is creating a better world. Ideally we would like to combine these two

techniques, so that we can sample more accurately from a larger class of distributions.

This method, called Markov Chain quasi-Monte Carlo (MCQMC), is the main topic

of this work.

**MARKOV CHAIN MONTE CARLO ALGORITHMS USING**

COMPLETELY UNIFORMLY DISTRIBUTED DRIVING

SEQUENCES

COMPLETELY UNIFORMLY DISTRIBUTED DRIVING

SEQUENCES

The advantage of low-discrepancy sequences in lieu of random numbers for simple

independent Monte Carlo sampling is well-known. This procedure, known as quasi-

Monte Carlo (QMC), yields an integration error that decays at a superior rate to

that obtained by IID sampling, by the Koksma-Hlawka inequality. For the class of

Markov chain Monte Carlo (MCMC) samplers, little literature has been produced

examining the use of low-discrepancy sequences, and previous experiments have of-

fered no theoretical validation for this practice. The central result in this work is

the establishment of conditions under which low-discrepancy sequences can be used

for consistent MCMC estimation. This condition of completely uniform distribution

(CUD) applies to a series of sequences that look like full outputs of a small random

number generator. A strategy for the incorporation of these sequences into a general

MCMC sampling scheme is thoroughly developed here, with attention to the preser-

vation of the CUD condition. The use of these sequences in a few MCMC examples

shows reductions in estimate error that are most significant in Gibbs samplers. From

these examples, the empirical benefits of CUD sequences in MCMC sampling are im-

mense, although no analog of the Koksma-Hlawka inequality has been produced for

MCMC to provide a general theoretical corroboration of these improvements.

**A Quasi-Monte Carlo Metropolis**

Algorithm

Algorithm

This paper presents a version of the Metropolis-Hastings algorithm using quasi-Monte Carlo inputs. We prove that the

method yields consistent estimates in some problems with

finite state spaces and completely uniformly distributed in-

puts. In some numerical examples, the proposed method is

much more accurate than ordinary Metropolis-Hastings sampling.

**Interacting Markov Chain Monte Carlo**

**Parallel hierarchical sampling: a general-purpose**

class of multiple-chains MCMC algorithms

class of multiple-chains MCMC algorithms

This paper introduces the Parallel Hierarchical Sampler (PHS), a

class of Markov chain Monte Carlo algorithms using several interact-

ing chains having the same target distribution but different mixing

properties.