Noisy Monte Carlo: convergence of Markov chains with approximate transition kernels

Monte Carlo algorithms often aim to draw from a distribution π by simulating a Markov chain with transition kernel P such that π is invariant under P. However, there are many situations for which it is impractical or impossible to draw from the transition kernel P. For instance, this is the case with massive datasets, where is it prohibitively expensive to calculate the likelihood and is also the case for intractable likelihood models arising from, for example, Gibbs random fields, such as those found in spatial statistics and network analysis. A natural approach in these cases is to replace P by an approximation Pˆ. Using theory from the stability of Markov chains we explore a variety of situations where it is possible to quantify how ’close’ the chain given by the transition kernel Pˆ is to the chain given by P . We apply these results to several examples from spatial statistics and network analysis.

Logistic regression with Markov chains as covariates

The tobacco-specific nitrosamine 4-(methylnitrosamino)-1-(3-pyridyl)-1-butanone (NNK) is an obvious carcinogen for lung cancer. Since CBMN (Cytokinesis-blocked micronucleus) has been found to be extremely sensitive to NNK-induced genetic damage, it is a potential important factor to predict the lung cancer risk. However, the association between lung cancer and NNK-induced genetic damage measured by CBMN assay has not been rigorously examined. ^ This research develops a methodology to model the chromosomal changes under NNK-induced genetic damage in a logistic regression framework in order to predict the occurrence of lung cancer. Since these chromosomal changes were usually not observed very long due to laboratory cost and time, a resampling technique was applied to generate the Markov chain of the normal and the damaged cell for each individual. A joint likelihood between the resampled Markov chains and the logistic regression model including transition probabilities of this chain as covariates was established. The Maximum likelihood estimation was applied to carry on the statistical test for comparison. The ability of this approach to increase discriminating power to predict lung cancer was compared to a baseline "non-genetic" model. ^ Our method offered an option to understand the association between the dynamic cell information and lung cancer. Our study indicated the extent of DNA damage/non-damage using the CBMN assay provides critical information that impacts public health studies of lung cancer risk. This novel statistical method could simultaneously estimate the process of DNA damage/non-damage and its relationship with lung cancer for each individual.^

The cutoff phenomenon in finite Markov chains.

Natural mixing processes modeled by Markov chains often show a sharp cutoff in their convergence to long-time behavior. This paper presents problems where the cutoff can be proved (card shuffling, the Ehrenfests' urn). It shows that chains with polynomial growth (drunkard's walk) do not show cutoffs. The best general understanding of such cutoffs (high multiplicity of second eigenvalues due to symmetry) is explored. Examples are given where the symmetry is broken but the cutoff phenomenon persists.

Finite Markov chains,

Mode of access: Internet.

Markov chains and manpower forecasts /

On cover: AD719413.

Aspects of Markov Chains and Particle Systems

Thesis (Ph.D.)--University of Washington, 2016-06

Uniqueness criteria for continuous-time Markov chains with general transition structures

We derive necessary and sufficient conditions for the existence of bounded or summable solutions to systems of linear equations associated with Markov chains. This substantially extends a famous result of G. E. H. Reuter, which provides a convenient means of checking various uniqueness criteria for birth-death processes. Our result allows chains with much more general transition structures to be accommodated. One application is to give a new proof of an important result of M. F. Chen concerning upwardly skip-free processes. We then use our generalization of Reuter's lemma to prove new results for downwardly skip-free chains, such as the Markov branching process and several of its many generalizations. This permits us to establish uniqueness criteria for several models, including the general birth, death, and catastrophe process, extended branching processes, and asymptotic birth-death processes, the latter being neither upwardly skip-free nor downwardly skip-free.

Sensor scheduling in electronic support using Markov chains

In electronic support, receivers must maintain surveillance over the very wide portion of the electromagnetic spectrum in which threat emitters operate. A common approach is to use a receiver with a relatively narrow bandwidth that sweeps its centre frequency over the threat bandwidth to search for emitters. The sequence and timing of changes in the centre frequency constitute a search strategy. The search can be expedited, if there is intelligence about the operational parameters of the emitters that are likely to be found. However, it can happen that the intelligence is deficient, untrustworthy or absent. In this case, what is the best search strategy to use? A random search strategy based on a continuous-time Markov chain (CTMC) is proposed. When the search is conducted for emitters with a periodic scan, it is shown that there is an optimal configuration for the CTMC. It is optimal in the sense that the expected time to intercept an emitter approaches linearity most quickly with respect to the emitter's scan period. A fast and smooth approach to linearity is important, as other strategies can exhibit considerable and abrupt variations in the intercept time as a function of scan period. In theory and numerical examples, the optimum CTMC strategy is compared with other strategies to demonstrate its superior properties.

Some Inequalities of the Uniform Ergodicity and Strong Stability of Homogeneous Markov Chains

2000 Mathematics Subject Classification: 60J45, 60K25

Resonance Energy Transfer-Based Molecular Switch Designed Using a Systematic Design Process Based on Monte Carlo Methods and Markov Chains

A RET network consists of a network of photo-active molecules called chromophores that can participate in inter-molecular energy transfer called resonance energy transfer (RET). RET networks are used in a variety of applications including cryptographic devices, storage systems, light harvesting complexes, biological sensors, and molecular rulers. In this dissertation, we focus on creating a RET device called closed-diffusive exciton valve (C-DEV) in which the input to output transfer function is controlled by an external energy source, similar to a semiconductor transistor like the MOSFET. Due to their biocompatibility, molecular devices like the C-DEVs can be used to introduce computing power in biological, organic, and aqueous environments such as living cells. Furthermore, the underlying physics in RET devices are stochastic in nature, making them suitable for stochastic computing in which true random distribution generation is critical.

In order to determine a valid configuration of chromophores for the C-DEV, we developed a systematic process based on user-guided design space pruning techniques and built-in simulation tools. We show that our C-DEV is 15x better than C-DEVs designed using ad hoc methods that rely on limited data from prior experiments. We also show ways in which the C-DEV can be improved further and how different varieties of C-DEVs can be combined to form more complex logic circuits. Moreover, the systematic design process can be used to search for valid chromophore network configurations for a variety of RET applications.

We also describe a feasibility study for a technique used to control the orientation of chromophores attached to DNA. Being able to control the orientation can expand the design space for RET networks because it provides another parameter to tune their collective behavior. While results showed limited control over orientation, the analysis required the development of a mathematical model that can be used to determine the distribution of dipoles in a given sample of chromophore constructs. The model can be used to evaluate the feasibility of other potential orientation control techniques.

Probabilistic approach of water residence time and connectivity using Markov Chains with application to tidal embayments

Markov Chain analysis was recently proposed to assess the time scales and preferential pathways into biological or physical networks by computing residence time, first passage time, rates of transfer between nodes and number of passages in a node. We propose to adapt an algorithm already published for simple systems to physical systems described with a high resolution hydrodynamic model. The method is applied to bays and estuaries on the Eastern Coast of Canada for their interest in shellfish aquaculture. Current velocities have been computed by using a 2 dimensional grid of elements and circulation patterns were summarized by averaging Eulerian flows between adjacent elements. Flows and volumes allow computing probabilities of transition between elements and to assess the average time needed by virtual particles to move from one element to another, the rate of transfer between two elements, and the average residence time of each system. We also combined transfer rates and times to assess the main pathways of virtual particles released in farmed areas and the potential influence of farmed areas on other areas. We suggest that Markov chain is complementary to other sets of ecological indicators proposed to analyse the interactions between farmed areas - e.g. depletion index, carrying capacity assessment. Markov Chain has several advantages with respect to the estimation of connectivity between pair of sites. It makes possible to estimate transfer rates and times at once in a very quick and efficient way, without the need to perform long term simulations of particle or tracer concentration.

A Bayesian‐Markov process for reliability prediction

Accurate reliability prediction for large-scale, long lived engineering is a crucial foundation for effective asset risk management and optimal maintenance decision making. However, a lack of failure data for assets that fail infrequently, and changing operational conditions over long periods of time, make accurate reliability prediction for such assets very challenging. To address this issue, we present a Bayesian-Marko best approach to reliability prediction using prior knowledge and condition monitoring data. In this approach, the Bayesian theory is used to incorporate prior information about failure probabilities and current information about asset health to make statistical inferences, while Markov chains are used to update and predict the health of assets based on condition monitoring data. The prior information can be supplied by domain experts, extracted from previous comparable cases or derived from basic engineering principles. Our approach differs from existing hybrid Bayesian models which are normally used to update the parameter estimation of a given distribution such as the Weibull-Bayesian distribution or the transition probabilities of a Markov chain. Instead, our new approach can be used to update predictions of failure probabilities when failure data are sparse or nonexistent, as is often the case for large-scale long-lived engineering assets.

Asymptotic behaviour of point processes with Markov-dependent intervals

This splitting techniques for MARKOV chains developed by NUMMELIN (1978a) and ATHREYA and NEY (1978b) are used to derive an imbedded renewal process in WOLD's point process with MARKOV-correlated intervals. This leads to a simple proof of renewal theorems for such processes. In particular, a key renewal theorem is proved, from which analogues to both BLACKWELL's and BREIMAN's forms of the renewal theorem can be deduced.