The Application of Riemann Manifold Langevin Monte Carlo Methods in Bayesian Risk Analysis

Location: Melbourne, VIC

Duration: 5 months

Keywords: Bayesian theory, programming, multidisciplinary, statistics, MCMC method, differential geometry

This internship is only available to Australian Citizens due to funding eligibility requirements. All applicants who are not Australian Citizens will be automatically deemed ineligible.

Project Background

The airworthiness of aircraft fleets is critical in ensuring the safety of personnel and assets, while maintaining availability of the fleet. Airworthiness assessments rely on the application of probabilistic risk analysis to determine the risk of failure of individual components and the impact on the entire fleet.

Bayesian theory has been widely used in probabilistic risk analysis, since it allows a prioiri information (i.e. physical, expert and/or statistical information) about the component to be included in the analysis. The application of Bayesian analysis often requires the use of Markov chain Monte Carlo (MCMC) methods to sample the posterior probability density function (PDF) for the unknown parameters and to determine the integrated likelihood function, which is necessary for model selection and averaging.

The Metropolis-Hastings algorithm is a typical MCMC method that is used in Bayesian analysis[1]. The efficiency of the Monte Carlo sampling is important to ensure that sufficient samples are produced that represent the underlying distributions being sampled, that are used in the subsequent risk analysis. For example, inefficient sampling can result considerable computational over-head and may produce samples which are not statistically representative of the underlying distributions. Such sampling issues can have serious consequences in any subsequent analysis and probabilistic decision making. In particular, the sampling efficiency of MCMC methods becomes an issue when analysing heavily censored data, which comprises of a few failures and many non-failed (censored) components.

Considerable research effort has attempted to develop efficient MCMC methods2. Recent analysis has shown that exploiting both diffusion and the geometric properties of the underlying distributions can improve the sampling efficiency of MCMC methods[2]. However, in order to adapt such algorithms to Bayesian risk and reliability analysis, further investigation is necessary to understand how the diffusion process and geometry influence the analyses of heavily censored data.

The aim of this project is to investigate how a new class of MCMC methods1, those that exploit both diffusion and geometric properties, can be adapted and applied to Bayesian risk and reliability analysis.

[1] Giromlami, M. and Calderhead, B. (2011), “Riemann manifold Langevin and Hamiltonian Monte Carlo methods”, J. R. Statist. Soc. B., 73, Part 2, pp. 123-214. Also see references therein.

[2] Gelman et al. (2004). Bayesian data analysis, 2nd ed. Chapman & Hall. See references therein.

Research to be Conducted

The research project will include:

  1. Theoretical investigation of Riemann manifold Metropolis adjusted Langevin algorithms, in particular how the censored data influences the Riemann manifold, metric tensor and sampling characteristics of the algorithm.
  2. Develop, write, test and fully document an R-package (or Mathematica) for Riemann manifold Metropolis adjusted Langevin algorithm that can be readily applied.
  3. Application of the algorithm to analysing failure and heavily censored datasets, including evaluating the algorithm against existing algorithms.
  4. Co-author and submit a peer-review journal article to an appropriate journal.

Skills Required

For this project, we are looking for PhD students with:

  1. PhD in Mathematics, Statistic or Physics or other relevant field
  2. Multidisciplinary understanding of MCMC methods, stochastic processes, differential geometry, Bayesian theory, statistic, and numerical analysis.
  3. Strong programming ability and understanding of modern programming techniques.

Please contact the Business Developer if you have any questions regarding the internship requirements.

Expected Outcomes

Specific aims include:

  1. Understand how the theoretical properties of Riemann manifold Metropolis adjusted Langevin algorithm2 are affected by datasets consisting of varying proportions of failure and censored data, including the overall geometry, stochastic differential equation and optimal scaling etc.
  2. Understanding how to calculate, or approximate, a numerically stable metric tensor for varying proportions of failure and censored data.
  3. Develop a numerically stable procedure for determining the optimal scaling/tuning of a Riemann manifold Metropolis adjusted Langevin algorithm.

Additional Details

The intern will receive $3,000 per month of the internship, usually in the form of stipend payments.

It is expected that the intern will primarily undertake this research project during regular business hours, spending at least 80% of their time on-site with the industry partner. The intern will be expected to maintain contact with their academic mentor throughout the internship either through face-to-face or phone meetings as appropriate.

The intern and their academic mentor will have the opportunity to negotiate the project’s scope, milestones and timeline during the project planning stage.

To participate in the AMSI Intern program, all applicants must satisfy the following criteria:

  • Be a PhD student currently enrolled at an Australian University.
  • PhD candidature must be confirmed.
  • Applicants must have the written approval of their Principal Supervisor to undertake the internship. This approval must be submitted at the time of application.
  • Have Australian Citizenship
  • Internships are also subject to any requirements stipulated by the student’s and the academic mentor’s university.
Applications Close

26 November 2017

Reference:
INT – 0368