A statistical test for Nested Sampling algorithms
- 484 Downloads
Nested sampling is an iterative integration procedure that shrinks the prior volume towards higher likelihoods by removing a “live” point at a time. A replacement point is drawn uniformly from the prior above an ever-increasing likelihood threshold. Thus, the problem of drawing from a space above a certain likelihood value arises naturally in nested sampling, making algorithms that solve this problem a key ingredient to the nested sampling framework. If the drawn points are distributed uniformly, the removal of a point shrinks the volume in a well-understood way, and the integration of nested sampling is unbiased. In this work, I develop a statistical test to check whether this is the case. This “Shrinkage Test” is useful to verify nested sampling algorithms in a controlled environment. I apply the shrinkage test to a test-problem, and show that some existing algorithms fail to pass it due to over-optimisation. I then demonstrate that a simple algorithm can be constructed which is robust against this type of problem. This RADFRIENDS algorithm is, however, inefficient in comparison to MULTINEST.
KeywordsNested sampling MCMC Bayesian inference Evidence Test Marginal likelihood
I would like to thank Frederik Beaujean and Udo von Toussaint for reading the initial manuscript. I acknowledge funding through a doctoral stipend by the Max Planck Society. This manuscript has greatly benefited from the comments of the two anonymous referees, whom I would also like to thank. I acknowledge financial support through a Max Planck society stipend.
- Beaujean, F., Caldwell, A.: Initializing adaptive importance sampling with Markov chains. ArXiv (e-prints) (2013)Google Scholar
- Betancourt, M.: Nested sampling with constrained Hamiltonian Monte Carlo. In Mohammad-Djafari, A., Bercher, J.-F., & Bessiére, P. (eds.) American Institute of Physics Conference Series, vol. 1305, pp. 165–172. American Institute of Physics Conference Series (2011)Google Scholar
- Cameron, E., Pettitt, A.: Recursive pathways to marginal likelihood estimation with prior-sensitivity analysis. ArXiv e-prints (2013)Google Scholar
- Chopin, N., Robert, C.: Comments on nested sampling by john skilling. Bayesian Stat. 8, 491–524 (2007)Google Scholar
- Chopin, N., Robert, C.P.: Properties of nested sampling. Biometrika. (2010)Google Scholar
- Evans, M.: Discussion of nested sampling for bayesian computations by john skilling. Bayesian Stat. 8, 491–524 (2007)Google Scholar
- Feroz, F., Hobson, M. P., Cameron, E., Pettitt, A. N.: Importance nested sampling and the MultiNest algorithm. ArXiv e-prints. (2013)Google Scholar
- Sivia, D., Skilling, J.: Data Analysis: A Bayesian Tutorial. Oxford science publications. Oxford University Press, Oxford (2006)Google Scholar
- Skilling, J.: Nested sampling. In: AIP Conference Proceedings, vol. 735, p. 395. (2004)Google Scholar
- Skilling, J.: Nested sampling’s convergence. In: BAYESIAN INFERENCE AND MAXIMUM ENTROPY METHODS IN SCIENCE ANDENGINEERING. The 29th International Workshop on Bayesian Inference andMaximum Entropy Methods in Science and Engineering, vol. 1193, pp. 277–291. AIP Publishing, New York (2009)Google Scholar