MontePython 3: boosted MCMC sampler and other features (1804.07261v2)

Abstract: MontePython is a parameter inference package for cosmology. We present the latest development of the code over the past couple of years. We explain, in particular, two new ingredients both contributing to improve the performance of Metropolis-Hastings sampling: an adaptation algorithm for the jumping factor, and a calculation of the inverse Fisher matrix, which can be used as a proposal density. We present several examples to show that these features speed up convergence and can save many hundreds of CPU-hours in the case of difficult runs, with a poor prior knowledge of the covariance matrix. We also summarise all the functionalities of MontePython in the current release, including new likelihoods and plotting options.

• The paper introduces an adaptive jumping factor that dynamically optimizes the acceptance rate in the MH algorithm.
• It leverages the inverse Fisher matrix to approximate the covariance matrix, accelerating convergence in complex parameter spaces.
• These enhancements reduce computational demands and enable robust exploration of advanced cosmological models.

MontePython 3: Enhancements in MCMC Sampling for Cosmology

The paper discusses recent advancements incorporated into MontePython, a Python-based Markov Chain Monte Carlo (MCMC) parameter inference tool for cosmological studies. The latest version, MontePython 3, introduces several enhancements to the Metropolis-Hastings (MH) sampling algorithm, aiming to improve efficiency and adaptability, particularly for difficult posterior distributions where prior knowledge of the covariance matrix is limited.

Key Features and Methodological Improvements

MontePython 3 enhances the MH sampling technique by implementing two notable features: an adaptive algorithm for the jumping factor and the utilization of the inverse Fisher matrix as a proposal density. These improvements target the acceleration of convergence and reduction of computational resources in complex MCMC runs.

1. Adaptive Jumping Factor: The paper outlines a new scheme, termed "superupdate," which dynamically adjusts the jumping factor in the MH algorithm to maintain an optimal acceptance rate. This adaptation alleviates the necessity for pre-tuning and aligns the proposal with the evolving characteristics of the posterior distribution. The adaptation process delays the start of adjustment until early convergence indicators—such as the Gelman-Rubin statistic—suggest that the sampling distribution is stabilizing.
2. Inverse Fisher Matrix: The authors propose the computation of the inverse Fisher matrix directly from the likelihood function. This matrix serves as an initial approximation to the covariance matrix for MH sampling. By encapsulating the local curvature of the likelihood surface, the Fisher matrix expedites convergence in scenarios where the maximum likelihood estimate is known or can be accurately approximated. This feature proves particularly beneficial in forecast studies with mock data.
3. Performance Enhancements: Together, these strategies significantly curtail the computational demands associated with MCMC sampling. The paper presents empirical evaluations, evidencing that these enhancements can save substantial CPU hours, particularly in extended parameter spaces or in the absence of reliable prior covariance matrices.

Implications for Cosmological Research

The improvements in MontePython 3 carry meaningful implications for cosmological parameter estimation. Efficient handling of complex, high-dimensional parameter spaces permits researchers to tackle increasingly sophisticated models and extensive datasets, leading to more robust inferences about cosmological phenomena.

By diminishing the need for extensive manual tuning and providing a mechanism for rapid adaptation, these advancements encourage exploration across a broader landscape of cosmological models. This adaptability is essential for staying abreast of new data from upcoming observational campaigns, such as those involving next-generation galaxy surveys and CMB measurements.

Future Trajectories

The developments outlined in MontePython 3 suggest a trajectory where MCMC tools increasingly leverage adaptive techniques and statistical approximations (such as the Fisher matrix) to enhance sampling efficiency. Future iterations may refine these methods further, incorporating more sophisticated sensitivity analysis and possibly integrating additional algorithms from machine learning to adaptively learn the structure of the parameter space.

In conclusion, MontePython 3 represents a significant step forward in the field of cosmological computation, providing sophisticated tools that align with the computational challenges presented by contemporary and future cosmology datasets. This work demonstrates the critical intersection of improved algorithmic techniques with scientific inquiry, paving the way for precision cosmology.