Nested sampling methods

Buchner, Johannes

doi:10.1214/23-ss144

Cited by 40 publications

(21 citation statements)

References 74 publications

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“…The algorithm returns a set of nested samples, with corresponding prior volumes and likelihoods, and an evidence estimate with a corresponding error. The stopping criterion is typically related to the fractional change in the evidence between iterations [7].…”

Section: Nested Samplingmentioning

confidence: 99%

“…In the original paper [2], Skilling proposes using MCMC over the prior and accepting only those points for which L(θ) > L * until the correlation with the starting point (one of the existing samples) has been lost. This method requires a random walk that can adapt to the continuously shrinking likelihood-constrained prior and a method for determining the number of steps to take [7]. Further modifications are often needed to handle multi-modality and complex correlations between parameters, for example, as implemented in Veitch et al [3].…”

Section: Nested Samplingmentioning

confidence: 99%

“…When implementing nested sampling, the main challenge is drawing new points from the likelihood-constrained prior at a given iteration. There are different approaches to this such as using Markov Chain Monte Carlo (MCMC), slice sampling or sampling from bounding distributions [7]. There have also been efforts to incorporate machine learning into nested sampling for approximating the likelihood [8], in the proposal process [9,10] and for sampling from arbitrary priors [11].…”

Section: Introductionmentioning

confidence: 99%

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Importance nested sampling with normalising flows

Williams

Veitch

Messenger

2023

Mach. Learn.: Sci. Technol.

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We present an improved version of the nested sampling algorithm nessai in which the core algorithm is modified to use importance weights. In the modified algorithm, samples are drawn from a mixture of normalising flows and the requirement for samples to be independently and identically distributed (i.i.d.) according to the prior is relaxed. Furthermore, it allows for samples to be added in any order, independently of a likelihood constraint, and for the evidence to be updated with batches of samples. We call the modified algorithm i-nessai. We first validate i-nessai using analytic likelihoods with known Bayesian evidences and show that the evidence estimates are unbiased in up to 32 dimensions. We compare i-nessai to standard nessai for the analytic likelihoods and the Rosenbrock likelihood, the results show that i-nessai is consistent with nessai whilst producing more precise evidence estimates. We then test i-nessai on 64 simulated gravitational-wave signals from binary black hole coalescence and show that it produces unbiased estimates of the parameters. We compare our results to those obtained using standard nessai and dynesty and find that i-nessai requires 2.68 and 13.3 times fewer likelihood evaluations to converge, respectively. We also test i-nessai of an 80-second simulated binary neutron star signal using a Reduced-Order-Quadrature (ROQ) basis and find that, on average, it converges in 24 minutes, whilst only requiring 1.01 × 106 likelihood evaluations compared to 1.42 × 106 for nessai and 4.30 × 107 for dynesty. These results demonstrate the i-nessai is consistent with nessai and dynesty whilst also being more efficient. 

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Section: Nested Samplingmentioning

confidence: 99%

Section: Nested Samplingmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Importance nested sampling with normalising flows

Williams

Veitch

Messenger

2023

Mach. Learn.: Sci. Technol.

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“…We employ the Bayesian inference method of nested sampling (Skilling 2004; for reviews see Ashton et al 2022;Buchner 2023). Conceptually, nested sampling algorithms converge on the best parameter estimates by iteratively removing regions of the prior volume with lower likelihood.…”

Section: Nested Samplingmentioning

confidence: 99%

“…In general, Monte Carlo methods like nested sampling and MCMC are computationally expensive because they require many likelihood evaluations to sample the converged posterior distributions. We choose nested sampling instead of MCMC because the former is designed to better constrain complex parameter spaces with banana-shaped curved degeneracies and/or multimodal distributions (Buchner 2023). Another likelihood-based method that has been applied to parameter estimation of the global 21 cm signal is Fisher matrix analysis (Liu et al 2013;Muñoz et al 2020;Hibbard et al 2022;Mason et al 2023a), which assumes multivariate Gaussian posterior distributions and requires only ( )  N likelihood evaluations for N parameters being sampled.…”

Section: Nested Samplingmentioning

confidence: 99%

Validating Posteriors Obtained by an Emulator When Jointly Fitting Mock Data of the Global 21 cm Signal and High-z Galaxy UV Luminosity Function

Dorigo Jones,

Rapetti,

Mirocha

et al. 2023

ApJ

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Although neural-network-based emulators enable efficient parameter estimation in 21 cm cosmology, the accuracy of such constraints is poorly understood. We employ nested sampling to fit mock data of the global 21 cm signal and high-z galaxy ultraviolet luminosity function (UVLF) and compare for the first time the emulated posteriors obtained using the global signal emulator globalemu to the “true” posteriors obtained using the full model on which the emulator is trained using ARES. Of the eight model parameters we employ, four control the star formation efficiency (SFE) and thus can be constrained by UVLF data, while the remaining four control UV and X-ray photon production and the minimum virial temperature of star-forming halos ( T min ) and thus are uniquely probed by reionization and 21 cm measurements. For noise levels of 50 and 250 mK in the 21 cm data being jointly fit, the emulated and “true” posteriors are consistent to within 1σ. However, at lower noise levels of 10 and 25 mK, globalemu overpredicts T min and underpredicts γ lo, an SFE parameter, by ≈3σ–4σ, while the “true” ARES posteriors capture their fiducial values within 1σ. We find that jointly fitting the mock UVLF and 21 cm data significantly improves constraints on the SFE parameters by breaking degeneracies in the ARES parameter space. Our results demonstrate the astrophysical constraints that can be expected for global 21 cm experiments for a range of noise levels from pessimistic to optimistic, as well as the potential for probing redshift evolution of SFE parameters by including UVLF data.

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Bayesian parameter estimation using truncated normal distributions as priors for parameters in fundamental models of chemical processes

Gibson,

McAuley

2024

Can J Chem Eng

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Modellers of chemical processes with knowledge about plausible parameter values use Bayesian parameter estimation methods to account for their prior beliefs. Some modellers specify prior distributions with finite parameter ranges, such as uniform distributions and truncated normal distributions, because they better account for knowledge about realistic parameter ranges than normal prior distributions with parameter values ranging between and . We derive closed‐form objective functions for Bayesian parameter estimation with truncated normal priors and uniform priors, for the first time, so that parameter estimation can be performed by solving simple optimization problems rather than using complex sampling‐based techniques. A parametric bootstrapping method that considers truncated normal priors and model nonlinearity is proposed to determine 95% confidence intervals and joint confidence regions. A pharmaceutical case study is used to show the effectiveness of the proposed objective functions and bootstrapping methodology. Confidence regions from bootstrapping are similar to linearization‐based confidence regions that do not account for truncation when truncated areas in normal prior distributions are relatively small. More truncation, which corresponds to more‐precise prior knowledge about the parameters, results in smaller joint confidence regions. The proposed methods will be attractive for parameter estimation in complex process models because they can be less computationally intensive than Markov chain Monte Carlo methods that provide similar results.

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Nested sampling methods

Cited by 40 publications

References 74 publications

Importance nested sampling with normalising flows

Importance nested sampling with normalising flows

Validating Posteriors Obtained by an Emulator When Jointly Fitting Mock Data of the Global 21 cm Signal and High-z Galaxy UV Luminosity Function

Bayesian parameter estimation using truncated normal distributions as priors for parameters in fundamental models of chemical processes

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