Since each term in the covariance matrix of this proposal distribution is an unspecified parameter, this method has N/2 tuning parameters (where N is the dimension of the parameter space). Each step in a M–H chain is proposed using a compact proposal distribution centered on the current position of the chain (normally a multivariate Gaussian or something similar). Most uses of MCMC in the astrophysics literature are based on slight modifications to the Metropolis–Hastings (M–H) method (introduced below in § 2). The methods presented here are designed for efficiency. In this regime, MCMC sampling is very valuable, but it is even more valuable if the MCMC algorithm is efficient, in the sense that it does not require many function evaluations to generate a statistically independent sample from the posterior PDF. In addition to the problem of marginalization, in many problems of interest the likelihood or the prior is the result of an expensive simulation or computation. However, a MCMC-generated sampling of values (Θ t,α t) of the model and nuisance parameters from the joint distribution p(Θ,α| D) automatically provides a sampling of values Θ t from the marginalized PDF p(Θ| D). Because the nuisance parameter set α can be very large, this integral is often extremely daunting. Where α is the set (list or vector) of nuisance parameters. The exact result of marginalization is the marginalized probability function p(Θ| D) of the set (list or vector) of model parameters Θ given the set of observations D Often we wish to marginalize over all nuisance parameters in a model. Marginalization is the process of integrating over all possible values of the parameter and hence propagating the effects of uncertainty about its value into the final result. A nuisance parameter is one that is required in order to model the process that generates the data, but is otherwise of little interest. 2005).Īrguably the most important advantage of Bayesian data analysis is that it is possible to marginalize over nuisance parameters. This has proven useful in too many research applications to list here but the results from the NASA Wilkinson Microwave Anisotropy Probe (WMAP) cosmology mission provide a dramatic example (for example, Dunkley et al. MCMC methods are designed to sample from-and thereby provide sampling approximations to-the posterior PDF efficiently even in parameter spaces with large numbers of dimensions. In some cases it is sufficient to find the maximum of one of these, but it is often necessary to understand the posterior PDF in detail. Probabilistic data analysis procedures involve computing and using either the posterior probability density function (PDF) for the parameters of the model or the likelihood function. For example, many problems in cosmology and astrophysics 6 have directly benefited from MCMC because the models are often expensive to compute, there are many free parameters, and the observations are usually low in signal-to-noise. Many of the most significant gains have come from numerical methods for approximate inference, especially Markov chain Monte Carlo (MCMC). Probabilistic data analysis-including Bayesian inference-has transformed scientific research in the past decade. The code is available online at under the GNU General Public License v2. Exploiting the parallelism of the ensemble method, emcee permits any user to take advantage of multiple CPU cores without extra effort. In this document, we describe the algorithm and the details of our implementation. One major advantage of the algorithm is that it requires hand-tuning of only 1 or 2 parameters compared to ∼ N 2 for a traditional algorithm in an N-dimensional parameter space. The algorithm behind emcee has several advantages over traditional MCMC sampling methods and it has excellent performance as measured by the autocorrelation time (or function calls per independent sample). The code is open source and has already been used in several published projects in the astrophysics literature. We introduce a stable, well tested Python implementation of the affine-invariant ensemble sampler for Markov chain Monte Carlo (MCMC) proposed by Goodman & Weare ( 2010).
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