Bayesian inference with Wolfram Mathematica
mBayes aims at helping researchers to effortlessly carry out Bayesian inference within Wolfram Mathematica. Much of the workflow has been automatized. At the moment, mBayes explores the posterior on a grid. Grid exploration is particularly convenient if one needs to compute 3+ sigma intervals. This works reasonably well for posteriors up to 4-6 dimensions. If the posterior is close to a multivariate Gaussian distribution, then mBayes can save a lot of time by exploring only a suitable ellipsoid obtained via the Fisher matrix. If dealing with high-dimensional degenerate posteriors please consider tools such as emcee or CosmoHammer.
Have fun, Valerio
At the moment, the following features are implemented:
- multivariate and flat priors
- variables can be easily fixed without editing the glue code
- Fisher matrix approximation for likelihood and posterior
- Mathematica experimental Hessian
- Fisher and fast numerical evidence
- grid optimization with Fisher
- optimized parallel computation and exportation
- automatized exportation of results with consistent labeling
- confidence levels (actual and gaussian)
- combinations of triangular plots
- an mcmc sampler is not yet implemented; coming...
- works seemlessly with MAGMA
To run the code you should use Mathematica 11; earlier version may show some incompatibilities. Do note move the Mathematica files from the relative folders.
Start by opening the notebook 1-exploration.nb. There you should customize Section 2 (Glue code) and Section 3 (Parameter space and prior). Please read the comments for help. The result is automatically saved in the folder results-chi2.
Once the posterior has been explored, you should open 2-analysis.nb, rename the relevant variables and run everything. The plots and the other products are automatically saved in the folder results-analysis.
1-exploration.nb and 2-analysis.nb are distinct so that you may associate a different kernel to them and, for example, run 2-analysis.nb while 1-exploration.nb is busy doing something else.
I will put asap a video for a quick overview of mBayes' functionalities and workflow.
You can use mBayes, or part of it, freely, provided that in your publications you acknowledge its use and cite the paper Camarena & Marra arXiv:1805.09900. If you modify parts of mBayes, you may append to the credit line the following text:
This code is released under the GPL license. Copyright by Valerio Marra (email@example.com)
This code has been modified by XXX (firstname.lastname@example.org).
If using the provided JLA and Pantheon catalogs, please cite Betoule et al 2014 and Scolnic et al 2018, respectively.
Mathematica 11.3 is much less forgiving than previous versions. When using Mathematica 11.3 the marginalization routines may give some (inconsequential, most of the times) errors.
Valerio Marra acknowledges contributions from Tiago Castro, Miguel Quartin and the awesome community of mathematica.stackexchange. The use of the color palettes developed by Paul Tol (personal.sron.nl/~pault) and colorbrewer2 is acknowledged.