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Examples to test and run tempered PMC.

If the cosmopmc conda enviroment is not running, activate it now:

conda activate cosmopmc

  1. 1_mvnorm_2D_temp_none

    No tempering. Test of Bayesian evidence and approximations.

    Run

    cosmo_pmc.pl -f "0.5 0.5"

    The likelihood is a 2D multi-variate normal. The prior volume is the unit square. Since the likelihood is normalised, the Bayesian evidence is the inverse prior volume, which is 1, computed and written to the file evidence_analytic..

    The file evidence should be consistent with 1.

    The Laplace approximation with the Fisher matrix replacing the input likelihood uses as maximum-likelihood a point very close to the true value (given as starting point for the maximum-search by -f "0.5 0.5"). The file evidence_fisher should thus also be consistent with 1.

    Another Laplace approximation is computed using a the sample covariance instead of the true input covariance matrix. The results are performed for each iteration, and written to iter_?/evidence_covariance. For the final iteration the result should be close to 1.

    The final triangle contour plot is iter_4/all_contour2D.pdf, which is produced automatically if R is installed and working. The contours are not smoothed, and can look quite noisy. To make a nicer plot with smoothed 68% and 95% contours, similar to GetDist change to iter_4 and run the command

    Rscript `which plot_confidence.R ` pmcsim -c ../config_pmc -g 10 -s 2

    The output should look like this:

    For comparison, this is the output of cobaya/GetDist of the same posterior (see the cobaya config file 1_mvnorm_2D_temp_none/gaussian.yaml) with MCMC:

  2. 2_mixmvnorm_2D_temp_none

    No tempering. Illustration of biased evidence if only one of multi-component posterior is found and sampled.

    Run

    cosmo_pmc.pl -A y

    The likelihood is a 2D mixture of multi-variate normals, with two non-overlapping components. The two components are far enough away that traditional PMC (or MCMC) does not always find both components.

    As in example 1, the likelihood and prior volume are normalised to one, and thus the Bayesian evidence is unity, see evidence_analytic.

    The Fisher matrix is necessarily computed at (or close) to one of the two component maxima. Thus, the corresponding approximate likelihood contains only half of the density, and the evidence is biased low by a factor of around 2.

    The evidence from PMC sampling, evidence, is likely to approach either 0.5 or 1, dependent whether one or two components have been found. Which is the case can be seen in the posterior contour plots, iter_9/all_cont2d.pdf. To test, run the code several times.

    Here is the output of three runs:

    Both peaks are found, evidence = 0.998802. Only the first peak is found, evidence = 0.50055. Only the second peak is found, evidence = 0.499927.

  3. 3_topolike_1D_temp_log

    For this example, the topology likelihood needs to be installed and linked with CosmoPMC. Run

    cosmo_pmc.pl

    In this 1D example, only one of the three angles is varied. The multi-modal posterior should be visible.