Add MCMC module for 1DLA parameter search

[jibanCat]

Additions:

1. add _pdf function in the dla_samples.py
2. add run_mcmc function in dla_gp; still very ugly written.
3. add a file log_posterior_mcmc.py with the likelihood functions completely separated from the class, which is required by the parallelization in the mcmc chains.

Todo

• make a test on finding parameters in comparison with MAP values.
• try to write it cleaner.

[jibanCat]

Current Usage:

For example, you want to do mcmc on this example in the paper Ho-Bird-Garnett 2020:

from tests.test_model import prepare_dla_model
from examples.plot_mcmc import plot_corner, plot_sample_this_mu

# prepare a DLA model conditioned on a given spectrum
dla_gp = prepare_dla_model(plate=5309, mjd=55929, fiber_id=362, z_qso=3.166)

# number of samples you would like to sample in MCMC chains
nsamples = 10000
# number of burn-in samples you would like to discard in your MCMC chains
discard = 2000

# sampling DLA parameters using Markov Chain Monte Carlo
# nwalkers is the number of chains you would like to have. details refer to emcee website.
# kth_dla now only supports 1
sampler = dla_gp.run_mcmc(nwalkers=32, kth_dla=1, nsamples=nsamples)

# plot the sampling results from your MCMC chain. You can set the desired zDLA range
# using min_z and max_z
plot_corner(sampler, discard=discard, min_z=2.6, max_z=5.0)

plot_corner(sampler, discard=discard, min_z=2.0, max_z=2.6)

# plot the dla model with multiple MCMC samples, showing the probabilistic detection of DLAs
plot_sample_this_mu(dla_gp, sampler, discard=discard)

[jibanCat]

Merge for now.

Added features:

1. MCMC for 1 DLA
2. MCMC for 1 CIV

Not implement:

• QMC for CIV. (there's no need to implement now. Wait for Reza's paper being published)

So CIV module only supports using MCMC. Details see in the notebook.