# MCMC Tutorial¶

This tutorial describes the available options when running an MCMC with mc3. The following sections make up a script meant to be run from the Python interpreter or in a Python script. At the bottom of this page you can see the entire script.

## Preamble¶

In this tutorial, we will use the following function to create and fit a synthetic dataset following a quadratic behavior:

def quad(p, x):
"""

Parameters
p: Polynomial constant, linear, and quadratic coefficients.
x: Array of dependent variables where to evaluate the polynomial.
Returns
y: Polinomial evaluated at x:  y(x) = p0 + p1*x + p2*x^2
"""
y = p[0] + p[1]*x + p[2]*x**2.0
return y


## Argument Inputs¶

From the shell, the arguments can be input as command-line arguments. However, in this case, the best option is to specify all inputs in a cconfiguration file. An mc3 configuration file follows the configparser standard format described here. To see all the available options, run:

mc3 --help


From the Python interpreter, the arguments must be input as function arguments. To see the available options, run:

import mc3
help(mc3.sample)


## Input Data¶

The data and uncert arguments (required) defines the dataset to be fitted and their $$1\sigma$$ uncertainties, respectively. Each one of these arguments is a 1D float ndarray:

# Preamble (create a synthetic dataset, in a real scenario you would
np.random.seed(314)
x  = np.linspace(0, 10, 1000)
p0 = [3, -2.4, 0.5]

uncert = np.sqrt(np.abs(y))
error  = np.random.normal(0, uncert)
data   = y + error


Note

Alternatively, the data argument can be a string specifying a Numpy npz filename containing the data and uncert arrays. See the Input Data File Section below.

## Modeling Function¶

The func argument (required) defines the parameterized modeling function fitting the data. The user can either set func as a callable, e.g.:

# Define the modeling function as a callable:


or as a tuple of strings pointing to the modeling function, e.g.:

# A three-element tuple indicates the function's name, the module
# name (without the '.py' extension), and the path to the module.

# If the module is already within the scope of the Python path,
# the user can set func as a two-elements tuple:


Note

The only requirement for the modeling function is that its arguments follow the same structure of the callable in scipy.optimize.leastsq, i.e., the first argument is a 1D iterable containing the fitting parameters.

The indparams argument (optional) contains any additional argument required by func:

# List of additional arguments of func (if necessary):
indparams = [x]


Note

Even if there is only one additional argument to func, indparams must be defined as a tuple (as in the example above). Eventually, the modeling function has to able to be called as: model = func(params, *indparams)

## Fitting Parameters¶

The params argument (required) is a 1D float ndarray containing the initial-guess values for the model fitting parameters.

# Array of initial-guess values of fitting parameters:
params = np.array([ 10.0, -2.0, 0.1])


The pmin and pmax arguments (optional) are 1D float ndarrays that set lower and upper boundaries explored by the MCMC, for each fitting parameter (same size as params).

# Lower and upper boundaries for the MCMC exploration:
pmin = np.array([-10.0, -20.0, -10.0])
pmax = np.array([ 40.0,  20.0,  10.0])


If a proposed step falls outside the set boundaries, that iteration is automatically rejected. The default values for each element of pmin and pmax are -np.inf and +np.inf, respectively.

## Parameters Stepping Behavior¶

The pstep argument (optional) is a 1D float ndarray that defines the stepping behavior of the fitting parameters over the parameter space. This argument has actually a dual purpose:

### Stepping Behavior¶

First, it can keep a fitting parameter fixed by setting its pstep value to zero, for example:

# Keep the third parameter fixed:
pstep = np.array([1.0, 0.5, 0.0])


It can force a fitting parameter to share its value with another parameter by setting its pstep value equal to the negative index of the sharing parameter, for example:

# Make the third parameter share the value of the second parameter:
pstep = np.array([1.0, 0.5, -2])


Otherwise, a positive pstep value leaves the parameter as a free fitting parameter:

# Parameters' stepping behavior:
pstep = np.array([1.0, 0.5, 0.1])


### Stepping Scale¶

pstep also sets the step size of the free parameters. For a differential-evolution run (e.g., sampler = 'snooker'), mc3 starts the MCMC drawing samples from a normal distribution for each parameter, whose standard deviation is set by the pstep values. For a classic Metropolis random walk (sampler = 'mrw'), the pstep values set the standard deviation of the Gaussian proposal jumps for each parameter.

For more details on the MCMC algorithms, see Sampler Algorithm.

## Parameter Priors¶

The prior, priorlow, and priorup arguments (optional) are 1D float ndarrays that set the prior estimate, lower uncertainty, and upper uncertainty of the fitting parameters. mc3 supports two types of priors:

A priorlow value of zero (default) defines a uniform prior between the parameter boundaries. This is appropriate when there is no prior knowledge for a parameter $$\theta$$:

$p(\theta) = \frac{1}{\theta_{\rm max} - \theta_{\rm min}},$

Positive priorlow and priorup values define a Gaussian prior for a parameter $$\theta$$:

$p(\theta) = A \exp\left(\frac{-(\theta-\theta_{p})^{2}}{2\sigma_{p}^{2}}\right),$

where prior sets the prior value $$\theta_{p}$$, and priorlow and priorup set the lower and upper $$1\sigma$$ prior uncertainties, $$\sigma_{p}$$, of the prior (depending if the proposed value $$\theta$$ is lower or higher than $$\theta_{p}$$, respectively). The leading factor is given by: $$A = 2/(\sqrt{2\pi}(\sigma_{\rm up}+\sigma_{\rm lo}))$$ (see [Wallis2014]), which reduces to the familiar Gaussian normal distribution when $$\sigma_{\rm up} = \sigma_{\rm lo}$$:

$p(\theta) = \frac{1}{\sqrt{2\pi}\sigma_{p}} \exp\left(\frac{-(\theta-\theta_{p})^{2}}{2\sigma_{p}^{2}}\right),$

For example, to explicitly set uniform priors for all parameters:

# Parameter prior probability distributions:
prior    = np.array([ 0.0, 0.0, 0.0])
priorlow = np.array([ 0.0, 0.0, 0.0])
priorup  = np.array([ 0.0, 0.0, 0.0])


## Parameter Names¶

The pnames argument (optional) define the names of the model parametes to be shown in the scren output and figure labels. The screen output will display up to 11 characters. For figures, the texnames argument (optional) enables names using LaTeX syntax, for example:

# Parameter names:
pnames   = ['y0', 'alpha', 'beta']
texnames = [r'$y_{0}$', r'$\alpha$', r'$\beta$']


If texnames = None, it defaults to pnames. If pnames = None, it defaults to texnames. If both arguments are None, they default to a generic [Param 1, Param 2, ...] list.

## Sampler Algorithm¶

The sampler argument (required) defines the sampler algorithm for the MCMC:

# Sampler algorithm, choose from: 'snooker', 'demc' or 'mrw'.
sampler = 'snooker'


The standard Differential-Evolution MCMC algorithm (sampler = 'demc', [terBraak2006]) proposes for each chain $$i$$ in state $$\mathbf{x}_{i}$$:

$\mathbf{x}^* = \mathbf{x}_i + \gamma (\mathbf{x}_{R1}-\mathbf{x}_{R2}) + \mathbf{e},$

where $$\mathbf{x}_{R1}$$ and $$\mathbf{x}_{R2}$$ are randomly selected without replacement from the population of current states except $$\mathbf{x}_{i}$$. This implementation adopts $$\gamma=f_{\gamma} 2.38/\sqrt{2 N_{\rm free}}$$, with $$N_{\rm free}$$ the number of free parameters; and $$\mathbf{e}\sim \mathcal{N}(0, \sigma^2)$$, with $$\sigma=f_{e}$$ pstep, where the scaling factors are defaulted to $$f_{\gamma}=1.0$$ and $$f_{e}=0.0$$ (see Fine-tuning).

If sampler = 'snooker' (recommended), mc3 will use the DEMC-zs algorithm with snooker propsals (see [terBraakVrugt2008]).

If sampler = 'mrw', mc3 will use the classical Metropolis-Hastings algorithm with Gaussian proposal distributions. I.e., in each iteration and for each parameter, $$\theta$$, the MCMC will propose jumps, drawn from Gaussian distributions centered at the current value, $$\theta_0$$, with a standard deviation, $$\sigma$$, given by the values in the pstep argument:

$q(\theta) = \frac{1}{\sqrt{2 \pi \sigma^2}} \exp \left( -\frac{(\theta-\theta_0)^2}{2 \sigma^2}\right)$

Note

For sampler=snooker, an MCMC works well with 3 chains or more. For sampler=demc, [terBraak2006] suggest using $$2 N_{\rm free}$$ chains. From experience, I recommend the snooker, as it is more efficient than most others MCMC random walks.

## MCMC Configuration¶

The following arguments set the MCMC chains configuration:

# MCMC setup:
nsamples =  1e5
burnin   = 1000
nchains  =   14
ncpu     =    7
thinning =    1


The nsamples argument (required for MCMC runs) sets the total number of MCMC samples to compute.

The burnin argument (optional, default: 0) sets the number of burned-in (removed) iterations at the beginning of each chain.

The nchains argument (optional, default: 7) sets the number of parallel chains to use.

The ncpu argument (optional, default: nchains) sets the number CPUs to use for the chains. mc3 runs in multiple processors through the mutiprocessing Python Standard-Library package (additionaly, the central MCMC hub will use one extra CPU. Thus, the total number of CPUs used is ncpu + 1).

Note

If ncpu+1 is greater than the number of available CPUs in the machine, mc3 will cap ncpu to the number of available CPUs minus one. To keep a good balance, I recommend setting nchains equal to a multiple of chains ncpu as in the example above.

The thinning argument (optional, default: 1) sets the chains thinning factor (discarding all but every thinning-th sample). To reduce the memory usage, when requested, only the thinned samples are stored (and returned).

Note

Thinning is often unnecessary for a DE run, since this algorithm reduces significatively the sampling autocorrelation.

### Pre-MCMC Setup¶

The following arguments set how the code set the initial values for the MCMC chains:

# MCMC initial draw, choose from: 'normal' or 'uniform'
kickoff = 'normal'
# DEMC snooker pre-MCMC sample size:
hsize   = 10


The starting point of the MCMC chains come from a random draw, set by the kickoff argument (optional, default: ‘normal’). This can be a Normal-distribution draw centered at params with standard deviation pstep; or it can be a uniform draw bewteen pmin and pmax.

The snooker DEMC, in particular, needs an initial sample, set by the hsize argument (optional, default: 10). The draws from this initial sample do not count for the posterior-distribution statistics.

Usually, these variables do not have a significant impact in the outputs. Thus, they can be left at their default values.

## Optimization¶

When not None, the leastsq argument (optional, default: None) run a least-squares optimization before the MCMC:

# Optimization before MCMC, choose from: 'lm' or 'trf':
leastsq    = 'lm'
chisqscale = False


Set leastsq='lm' to use the Levenberg-Marquardt algorithm via Scipy’s leastsq, or set leastsq='trf' to use the Trust Region Reflective algorithm via Scipy’s least_squares. Fixed and shared-values apply during the optimization (see Stepping Behavior), as well as the priors (see Parameter Priors).

Note

From the scipy documentation: Levenberg-Marquardt ‘doesn’t handle bounds’ but is ‘the most efficient method for small unconstrained problems’; whereas the Trust Region Reflective algorithm is a ‘Generally robust method, suitable for large sparse problems with bounds’.

The chisqscale argument (optional, default: False) is a flag to scale the data uncertainties to enforce a reduced $$\chi^{2}$$ equal to $$1.0$$. The scaling applies by multiplying all uncertainties by a common scale factor.

## Convergence¶

mc3 checks for convergence through the Gelman-Rubin test ([GelmanRubin1992]):

# MCMC Convergence:
grtest  = True
grbreak = 1.01
grnmin  = 0.5


The grtest argument (optional, default: False), when True, runs the Gelman-Rubin convergence test. Values larger than 1.01 are indicative of non-convergence. See [GelmanRubin1992] for further information. The Gelman-Rubin test is computed every 10% of the MCMC exploration.

The grbreak argument (optional, default: 0.0) sets a convergence threshold to stop an MCMC when GR drops below grbreak. Reasonable values seem to be $${\rm grbreak} \lesssim 1.01$$. The default behavior is not to break (grbreak = 0.0).

The grnmin argument (optional, default: 0.5) sets a minimum number of valid samples (after burning and thinning) required for grbreak. If grnmin is greater than one, it defines the minimum number of samples to run before breaking out of the MCMC. If grnmin is lower than one, it defines the fraction of the total samples to run before breaking out of the MCMC.

## Wavelet-Likelihood MCMC¶

The wlike argument (optional, default: False) allows mc3 to implement the Wavelet-based method to account for time-correlated noise. When using this method, the used must append the three additional fitting parameters ($$\gamma, \sigma_{r}, \sigma_{w}$$) from [CarterWinn2009] to the end of the params array. Likewise, add the correspoding values to the pmin, pmax, stepsize, prior, priorlow, and priorup arrays. For further information see [CarterWinn2009].

This tutorial won’t use the wavelet method:

# Carter & Winn (2009) Wavelet-likelihood method:
wlike = False


## Fine-tuning¶

The $$f_{\gamma}$$ and $$f_{e}$$ factors scale the DEMC proposal distributions.

fgamma   = 1.0  # Scale factor for DEMC's gamma jump.
fepsilon = 0.0  # Jump scale factor for DEMC's "e" distribution


The default $$f_{\gamma} = 1.0$$ value is set such that the MCMC acceptance rate approaches 25–40%. Therefore, most of the time, the user won’t need to modify this. Only if the acceptance rate is very low, we recommend to set $$f_{\gamma} < 1.0$$. The $$f_{e}$$ factor sets the jump scale for the $$\mathbf e$$ distribution, which has to have a small variance compared to the posterior. For further information, see [terBraak2006].

## Logging¶

If not None, the log argument (optional, default: None) stores the screen output into a log file. log can either be a string of the filename where to store the log, or an mc3.utils.Log object (see API).

# Logging:
log = 'MCMC_tutorial.log'


## Outputs¶

The following arguments set the output files produced by mc3:

# File outputs:
savefile = 'MCMC_tutorial.npz'
plots    = True
rms      = True


The savefile argument (optional, default: None) defines an .npz file names where to store the MCMC outputs. This file contains the following key–items:

• posterior: thinned posterior distribution of shape [nsamples, nfree], including burn-in phase.
• zchain: chain indices for the posterior samples.
• zmask: posterior mask to remove the burn-in.
• chisq: $$\chi^2$$ values for the posterior samples.
• log_post: log of the posterior for the sample (as defined here).
• burnin: number of burned-in samples per chain.
• ifree: Indices of the free parameters.
• pnames: Parameter names.
• texnames: Parameter names in Latex format.
• meanp: mean of the marginal posteriors for each model parameter.
• stdp: standard deviation of the marginal posteriors for each model parameter.
• CRlo: lower boundary of the marginal 68%-highest posterior density (the credible region) for each model parameter.
• CRhi: upper boundary of the marginal 68%-HPD.
• stddev_residuals: standard deviation of the residuals.
• acceptance_rate: sample’s acceptance rate.
• best_log_post: optimal log of the posterior in the sample (see here).
• bestp: model parameters for the best_log_post sample.
• best_model: model evaluated at bestp.
• best_chisq: $$\chi^2$$ for the best_log_post sample.
• red_chisq: reduced chi-squared: $$\chi^2/(N_{\rm data}-N_{\rm free})$$ for the best_log_post sample.
• BIC: Bayesian Information Criterion: $$\chi^2 -N_{\rm free} \log(N_{\rm data})$$ for the best_log_post sample.
• chisq_factor: Uncertainties scale factor to enforce $$\chi^2_{\rm red} \equiv 1$$.

Note

Notice that if there are fixed or shared parameters, then the number of free parameters won’t be the same as the number of model parameters. The output posterior Z includes only the free parameters, whereas the CRlo, CRhi, stdp, meanp, and bestp outputs include all model parameters.

Setting the plots argument (optional, default: False) to True will generate data (along with the best-fitting model) plot, the MCMC-chain trace plot for each parameter, and the marginalized and pair-wise posterior plots. Setting the ioff argument to True (optional, default: False) will turn the display interactive mode off.

Set the rms argument (optional, default: False) to True to compute and plot the time-averaging test for time-correlated noise (see [Winn2008]).

## MCMC Run¶

Putting it all together, here’s a Python script to run an mc3 retrieval explicitly defining all the variables described above:

import sys
import numpy as np
import mc3

"""

Parameters
p: Polynomial constant, linear, and quadratic coefficients.
x: Array of dependent variables where to evaluate the polynomial.
Returns
y: Polinomial evaluated at x:  y(x) = p0 + p1*x + p2*x^2
"""
y = p[0] + p[1]*x + p[2]*x**2.0
return y

# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# Preamble (create a synthetic dataset, in a real scenario you would
np.random.seed(314)
x  = np.linspace(0, 10, 1000)
p0 = [3, -2.4, 0.5]

uncert = np.sqrt(np.abs(y))
error  = np.random.normal(0, uncert)
data   = y + error
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

# Define the modeling function as a callable:

# List of additional arguments of func (if necessary):
indparams = [x]

# Array of initial-guess values of fitting parameters:
params = np.array([ 10.0, -2.0, 0.1])
# Lower and upper boundaries for the MCMC exploration:
pmin = np.array([-10.0, -20.0, -10.0])
pmax = np.array([ 40.0,  20.0,  10.0])
# Parameters' stepping behavior:
pstep = np.array([1.0, 0.5, 0.1])

# Parameter prior probability distributions:
prior    = np.array([ 0.0, 0.0, 0.0])
priorlow = np.array([ 0.0, 0.0, 0.0])
priorup  = np.array([ 0.0, 0.0, 0.0])

# Parameter names:
pnames   = ['y0', 'alpha', 'beta']
texnames = [r'$y_{0}$', r'$\alpha$', r'$\beta$']

# Sampler algorithm, choose from: 'snooker', 'demc' or 'mrw'.
sampler = 'snooker'

# MCMC setup:
nsamples =  1e5
burnin   = 1000
nchains  =   14
ncpu     =    7
thinning =    1

# MCMC initial draw, choose from: 'normal' or 'uniform'
kickoff = 'normal'
# DEMC snooker pre-MCMC sample size:
hsize   = 10

# Optimization before MCMC, choose from: 'lm' or 'trf':
leastsq    = 'lm'
chisqscale = False

# MCMC Convergence:
grtest  = True
grbreak = 1.01
grnmin  = 0.5

# Logging:
log = 'MCMC_tutorial.log'

# File outputs:
savefile = 'MCMC_tutorial.npz'
plots    = True
rms      = True

# Carter & Winn (2009) Wavelet-likelihood method:
wlike = False

# Run the MCMC:
mc3_output = mc3.sample(data=data, uncert=uncert, func=func, params=params,
indparams=indparams, pmin=pmin, pmax=pmax, pstep=pstep,
pnames=pnames, texnames=texnames,
prior=prior, priorlow=priorlow, priorup=priorup,
sampler=sampler, nsamples=nsamples,  nchains=nchains,
ncpu=ncpu, burnin=burnin, thinning=thinning,
leastsq=leastsq, chisqscale=chisqscale,
grtest=grtest, grbreak=grbreak, grnmin=grnmin,
hsize=hsize, kickoff=kickoff,
wlike=wlike, log=log,
plots=plots, savefile=savefile, rms=rms)


This routine returns a dictionary containing the outputs listed in Outputs. The screen output should look like this:

::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Multi-core Markov-chain Monte Carlo (mc3).
Version 3.0.0.
Copyright (c) 2015-2019 Patricio Cubillos and collaborators.
::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Least-squares best-fitting parameters:
[ 3.02203328 -2.3897706   0.49543328]

Yippee Ki Yay Monte Carlo!
Start MCMC chains  (Thu Aug  8 11:23:20 2019)

[:         ]  10.0% completed  (Thu Aug  8 11:23:21 2019)
Out-of-bound Trials:
[0 0 0]
Best Parameters: (chisq=1035.2269)
[ 3.02203328 -2.3897706   0.49543328]

[::        ]  20.0% completed  (Thu Aug  8 11:23:21 2019)
Out-of-bound Trials:
[0 0 0]
Best Parameters: (chisq=1035.2269)
[ 3.02203328 -2.3897706   0.49543328]
[1.02204221 1.02386902 1.02470492]

[:::       ]  30.0% completed  (Thu Aug  8 11:23:21 2019)
Out-of-bound Trials:
[0 0 0]
Best Parameters: (chisq=1035.2269)
[ 3.02203328 -2.3897706   0.49543328]
[1.00644059 1.00601973 1.00644078]
All parameters converged to within 1% of unity.

[::::      ]  40.0% completed  (Thu Aug  8 11:23:22 2019)
Out-of-bound Trials:
[0 0 0]
Best Parameters: (chisq=1035.2269)
[ 3.02203328 -2.3897706   0.49543328]
[1.00332153 1.00383779 1.00326743]
All parameters converged to within 1% of unity.

[:::::     ]  50.0% completed  (Thu Aug  8 11:23:22 2019)
Out-of-bound Trials:
[0 0 0]
Best Parameters: (chisq=1035.2269)
[ 3.02203328 -2.3897706   0.49543328]
[1.00286025 1.00297467 1.00258288]
All parameters converged to within 1% of unity.

[::::::    ]  60.0% completed  (Thu Aug  8 11:23:22 2019)
Out-of-bound Trials:
[0 0 0]
Best Parameters: (chisq=1035.2269)
[ 3.02203328 -2.3897706   0.49543328]
[1.00169127 1.0016499  1.0013014 ]
All parameters converged to within 1% of unity.

All parameters satisfy the GR convergence threshold of 1.01, stopping
the MCMC.

MCMC Summary:
-------------
Number of evaluated samples:        60506
Number of parallel chains:             14
Average iterations per chain:        4321
Burned-in iterations per chain:      1000
Thinning factor:                        1
MCMC sample size (thinned, burned): 46506
Acceptance rate:   28.85%

Param name     Best fit   Lo HPD CR   Hi HPD CR        Mean    Std dev       S/N
----------- ----------------------------------- ---------------------- ---------
y0           3.0220e+00 -1.2142e-01  1.2574e-01  3.0223e+00 1.2231e-01      24.7
alpha       -2.3898e+00 -7.2210e-02  6.8853e-02 -2.3904e+00 7.0381e-02      34.0
beta         4.9543e-01 -8.3569e-03  8.9226e-03  4.9557e-01 8.6295e-03      57.4

Best-parameter's chi-squared:       1035.2269
Best-parameter's -2*log(posterior): 1035.2269
Bayesian Information Criterion:     1055.9502
Reduced chi-squared:                   1.0383
Standard deviation of residuals:  2.77253

Output MCMC files:
'MCMC_tutorial.npz'
'MCMC_tutorial_trace.png'
'MCMC_tutorial_pairwise.png'
'MCMC_tutorial_posterior.png'
'MCMC_tutorial_RMS.png'
'MCMC_tutorial_model.png'
'MCMC_tutorial.log'


## Resume a Previous Run¶

It is also possible to add more samples to a previous run (identified by the .npz output file name). To do this, set the resume = True. Ideally, keep the same number of MCMC chains from the previous run to avoid any conflict with the shape of the posterior. This resumed run will append nsamples samples into the posterior output from the previous run (overwritting all output files).

## Inputs from Files¶

The data, uncert, indparams, params, pmin, pmax, stepsize, prior, priorlow, and priorup input arrays can be optionally be given as input file. Furthermore, multiple input arguments can be combined into a single file.

### Input Data File¶

The data, uncert, and indparams inputs can be provided as binary numpy .npz files. data and uncert can be stored together into a single file. An indparams input file contain the list of independent variables (must be a list, even if there is a single independent variable).

The utils sub-package of mc3 provide utility functions to save and load these files. This script shows how to create data and indparams input files:

import numpy as np
import mc3

# Create a synthetic dataset using a quadratic polynomial curve:
x  = np.linspace(0.0, 10, 1000)
p0 = [3, -2.4, 0.5]
error  = np.random.normal(0, uncert)

data   = y + error
uncert = np.sqrt(np.abs(y))

# data.npz contains the data and uncertainty arrays:
mc3.utils.savebin([data, uncert], 'data.npz')
# indp.npz contains a list of variables:
mc3.utils.savebin([x], 'indp.npz')


### Model Parameters¶

The params, pmin, pmax, stepsize, prior, priorlow, and priorup inputs can be provided as plain ASCII files. For simplycity all of these input arguments can be combined into a single file.

In the params file, each line correspond to one model parameter, whereas each column correspond to one of the input array arguments. This input file can hold as few or as many of these argument arrays, as long as they are provided in that exact order. Empty or comment lines are allowed (and ignored by the reader). A valid params file look like this:

#       params            pmin            pmax        stepsize
10             -10              40             1.0
-2.0             -20              20             0.5
0.1             -10              10             0.1


Alternatively, the utils sub-package of mc3 provide utility functions to save and load these files:

params   = [ 10, -2.0,  0.1]
pmin     = [-10,  -20, -10]
pmax     = [ 40,   20,  10]
stepsize = [  1,  0.5,  0.1]

# Store ASCII arrays:
mc3.utils.saveascii([params, pmin, pmax, stepsize], 'params.txt')


Then, to run the MCMC simply provide the input file names to the mc3 routine:

# Set arguments as the file names:
data      = 'data.npz'
indparams = 'indp.npz'
params    = 'params.txt'

# Run the MCMC:
mc3_output = mc3.sample(data=data, func=func, params=params,
indparams=indparams, sampler=sampler, nsamples=nsamples,  nchains=nchains,
ncpu=ncpu, burnin=burnin, leastsq=leastsq, chisqscale=chisqscale,
grtest=grtest, grbreak=grbreak, grnmin=grnmin,
log=log, plots=plots, savefile=savefile, rms=rms)