MCMC Sampling a Maxwellian Curve Using Python'

Question

I am trying to introduce myself to MCMC sampling with emcee. I want to simply take a sample from a Maxwell Boltzmann distribution using a set of example code on github, https://github.com/dfm/emcee/blob/master/examples/quickstart.py.

The example code is really excellent, but when I change the distribution from a Gaussian to a Maxwellian, I receive the error, TypeError: lnprob() takes exactly 2 arguments (3 given)

However it is not called anywhere where it is not given the appropriate parameters? In need of some guidance as to how to define a Maxwellian Curve and have it fit into this example code.

Here is what I have;

    from __future__ import print_function
import numpy as np
import emcee

try:
    xrange
except NameError:
    xrange = range
def lnprob(x, a, icov):
    pi = np.pi
    return np.sqrt(2/pi)*x**2*np.exp(-x**2/(2.*a**2))/a**3

ndim = 2
means = np.random.rand(ndim)

cov  = 0.5-np.random.rand(ndim**2).reshape((ndim, ndim))
cov  = np.triu(cov)
cov += cov.T - np.diag(cov.diagonal())
cov  = np.dot(cov,cov)


icov = np.linalg.inv(cov)


nwalkers = 50


p0 = [np.random.rand(ndim) for i in xrange(nwalkers)]


sampler = emcee.EnsembleSampler(nwalkers, ndim, lnprob, args=[means, icov])

pos, prob, state = sampler.run_mcmc(p0, 5000)

sampler.reset()

sampler.run_mcmc(pos, 100000, rstate0=state)

Thanks

Answer 1

I think there are a couple of problems that I see. The main one is that emcee wants you to give it the natural logarithm of the probability distribution function that you want to sample. So, rather than having:

def lnprob(x, a, icov):
    pi = np.pi
    return np.sqrt(2/pi)*x**2*np.exp(-x**2/(2.*a**2))/a**3

you would instead want, e.g.

def lnprob(x, a):
    pi = np.pi
    if x < 0:
        return -np.inf
    else:
        return 0.5*np.log(2./pi) + 2.*np.log(x) - (x**2/(2.*a**2)) - 3.*np.log(a)

where the if...else... statement is to explicitly say that negative values of x have zero probability (or -infinity in log-space).

You also shouldn't have to calculate icov and pass it to lnprob as that's only needed for the Gaussian case in the example you link to.

When you call:

sampler = emcee.EnsembleSampler(nwalkers, ndim, lnprob, args=[means, icov])

the args value should just be any additional arguments that your lnprob function requires, so in your case this would be the value of a that you want to set your Maxwell-Boltxmann distribution up with. This should be a single value rather than the two randomly initialised values you have set when creating mean.

Overall, the following should work for you:

from __future__ import print_function

import emcee
import numpy as np
from numpy import pi as pi

# define the natural log of the Maxwell-Boltzmann distribution
def lnprob(x, a):               
    if x < 0:                                                                
        return -np.inf
    else:
        return 0.5*np.log(2./pi) + 2.*np.log(x) - (x**2/(2.*a**2)) - 3.*np.log(a)

# choose a value of 'a' for the distributions
a = 5. # I'm choosing 5!

# choose the number of walkers
nwalkers = 50

# set some initial points at which to calculate the lnprob
p0 = [np.random.rand(1) for i in xrange(nwalkers)]

# initialise the sampler
sampler = emcee.EnsembleSampler(nwalkers, 1, lnprob, args=[a])

# Run 5000 steps as a burn-in.
pos, prob, state = sampler.run_mcmc(p0, 5000)

# Reset the chain to remove the burn-in samples.
sampler.reset()

# Starting from the final position in the burn-in chain, sample for 100000 steps.
sampler.run_mcmc(pos, 100000, rstate0=state)

# lets check the samples look right
mbmean = 2.*a*np.sqrt(2./pi) # mean of Maxwell-Boltzmann distribution
print("Sample mean = {}, analytical mean = {}".format(np.mean(sampler.flatchain[:,0]), mbmean))
mbstd = np.sqrt(a**2*(3*np.pi-8.)/np.pi) # std. dev. of M-B distribution
print("Sample standard deviation = {}, analytical = {}".format(np.std(sampler.flatchain[:,0]), mbstd))