SMC-python

Python codes for Sequential Monte Carlo sampling Technique.

This technique is robust in sampling close to 1000-dimensions posterior probability densities.

Uses numpy, scipy and collections libraries.

Demonstration

Example 1: Bivariate bimodal Gaussian probability density

We draw samples from a bivariate bimodal Gaussian probability density as shown below:
The two modes for first parameter (par 1) is at [2 10], while for the second parameter (par 2) is at [6 12]. The corresponding standard deviation are [2 1] for par 1 and [2 1] for par 2, respectively.

Codes to calculate target density for the two parameters

postt calculates the log of the target distribution.

import numpy as np
from scipy.stats import multivariate_normal

mu1 = np.array([2,12])
sigma1 = np.array([2,1])
mvn1 = multivariate_normal(mu1,sigma1) 

mu2 = np.array([10,6])
sigma2 = np.array([1,2])
mvn2 = multivariate_normal(mu2,sigma2)

postt = lambda x: np.log((mvn1.pdf(x)+mvn2.pdf(x))/ \
                         (mvn1.pdf(mu1)+mvn2.pdf(mu2))) 

Generate samples corresponding to the target distribution

Import the collections library.

from collections import namedtuple

Create two named tuple objects: opt and samples.

opt corresponds to input parameters of the sampling technique. opt.N is the number of Markov chains, opt.Neff is the chain length, opt.LB is the lower bound of the target distribution parameters, and opt.UB is the upper bound.

samples corresponds to the information of the samples at each intermediate stage. samples.allsamples is the ensemble of samples at final stage, samples.postval is the log of target distribution values, samples.stage is the array of all the stages, samples.beta is the array of beta parameters, samples.covsmpl is the sample covariance at final stage, samples.resmpl is the resampled samples at the final stage.

NT1 = namedtuple('NT1', 'N Neff target LB UB')
opt = NT1(2000, 50, postt, np.array([0,0]), np.array([15,15]))

NT2 = namedtuple('NT2', 'allsamples postval beta stage covsmpl resmpl')
samples = NT2(None, None, None, None, None, None)

Run this line to get all the samples.

final = SMC_samples(opt,samples, NT1, NT2)

Plot the histograms of the resulting samples

plt.subplot(1, 2, 1)
n, bins, patches = plt.hist(final.allsamples[:,0], 50, density=True, \
                            facecolor='b', alpha=0.75)
plt.subplot(1, 2, 2)
n, bins, patches = plt.hist(final.allsamples[:,1], 50, density=True, \
                            facecolor='b', alpha=0.75)