Using a Bayesian fit is totally different from a least-squared fit. Doing it is also more complicated. In a chi-squared fit, we minimize a merit function. In a Bayesian fit, we have a set of priors, and a set of observations. To fit a model to those observations, we calculate a likelihood function. Unlike a least-squared problem, here basically each variable has a distribution. and we find the peak of distribution as the solution for unknown parameters. It is way better to read a review or something like this nice paper:

http://adsabs.harvard.edu/abs/1992ApJ…398..146G

Well, now that we know what is it, how can we do simplest things with that: i.e., fitting a straight line to some data points? Pymc made it easy. It is a python package which contains three different solvers for Bayesian statistics including a Markov chain Monte Carlo (MCMC) estimator.

You can not only use it to do simple fitting stuff like this, but also do more complicated things. However, I try to show some simple examples of its usage and comparison to a traditional fit in a separate post. Basically I compare fitting a parabola using chi-square and Bayesian method.

save the following in a file (I call it test.py):

# A quadratic fit

#———————————————————–

import numpy, pymc

# create some test data

x = numpy.arange(100) * 0.3

f = 0.1 * x**2 – 2.6 * x – 1.5

numpy.random.seed(76523654)

noise = numpy.random.normal(size=100) * .1 # create some Gaussian noise

f = f + noise # add noise to the data

z = numpy.polyfit(x, f, 2) # the traditional chi-square fit

print ‘The chi-square result: ‘, z

#priors

sig = pymc.Uniform(‘sig’, 0.0, 100.0, value=1.)

a = pymc.Uniform(‘a’, -10.0, 10.0, value= 0.0)

b = pymc.Uniform(‘b’, -10.0, 10.0, value= 0.0)

c = pymc.Uniform(‘c’, -10.0, 10.0, value= 0.0)

#model

@pymc.deterministic(plot=False)

def mod_quadratic(x=x, a=a, b=b, c=c):

return a*x**2 + b*x + c

#likelihood

y = pymc.Normal(‘y’, mu=mod_quadratic, tau=1.0/sig**2, value=f, observed=True)

#———————————————————–

Now, go to command line and run the following (or alternatively put them in a file):

import pymc, test # load the model file

R = pymc.MCMC(test) # build the model

R.sample(10000) # populate and run it

print ‘a ‘, R.a.value # print outputs

print ‘b ‘, R.b.value

print ‘c ‘, R.c.value

The output looks like this:

The chi-square result: [ 0.0999244 -2.59879643 -1.49278601]

Sampling: 100% [00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000] Iterations: 10000

a 0.0981867109293

b -2.54514077442

c -1.77016283024

The correct results are a=0.1, b=2.6, and c=1.5. Both methods gave satisfactory results. Indeed, if you increase number of iterations from 10000 to one million, the answer will be like this:

Sampling: 100% [000000000000000000000000000000000000000000000000000000000000000000000000000000000000000] Iterations: 1000000

a 0.0997945467529

b -2.59731835923

c -1.47308938314

As you see, the results are more accurate compared to the former case.

If you like to have more information, you can use the following command:

print ‘a ‘, R.a.stats()

a {‘95% HPD interval’: array([ 0.09956635, 0.10027448]), ‘n’: 1000000, ‘quantiles’: {2.5: 0.099545071298350371, 25: 0.099805050609863097, 50: 0.099919386231275664, 75: 0.10003496316945379, 97.5: 0.10025986774435611}, ‘standard deviation’: 0.0050467522199834905, ‘mc error’: 0.0003721035009025152, ‘mean’: 0.099548164235412406}

Recall that in a Bayesian fit, each variable has a distribution. The following plot shows the data points as well as the two Bayesian fits.

To compare it with a least-square fit, I repeated the experiment with a sample data which has more noise. The resulting fits are compared in the following panel.

There are much more you can learn from the examples of Pymc. I hope in a future post, I can explain other types of fit, like a weighted-least-square fit or a bisector fit.

on July 5, 2012 at 2:50 pm |SamThank you for the nice example. I am having trouble to understand how likelihood is handled in PyMc, On Abraham Flaxman very interesting Healthy Algorithms blog a Poisson distribution is used. And also some lambda functions for the model variables I don’t grasp. Your example is simpler for someone used to least-square minimization methods.

Yet I still do not understand the sig distribution that is also fitted by the MCMC algorithm and used in the likelihood.

Does this “sig” mean an unknown incertitude to be taken into account on the data values?

If errors are known on the data, could they be used instead of sig?

Thank you

Samuel

on September 17, 2013 at 4:12 pm |CarlosHi, thank you for the very neat example!

on September 29, 2013 at 8:41 pm |Accuracy of curve fitting using Bayesian method | Micropore[…] an earlier post, I have explained how a Bayesian based method can replace standard chi-square technique. Since a […]

on July 10, 2015 at 2:34 am |EliThanks so much for putting it all into context for me. Keep up the good work!

on February 4, 2017 at 10:39 pm |Python: fit with error on Y-axis | Micropore[…] some earlier post, I have discussed statistical fits with PyMC and EMCEE. Advantage of statistical methods is that they are not sensitive to the form of […]