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The problem

I want fit the model parameters of a simple 2-Gaussian mixture population. Given all the hype around Bayesian methods I want to understand if for this problem Bayesian inference is a better tool that traditional fitting methods.

So far MCMC performs very poorly in this toy example, but maybe I just overlooked something. So let see the code.

The tools

I will use python (2.7) + scipy stack, lmfit 0.8 and PyMC 2.3.

A notebook to reproduce the analysis can be found here

Generate the data

First let generate the data:

from scipy.stats import distributions

# Sample parameters
nsamples = 1000
mu1_true = 0.3
mu2_true = 0.55
sig1_true = 0.08
sig2_true = 0.12
a_true = 0.4

# Samples generation
np.random.seed(3)  # for repeatability
s1 = distributions.norm.rvs(mu1_true, sig1_true, size=round(a_true*nsamples))
s2 = distributions.norm.rvs(mu2_true, sig2_true, size=round((1-a_true)*nsamples))
samples = np.hstack([s1, s2])

The histogram of samples looks like this:

data histogram

a "broad peak", the components are hard to spot by eye.

Classical approach: fit the histogram

Let's try the classical approach first. Using lmfit it's easy to define a 2-peaks model:

import lmfit

peak1 = lmfit.models.GaussianModel(prefix='p1_')
peak2 = lmfit.models.GaussianModel(prefix='p2_')
model = peak1 + peak2

model.set_param_hint('p1_center', value=0.2, min=-1, max=2)
model.set_param_hint('p2_center', value=0.5, min=-1, max=2)
model.set_param_hint('p1_sigma', value=0.1, min=0.01, max=0.3)
model.set_param_hint('p2_sigma', value=0.1, min=0.01, max=0.3)
model.set_param_hint('p1_amplitude', value=1, min=0.0, max=1)
model.set_param_hint('p2_amplitude', expr='1 - p1_amplitude')
name = '2-gaussians'

Finally we fit the model with the simplex algorithm:

fit_res = model.fit(data, x=x_data, method='nelder')
print fit_res.fit_report()

The result is the following image (red dashed lines are fitted centers):

NLS fit results

Even if the problem is kind of hard, with proper initial values and constraints the models converged to quite a reasonable estimate.

Bayesian approach: MCMC

I define the model in PyMC in hierarchical fashion. centers and sigmas are the priors distribution for the hyperparameters representing the 2 centers and 2 sigmas of the 2 Gaussians. alpha is the fraction of the first population and the prior distribution is here a Beta.

A categorical variable chooses between the two populations. It is my understanding that this variable needs to be same size as the data (samples).

Finally mu and tau are deterministic variables that determine the parameters of the Normal distribution (they depend on the category variable so they randomly switch between the two values for the two populations).

sigmas = pm.Normal('sigmas', mu=0.1, tau=1000, size=2)
centers = pm.Normal('centers', [0.3, 0.7], [1/(0.1)**2, 1/(0.1)**2], size=2)
#centers = pm.Uniform('centers', 0, 1, size=2)

alpha  = pm.Beta('alpha', alpha=2, beta=3)
category = pm.Categorical("category", [alpha, 1 - alpha], size=nsamples)

@pm.deterministic
def mu(category=category, centers=centers):
    return centers[category]

@pm.deterministic
def tau(category=category, sigmas=sigmas):
    return 1/(sigmas[category]**2)

observations = pm.Normal('samples_model', mu=mu, tau=tau, value=samples, observed=True)
model = pm.Model([observations, mu, tau, category, alpha, sigmas, centers])

Then I run the MCMC with quite long number of iterations (1e5, ~60s on my machine):

mcmc = pm.MCMC(model)
mcmc.sample(100000, 30000)

However the results are very odd. For example $\alpha$ trace (the fraction of the first population) tends to 0 instead to converge to 0.4 and has a very strong autocorrelation:

MCMC alpha summary

Also the centers of the Gaussians do not converge as well. For example:

MCMC centers_0 summary

As you see in the prior choice, I tried to "help" the MCMC algorithm using a Beta distribution for the prior population fraction $\alpha$. Also the prior distributions for the centers and sigmas are quite reasonable (I think).

So what's going on here? Am I doing something wrong or MCMC is not suitable for this problem?

I understand that MCMC method will be slower, but the trivial histogram fit seems to perform immensely better in resolving the populations.

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The problem is caused by the way that PyMC draws samples for this model. As explained in section 5.8.1 of the PyMC documentation, all elements of an array variable are updated together. For small arrays like center this is not a problem, but for a large array like category it leads to a low acceptance rate. You can see the acceptance rate via

print mcmc.step_method_dict[category][0].ratio

The solution suggested in the documentation is to use an array of scalar-valued variables. In addition, you need to configure some of the proposal distributions since the default choices are bad. Here is the code that works for me:

import pymc as pm
sigmas = pm.Normal('sigmas', mu=0.1, tau=1000, size=2)
centers = pm.Normal('centers', [0.3, 0.7], [1/(0.1)**2, 1/(0.1)**2], size=2)
alpha  = pm.Beta('alpha', alpha=2, beta=3)
category = pm.Container([pm.Categorical("category%i" % i, [alpha, 1 - alpha]) 
                         for i in range(nsamples)])
observations = pm.Container([pm.Normal('samples_model%i' % i, 
                   mu=centers[category[i]], tau=1/(sigmas[category[i]]**2), 
                   value=samples[i], observed=True) for i in range(nsamples)])
model = pm.Model([observations, category, alpha, sigmas, centers])
mcmc = pm.MCMC(model)
# initialize in a good place to reduce the number of steps required
centers.value = [mu1_true, mu2_true]
# set a custom proposal for centers, since the default is bad
mcmc.use_step_method(pm.Metropolis, centers, proposal_sd=sig1_true/np.sqrt(nsamples))
# set a custom proposal for category, since the default is bad
for i in range(nsamples):
    mcmc.use_step_method(pm.DiscreteMetropolis, category[i], proposal_distribution='Prior')
mcmc.sample(100)  # beware sampling takes much longer now
# check the acceptance rates
print mcmc.step_method_dict[category[0]][0].ratio
print mcmc.step_method_dict[centers][0].ratio
print mcmc.step_method_dict[alpha][0].ratio

The proposal_sd and proposal_distribution options are explained in section 5.7.1. For the centers, I set the proposal to roughly match the standard deviation of the posterior, which is much smaller than the default due to the amount of data. PyMC does attempt to tune the width of the proposal, but this only works if your acceptance rate is sufficiently high to begin with. For category, the default proposal_distribution = 'Poisson' which gives poor results (I don't know why this is, but it certainly doesn't sound like a sensible proposal for a binary variable).

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  • $\begingroup$ Thanks, this is really useful, although it becomes almost unbearably slow. Can you briefly explain of the meaning of proposal_distribution and proposal_sd and why using Prior is better for the categorical variables? $\endgroup$ – user2304916 Oct 17 '14 at 17:36
  • $\begingroup$ Thanks, Tom. I agree that Poisson is an odd choice here. I opened up an issue: github.com/pymc-devs/pymc/issues/627 $\endgroup$ – twiecki Oct 17 '14 at 20:47
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Also, non-identifiability is a big problem for using MCMC for mixture models. Basically, if you switch labels on your cluster means and cluster assignments, the likelihood doesn't change, and this can confuse the sampler (between chains or within chains). One thing you might try to mitigate this is using Potentials in PyMC3. A good implementation of a GMM with Potentials is here. The posterior for these kinds of problems is also generally highly multimodal, which also presents a big problem. You might instead want to use EM (or Variational inference).

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You should not model $\sigma$ with a Normal, that way you are allowing negative values for the standard variation. Use instead something like:

sigmas = pm.Exponential('sigmas', 0.1, size=2)

update:

I got near the initial parameters of the data by changing these parts of your model:

sigmas = pm.Exponential('sigmas', 0.1, size=2)
alpha  = pm.Beta('alpha', alpha=1, beta=1)

and by invoking the mcmc with some thinning:

mcmc.sample(200000, 3000, 10)

results:

alpha

centers

sigmas

The posteriors are not very nice thou... In section 11.6 of the BUGS Book they discuss this type of model and state that there are convergence problems with no obvious solution. Check also here.

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  • $\begingroup$ That's a good point, I'm using a Gamma now. Nonetheless alpha trace always tends to 0 (instead of 0.4). I wonder if there is some stupid bug lurking in my example. $\endgroup$ – user2304916 Oct 15 '14 at 16:27
  • $\begingroup$ I tried Gamma(.1,.1) but Exp(.1) seems to work better. Also, when the autocorrelation is high, you could include some thinning, eg, mcmc.sample(60000, 3000, 3) $\endgroup$ – jpneto Oct 16 '14 at 9:02

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