How to model a mixture of 3 Normals in PyMC?

问题:

There is a question on CrossValidated on how to use PyMC to fit two Normal distributions to data. The answer of Cam.Davidson.Pilon was to use a Bernoulli distribution to assign data to one of the two Normals:

size = 10
p = Uniform( "p", 0 , 1) #this is the fraction that come from mean1 vs mean2
ber = Bernoulli( "ber", p = p, size = size) # produces 1 with proportion p.
precision = Gamma('precision', alpha=0.1, beta=0.1)

mean1 = Normal( "mean1", 0, 0.001 )
mean2 = Normal( "mean2", 0, 0.001 )

@deterministic
def mean( ber = ber, mean1 = mean1, mean2 = mean2):
    return ber*mean1 + (1-ber)*mean2

Now my question is: how to do it with three Normals?

Basically, the issue is that you can't use a Bernoulli distribution and 1-Bernoulli anymore. But how to do it then?

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edit: With the CDP's suggestion, I wrote the following code:

import numpy as np
import pymc as mc

n = 3
ndata = 500

dd = mc.Dirichlet('dd', theta=(1,)*n)
category = mc.Categorical('category', p=dd, size=ndata)

precs = mc.Gamma('precs', alpha=0.1, beta=0.1, size=n)
means = mc.Normal('means', 0, 0.001, size=n)

@mc.deterministic
def mean(category=category, means=means):
    return means[category]

@mc.deterministic
def prec(category=category, precs=precs):
    return precs[category]

v = np.random.randint( 0, n, ndata)
data = (v==0)*(50+ np.random.randn(ndata)) \
       + (v==1)*(-50 + np.random.randn(ndata)) \
       + (v==2)*np.random.randn(ndata)
obs = mc.Normal('obs', mean, prec, value=data, observed = True)

model = mc.Model({'dd': dd,
              'category': category,
              'precs': precs,
              'means': means,
              'obs': obs})

The traces with the following sampling procedure look good as well. Solved!

mcmc = mc.MCMC( model )
mcmc.sample( 50000,0 )
mcmc.trace('means').gettrace()[-1,:]

回答1:

there is a mc.Categorical object that does just this.

p =  [0.2, 0.3, .5]
t = mc.Categorical('test', p )
t.random()
#array(2, dtype=int32)

It returns an int between 0 and len(p)-1. To model the 3 Normals, you make p a mc.Dirichlet object (it accepts a k length array as the hyperparameters; setting the values in the array to be the same is setting the prior probabilities to be equal). The rest of the model is nearly identical.

This is a generalization of the model I suggested above.

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Update:

Okay, so instead of having different means, we can collapse them all into 1:

means = Normal( "means", 0, 0.001, size=3 )

...

@mc.deterministic
def mean(categorical=categorical, means = means):
   return means[categorical]