Tag Archives: EM

Classic EM in Python: Multinomial sampling

In the classic paper on the EM algorithm, the extensive example section begins with a multinomial modeling example that is theoretically very similar to the warm-up problem on 197 animals:

We can think of the complete data as an n \times p matrix x whose (i,j) element is unity if the i-th unit belongs in the j-th of p possible cells, and is zero otherwise. The i-th row of x contains p-1 zeros and one unity, but if the i-th unit has incomplete data, some of the indicators in the i-th row of x are observed to be zero, while the others are missing and we know only that one of them must be unity. The E-step then assigns to the missing indicators fractions that sum to unity within each unit, the assigned values being expectations given the current estimate of \phi. The M-step then becomes the usual estimation of \phi from the observed and assigned values of the indicators summed over the units.

In practice, it is convenient to collect together those units with the same pattern of missing indicators, since the filled in fractional counts will be the same for each; hence one may think of the procedure as filling in estimated counts for each of the missing cells within each group of units having the same pattern of missing data.

When I first made some data to try this out, it looked like this:

import pymc as mc, numpy as np, pandas as pd, random

n = 100000
p = 5

pi_true = mc.rdirichlet(np.ones(p))
pi_true = np.hstack([pi_true, 1-pi_true.sum()])
x_true = mc.rmultinomial(1, pi_true, size=n)

x_obs = array(x_true, dtype=float)
for i in range(n):
    for j in random.sample(range(p), 3):
        x_obs[i,j] = np.nan

At first, I was pretty pleased with myself when I managed to make a PyMC model and an E-step and M-step that converged to something like the true value of \pi. The model is not super slick:

pi = mc.Uninformative('pi', value=np.ones(p)/p)

x_missing = np.isnan(x_obs)
x_initial = x_obs.copy()
x_initial[x_missing] = 0.
for i in range(n):
    if x_initial[i].sum() == 0:
        j = np.where(x_missing[i])[0][0]
        x_initial[i,j] = 1.
@mc.stochastic
def x(pi=pi, value=x_initial):
    return mc.multinomial_like(value, 1, pi)

@mc.observed
def y(x=x, value=x_obs):
    if np.allclose(x[~x_missing], value[~x_missing]):
        return 0
    else:
        return -np.inf

And the E-step/M-step parts are pretty simple:

def E_step():
    x_new = array(x_obs, dtype=float)
    for i in range(n):
        if x_new[i, ~x_missing[i]].sum() == 0:
            conditional_pi_sum = pi.value[x_missing[i]].sum()
            for j in np.where(x_missing[i])[0]:
                x_new[i,j] = pi.value[j] / conditional_pi_sum
        else:
            x_new[i, x_missing[i]] = 0.
    x.value = x_new

def M_step():
    counts = x.value.sum(axis=0)
    pi.value = (counts / counts.sum())

But the way the values converge does look nice:
em

The thing that made me feel silly was comparing this fancy-pants approach to the result of averaging all of the non-empty cells of x_obs:

ests = pd.DataFrame(dict(pr=pi_true, true=x_true.mean(0),
                    naive=pd.DataFrame(x_obs).mean(), em=pi.value),
                    columns=['pr', 'true', 'naive', 'em']).sort('true')
print np.round_(ests, 3)
      pr   true  naive     em
2  0.101  0.101  0.100  0.101
0  0.106  0.106  0.108  0.108
3  0.211  0.208  0.209  0.208
1  0.269  0.271  0.272  0.271
4  0.313  0.313  0.314  0.313

Simple averages are just as good as EM, for the simplest distribution I could think of based on the example, anyways.

To see why this EM business is worth the effort requires a more elaborate model of missingness. I made one, but it is a little bit messy. Can you make one that is nice and neat?

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Classic EM in Python: Warm-up problem of 197 animals in PyMC

The classic paper on the EM algorithm begins with a little application in multinomial modeling:

Rao (1965, pp. 368-369) presents data in which 197 animals are distributed multinomially into four categories, so that the observed data consist of y = (y_1, y_2, y_3, y_4) = (125, 18, 20, 34).

A genetic model for the population specifies cell probabilities \left(\frac12 + \frac14\pi, \frac14(1-\pi), \frac14(1 -\pi), \frac14\pi\right) for some \pi with 0\leq \pi \leq 1.

Solving this analytically sounds very much like a 1960′s pastime to me (answer: \pi = \frac{15 + \sqrt{53809}}{394}), and a modern computer with PyMC can make short work of numerically approximating this maximum likelihood estimation. It takes no more than this:

import pymc as mc, numpy as np

pi = mc.Uniform('pi', 0, 1)
y = mc.Multinomial('y', n=197, 
                   p=[.5 + .25*pi, .25*(1-pi), .25*(1-pi), .25*pi],
                   value=[125, 18, 20, 34], observed=True)

mc.MAP([pi,y]).fit()
print pi.value

But the point is to warm-up, not to innovate. The EM way is to introduce a latent variable x = (x_1, x_2, x_3, x_4, x_5) and set y_1 = x_1+x_2, y_2=x_3, y_3=x_4, y_4=x_5, and work with a multinomial distribution for x that induces the multinomial distribution above on y. The art is determining a good latent variable, mapping, and distribution, and “good” is meant in terms of the computation it yields:

import pymc as mc, numpy as np

pi = mc.Uniform('pi', 0, 1)

@mc.stochastic
def x(n=197, p=[.5, .25*pi, .25*(1-pi), .25*(1-pi), .25*pi], value=[125.,0.,18.,20.,34.]):
    return mc.multinomial_like(np.round_(value), n, p)

@mc.observed
def y(x=x, value=[125, 18, 20, 34]):
    if np.allclose([x[0]+x[1], x[2], x[3], x[4]], value):
        return 0
    else:
        return -np.inf

It is no longer possible to get a good fit to an mc.MAP object for this model (why?), but EM does not need to. The EM approach is to alternate between two steps:

  • Update x (E-step, because it is set to its conditional expectation based on the current value of \pi)
  • Update \pi (M-step, because it is chosen to maximize the likelihood for the current value of x)

This is simply implemented in PyMC and quick to get the goods:

def E_step():
    x.value = [125 * .5 / (.5 + .25*pi.value), 125 * .25*pi.value / (.5 + .25*pi.value), 18, 20, 34]
def M_step():
    pi.value = (x.value[1] + 34.) / (x.value[1] + 34. + 18. + 20.)
for i in range(5):
    print 'iter %2d: pi=%0.4f, X=%s' %(i, pi.value, x.value)
    E_step()
    M_step()

It is not fair to compare speeds without making both run to the same tolerance, but when that is added EM is 2.5 times faster. That could be important sometimes, I suppose.

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Rewind Data Augmentation to EM

The original paper on Data Augmentation (DA) got me thinking it was time to have a careful look at the original paper on EM. These are both highly cited papers, and Google scholar says the DA paper has been cited 2538 times (a lot!) and the EM paper has been cited 31328 times (is that possible?).

EM (which stands of Expectation-Maximization) is something I have heard about a lot, and, as an algorithm, it is even something I understood when I read a blog post relating it to a clustering algorithm that I now cannot find (maybe on the Geomblog). The arguments from the EM to DA paper, which I read recently, makes me think that there is something more than the algorithm going on here. In this later paper, Tanner and Wong identify the importance of Section 4 of the EM paper, which provides a collection of example applications of their EM algorithm. Is there something about this way of thinking that is even more important than the algorithm itself?

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