Notes on the Dirichlet Distribution and Dirichlet Process

2023-05-19,,

Notes on the Dirichlet Distribution and Dirichlet Process

In [3]:

%matplotlib inline

 

Note: I wrote this post in an IPython notebook. It might be rendered better on NBViewer.

Dirichlet Distribution

The symmetric Dirichlet distribution (DD) can be considered a distribution of distributions. Each sample from the DD is acategorial distribution over K categories. It is parameterized G0, a distribution over K categories and α, a scale factor.

The expected value of the DD is G0. The variance of the DD is a function of the scale factor. When α is large, samples from DD(α⋅G0) will be very close to G0. When α is small, samples will vary more widely.

We demonstrate below by setting G0=[.2,.2,.6] and varying α from 0.1 to 1000. In each case, the mean of the samples is roughly G0, but the standard deviation is decreases as α increases.

In [10]:

import numpy as np
from scipy.stats import dirichlet
np.set_printoptions(precision=2) def stats(scale_factor, G0=[.2, .2, .6], N=10000):
samples = dirichlet(alpha = scale_factor * np.array(G0)).rvs(N)
print " alpha:", scale_factor
print " element-wise mean:", samples.mean(axis=0)
print "element-wise standard deviation:", samples.std(axis=0)
print for scale in [0.1, 1, 10, 100, 1000]:
stats(scale)

 

                          alpha: 0.1
element-wise mean: [ 0.2 0.2 0.6]
element-wise standard deviation: [ 0.38 0.38 0.47] alpha: 1
element-wise mean: [ 0.2 0.2 0.6]
element-wise standard deviation: [ 0.28 0.28 0.35] alpha: 10
element-wise mean: [ 0.2 0.2 0.6]
element-wise standard deviation: [ 0.12 0.12 0.15] alpha: 100
element-wise mean: [ 0.2 0.2 0.6]
element-wise standard deviation: [ 0.04 0.04 0.05] alpha: 1000
element-wise mean: [ 0.2 0.2 0.6]
element-wise standard deviation: [ 0.01 0.01 0.02]

 

Dirichlet Process

 

The Dirichlet Process can be considered a way to generalizethe Dirichlet distribution. While the Dirichlet distribution is parameterized by a discrete distribution G0 and generates samples that are similar discrete distributions, the Dirichlet process is parameterized by a generic distribution H0 and generates samples which are distributions similar to H0. The Dirichlet process also has a parameter α that determines how similar how widely samples will vary from H0.

We can construct a sample H (recall that H is a probability distribution) from a Dirichlet process DP(αH0) by drawing a countably infinite number of samples θk from H0) and setting:

H=∑k=1∞πk⋅δ(x−θk)

where the πk are carefully chosen weights (more later) that sum to 1. (δ is the Dirac delta function.)

H, a sample from DP(αH0), is a probability distributionthat looks similar to H0 (also a distribution). In particular, His a discrete distribution that takes the value θk with probability πk. This sampled distribution H is a discrete distribution even if H0 has continuous support; the support ofH is a countably infinite subset of the support H0.

The weights (pk values) of a Dirichlet process sample related the Dirichlet process back to the Dirichlet distribution.

Gregor Heinrich writes:

The defining property of the DP is that its samples have weights πk and locations θk distributed in such a way that when partitioning S(H) into finitely many arbitrary disjoint subsets S1,…,Sj J<∞, the sums of the weights πk in each of these J subsets are distributed according to a Dirichlet distribution that is parameterized by α and a discrete base distribution (likeG0) whose weights are equal to the integrals of the base distribution H0 over the subsets Sn.

As an example, Heinrich imagines a DP with a standard normal base measure H0∼N(0,1). Let H be a sample fromDP(H) and partition the real line (the support of a normal distribution) as S1=(−∞,−1], S2=(−1,1], and S3=(1,∞] then

H(S1),H(S2),H(S3)∼Dir(αerf(−1),α(erf(1)−erf(−1)),α(1−erf(1)))

where H(Sn) be the sum of the πk values whose θk lie in Sn.

These Sn subsets are chosen for convenience, however similar results would hold for any choice of Sn. For any sample from a Dirichlet process, we can construct a sample from a Dirichletdistribution by partitioning the support of the sample into a finite number of bins.

There are several equivalent ways to choose the πk so that this property is satisfied: the Chinese restaurant process, the stick-breaking process, and the Pólya urn scheme.

To generate {πk} according to a stick-breaking process we define βk to be a sample from Beta(1,α). π1 is equal to β1. Successive values are defined recursively as

πk=βk∏j=1k−1(1−βj).

Thus, if we want to draw a sample from a Dirichlet distribution, we could, in theory, sample an infinite number of θk values from the base distribution H0, an infinite number of βk values from the Beta distribution. Of course, sampling an infinite number of values is easier in theory than in practice.

However, by noting that the πk values are positive values summing to 1, we note that, in expectation, they must get increasingly small as k→∞. Thus, we can reasonably approximate a sample H∼DP(αH0) by drawing enoughsamples such that ∑Kk=1πk≈1.

We use this method below to draw approximate samples from several Dirichlet processes with a standard normal (N(0,1)) base distribution but varying α values.

Recall that a single sample from a Dirichlet process is a probability distribution over a countably infinite subset of the support of the base measure.

The blue line is the PDF for a standard normal. The black lines represent the θk and πk values; θk is indicated by the position of the black line on the x-axis; πk is proportional to the height of each line.

We generate enough πk values are generated so their sum is greater than 0.99. When α is small, very few θk's will have corresponding πk values larger than 0.01. However, as αgrows large, the sample becomes a more accurate (though still discrete) approximation of N(0,1).

In [13]:

import matplotlib.pyplot as plt
from scipy.stats import beta, norm def dirichlet_sample_approximation(base_measure, alpha, tol=0.01):
betas = []
pis = []
betas.append(beta(1, alpha).rvs())
pis.append(betas[0])
while sum(pis) < (1.-tol):
s = np.sum([np.log(1 - b) for b in betas])
new_beta = beta(1, alpha).rvs()
betas.append(new_beta)
pis.append(new_beta * np.exp(s))
pis = np.array(pis)
thetas = np.array([base_measure() for _ in pis])
return pis, thetas def plot_normal_dp_approximation(alpha):
plt.figure()
plt.title("Dirichlet Process Sample with N(0,1) Base Measure")
plt.suptitle("alpha: %s" % alpha)
pis, thetas = dirichlet_sample_approximation(lambda: norm().rvs(), alpha)
pis = pis * (norm.pdf(0) / pis.max())
plt.vlines(thetas, 0, pis, )
X = np.linspace(-4,4,100)
plt.plot(X, norm.pdf(X)) plot_normal_dp_approximation(.1)
plot_normal_dp_approximation(1)
plot_normal_dp_approximation(10)
plot_normal_dp_approximation(1000)

 
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" 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" 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" 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" alt="" />

 

Often we want to draw samples from a distribution sampled from a Dirichlet process instead of from the Dirichlet process itself. Much of the literature on the topic unhelpful refers to this as sampling from a Dirichlet process.

Fortunately, we don't have to draw an infinite number of samples from the base distribution and stick breaking process to do this. Instead, we can draw these samples as they are needed.

Suppose, for example, we know a finite number of the θk and πk values for a sample H∼Dir(αH0). For example, we know

π1=0.5,π3=0.3,θ1=0.1,θ2=−0.5.

To sample from H, we can generate a uniform random unumber between 0 and 1. If u is less than 0.5, our sample is 0.1. If 0.5<=u<0.8, our sample is −0.5. If u>=0.8, our sample (from H will be a new sample θ3 from H0. At the same time, we should also sample and store π3. When we draw our next sample, we will again draw u∼Uniform(0,1) but will compare against π1,π2, AND π3.

The class below will take a base distribution H0 and α as arguments to its constructor. The class instance can then be called to generate samples from H∼DP(αH0).

In [20]:

from numpy.random import choice

class DirichletProcessSample():
def __init__(self, base_measure, alpha):
self.base_measure = base_measure
self.alpha = alpha self.cache = []
self.weights = []
self.total_stick_used = 0. def __call__(self):
remaining = 1.0 - self.total_stick_used
i = DirichletProcessSample.roll_die(self.weights + [remaining])
if i is not None and i < len(self.weights) :
return self.cache[i]
else:
stick_piece = beta(1, self.alpha).rvs() * remaining
self.total_stick_used += stick_piece
self.weights.append(stick_piece)
new_value = self.base_measure()
self.cache.append(new_value)
return new_value @staticmethod
def roll_die(weights):
if weights:
return choice(range(len(weights)), p=weights)
else:
return None

 

This Dirichlet process class could be called stochastic memoization. This idea was first articulated in somewhat abstruse terms by Daniel Roy, et al.

Below are histograms of 10000 samples drawn from samplesdrawn from Dirichlet processes with standard normal base distribution and varying α values.

In [22]:

import pandas as pd

base_measure = lambda: norm().rvs()
n_samples = 10000
samples = {}
for alpha in [1, 10, 100, 1000]:
dirichlet_norm = DirichletProcessSample(base_measure=base_measure, alpha=alpha)
samples["Alpha: %s" % alpha] = [dirichlet_norm() for _ in range(n_samples)] _ = pd.DataFrame(samples).hist()

 
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" alt="" />

 

Note that these histograms look very similar to the corresponding plots of sampled distributions above. However, these histograms are plotting points sampled from a distribution sampled from a Dirichlet process while the plots above were showing approximate distributions samples from the Dirichlet process. Of course, as the number of samples from each H grows large, we would expect the histogram to be a very good empirical approximation of H.

In a future post, I will look at how this DirichletProcessSampleclass can be used to draw samples from a hierarchicalDirichlet process.

Notes on the Dirichlet Distribution and Dirichlet Process的相关教程结束。

《Notes on the Dirichlet Distribution and Dirichlet Process.doc》

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