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%matplotlib inline

Note: This is best viewed on NBViewer. It is part of a series on Dirichlet Processes and Nonparametric Bayes.

Dirichlet Distribution

The symmetric Dirichlet distribution (DD) can be considered a distribution of distributions. Each sample from the DD is a categorial distribution over $K$ categories. It is parameterized $G_0$, a distribution over $K$ categories and $\alpha$, a scale factor.

The expected value of the DD is $G_0$. The variance of the DD is a function of the scale factor. When $\alpha$ is large, samples from $DD(\alpha\cdot G_0)$ will be very close to $G_0$. When $\alpha$ is small, samples will vary more widely.

We demonstrate below by setting $G_0=[.2, .2, .6]$ and varying $\alpha$ from 0.1 to 1000. In each case, the mean of the samples is roughly $G_0$, but the standard deviation is decreases as $\alpha$ increases.

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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 generalize the Dirichlet distribution. While the Dirichlet distribution is parameterized by a discrete distribution $G_0$ and generates samples that are similar discrete distributions, the Dirichlet process is parameterized by a generic distribution $H_0$ and generates samples which are distributions similar to $H_0$. The Dirichlet process also has a parameter $\alpha$ that determines how similar how widely samples will vary from $H_0$.

We can construct a sample $H$ (recall that $H$ is a probability distribution) from a Dirichlet process $\text{DP}(\alpha H_0)$ by drawing a countably infinite number of samples $\theta_k$ from $H_0$) and setting:

$$H=\sum_{k=1}^\infty \pi_k \cdot\delta(x-\theta_k)$$

where the $\pi_k$ are carefully chosen weights (more later) that sum to 1. ($\delta$ is the Dirac delta function.)

$H$, a sample from $DP(\alpha H_0)$, is a probability distribution that looks similar to $H_0$ (also a distribution). In particular, $H$ is a discrete distribution that takes the value $\theta_k$ with probability $\pi_k$. This sampled distribution $H$ is a discrete distribution _even if $H_0$ has continuous support_; the support of $H$ is a countably infinite subset of the support $H_0$.

The weights ($\pi_k$ 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 $\pi_k$ and locations $\theta_k$ distributed in such a way that when partitioning $S(H)$ into finitely many arbitrary disjoint subsets $S_1, \ldots, S_j$ $J<\infty$, the sums of the weights $\pi_k$ in each of these $J$ subsets are distributed according to a Dirichlet distribution that is parameterized by $\alpha$ and a discrete base distribution (like $G_0$) whose weights are equal to the integrals of the base distribution $H_0$ over the subsets $S_n$.

As an example, Heinrich imagines a DP with a standard normal base measure $H_0\sim \mathcal{N}(0,1)$. Let $H$ be a sample from $DP(H_0)$ and partition the real line (the support of a normal distribution) as $S_1=(-\infty, -1]$, $S_2=(-1, 1]$, and $S_3=(1, \infty]$ then

$$H(S_1),H(S_2), H(S_3) \sim \text{Dir}\left(\alpha\,\text{erf}(-1), \alpha\,(\text{erf}(1) - \text{erf}(-1)), \alpha\,(1-\text{erf}(1))\right)$$

where $H(S_n)$ be the sum of the $\pi_k$ values whose $\theta_k$ lie in $S_n$.

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

There are several equivalent ways to choose the $\pi_k$ so that this property is satisfied: the Chinese restaurant process, the stick-breaking process, and the PĆ³lya urn scheme.

To generate $\left\{\pi_k\right\}$ according to a stick-breaking process we define $\beta_k$ to be a sample from $\text{Beta}(1,\alpha)$. $\pi_1$ is equal to $\beta_1$. Successive values are defined recursively as

$$\pi_k=\beta_k \prod_{j=1}^{k-1}(1-\beta_j).$$

Thus, if we want to draw a sample from a Dirichlet process, we could, in theory, sample an infinite number of $\theta_k$ values from the base distribution $H_0$, an infinite number of $\beta_k$ 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 $\pi_k$ values are positive values summing to 1, we note that, in expectation, they must get increasingly small as $k\rightarrow\infty$. Thus, we can reasonably approximate a sample $H\sim DP(\alpha H_0)$ by drawing enough samples such that $\sum_{k=1}^K \pi_k\approx 1$.

We use this method below to draw approximate samples from several Dirichlet processes with a standard normal ($\mathcal{N}(0,1)$) base distribution but varying $\alpha$ 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 $\theta_k$ and $\pi_k$ values; $\theta_k$ is indicated by the position of the black line on the $x$-axis; $\pi_k$ is proportional to the height of each line.

We generate enough $\pi_k$ values are generated so their sum is greater than 0.99. When $\alpha$ is small, very few $\theta_k$'s will have corresponding $\pi_k$ values larger than $0.01$. However, as $\alpha$ grows large, the sample becomes a more accurate (though still discrete) approximation of $\mathcal{N}(0,1)$.

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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)

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 $\theta_k$ and $\pi_k$ values for a sample $H\sim \text{Dir}(\alpha H_0)$. For example, we know

$$\pi_1=0.5,\; \pi_3=0.3,\; \theta_1=0.1,\; \theta_2=-0.5.$$

To sample from $H$, we can generate a uniform random $u$ number 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 $\theta_3$ from $H_0$. At the same time, we should also sample and store $\pi_3$. When we draw our next sample, we will again draw $u\sim\text{Uniform}(0,1)$ but will compare against $\pi_1, \pi_2$, AND $\pi_3$.

The class below will take a base distribution $H_0$ and $\alpha$ as arguments to its constructor. The class instance can then be called to generate samples from $H\sim \text{DP}(\alpha H_0)$.

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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 samples drawn from Dirichlet processes with standard normal base distribution and varying $\alpha$ values.

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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()

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 DirichletProcessSample class can be used to draw samples from a hierarchical Dirichlet process.

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