Dirichlet process

Animation of a Chinese restaurant process with scaling parameter ${\displaystyle \alpha =0.5}$. Tables are hidden once the customers of a table can not be displayed anymore; however, every table has infinitely many seats. (Recording of an interactive animation.[3])

A widely employed metaphor for the Dirichlet process is based on the so-called Chinese restaurant process. The metaphor is as follows:

Imagine a Chinese restaurant in which customers enter. A new customer sits down at a table with a probability proportional to the number of customers already sitting there. Additionally, a customer opens a new table with a probability proportional to the scaling parameter ${\displaystyle \alpha }$. After infinitely many customers entered, one obtains a probability distribution over infinitely many tables to be chosen. This probability distribution over the tables is a random sample of the probabilities of observations drawn from a Dirichlet process with scaling parameter ${\displaystyle \alpha }$.

If one associates draws from the base measure ${\displaystyle H}$ with every table, the resulting distribution over the sample space ${\displaystyle S}$ is a random sample of a Dirichlet process. The Chinese restaurant process is related to the Pólya urn sampling scheme which yields samples from finite Dirichlet distributions.

Because customers sit at a table with a probability proportional to the number of customers already sitting at the table, two properties of the DP can be deduced:

1. The Dirichlet process exhibits a self-reinforcing property: The more often a given value has been sampled in the past, the more likely it is to be sampled again.
2. Even if ${\displaystyle H}$ is a distribution over an uncountable set, there is a nonzero probability that two samples will have exactly the same value because the probability mass will concentrate on a small number of tables.

A third approach to the Dirichlet process is the so-called stick-breaking process view. Conceptually, this involves repeatedly breaking off and discarding a random fraction (sampled from a Beta distribution) of a "stick" that is initially of length 1. Remember that draws from a Dirichlet process are distributions over a set ${\displaystyle S}$. As noted previously, the distribution drawn is discrete with probability 1. In the stick-breaking process view, we explicitly use the discreteness and give the probability mass function of this (random) discrete distribution as:

${\displaystyle f(\theta )=\sum _{k=1}^{\infty }\beta _{k}\cdot \delta _{\theta _{k}}(\theta )}$

where ${\displaystyle \delta _{\theta _{k}}}$ is the indicator function which evaluates to zero everywhere, except for ${\displaystyle \delta _{\theta _{k}}(\theta _{k})=1}$. Since this distribution is random itself, its mass function is parameterized by two sets of random variables: the locations ${\displaystyle \left\{\theta _{k}\right\}_{k=1}^{\infty }}$ and the corresponding probabilities ${\displaystyle \left\{\beta _{k}\right\}_{k=1}^{\infty }}$. In the following, we present without proof what these random variables are.

The locations ${\displaystyle \theta _{k}}$ are independent and identically distributed according to ${\displaystyle H}$, the base distribution of the Dirichlet process. The probabilities ${\displaystyle \beta _{k}}$ are given by a procedure resembling the breaking of a unit-length stick (hence the name):

${\displaystyle \beta _{k}=\beta '_{k}\cdot \prod _{i=1}^{k-1}\left(1-\beta '_{i}\right)}$

where ${\displaystyle \beta '_{k}}$ are independent random variables with the beta distribution ${\displaystyle \operatorname {Beta} (1,\alpha )}$. The resemblance to 'stick-breaking' can be seen by considering ${\displaystyle \beta _{k}}$ as the length of a piece of a stick. We start with a unit-length stick and in each step we break off a portion of the remaining stick according to ${\displaystyle \beta '_{k}}$ and assign this broken-off piece to ${\displaystyle \beta _{k}}$. The formula can be understood by noting that after the first k − 1 values have their portions assigned, the length of the remainder of the stick is ${\displaystyle \prod _{i=1}^{k-1}\left(1-\beta '_{i}\right)}$ and this piece is broken according to ${\displaystyle \beta '_{k}}$ and gets assigned to ${\displaystyle \beta _{k}}$.

The smaller ${\displaystyle \alpha }$ is, the less of the stick will be left for subsequent values (on average), yielding more concentrated distributions.

The stick-breaking process is similar to the construction where one samples sequentially from marginal beta distributions in order to generate a sample from a Dirichlet distribution.[4]

Yet another way to visualize the Dirichlet process and Chinese restaurant process is as a modified Pólya urn scheme sometimes called the Blackwell-MacQueen sampling scheme. Imagine that we start with an urn filled with ${\displaystyle \alpha }$ black balls. Then we proceed as follows:

1. Each time we need an observation, we draw a ball from the urn.
2. If the ball is black, we generate a new (non-black) colour uniformly, label a new ball this colour, drop the new ball into the urn along with the ball we drew, and return the colour we generated.
3. Otherwise, label a new ball with the colour of the ball we drew, drop the new ball into the urn along with the ball we drew, and return the colour we observed.

The resulting distribution over colours is the same as the distribution over tables in the Chinese restaurant process. Furthermore, when we draw a black ball, if rather than generating a new colour, we instead pick a random value from a base distribution ${\displaystyle H}$ and use that value to label the new ball, the resulting distribution over labels will be the same as the distribution over the values in a Dirichlet process.

In this model, the posterior distribution is again a Dirichlet process. This means that the Dirichlet process is a conjugate prior for this model. The posterior distribution is given by

${\displaystyle {\begin{aligned}P|X_{1},\cdots ,X_{n}&\sim {\textrm {DP}}\left({\frac {\alpha }{\alpha +n}}H+{\frac {1}{\alpha +n}}\sum _{i=1}^{n}\delta _{X_{i}},\;\alpha +n\right)\\&={\textrm {DP}}\left({\frac {\alpha }{\alpha +n}}H+{\frac {n}{\alpha +n}}\mathbb {P} _{n},\;\alpha +n\right)\end{aligned}}}$

where ${\displaystyle \mathbb {P} _{n}}$ is defined below.

If we take the frequentist view of probability, we believe there is a true probability distribution ${\displaystyle P_{0}}$ that generated the data. Then it turns out that the Dirichlet process is consistent in the weak topology, which means that for every weak neighbourhood ${\displaystyle U}$ of ${\displaystyle P_{0}}$, the posterior probability of ${\displaystyle U}$ converges to ${\displaystyle 1}$.

In order to interpret the credible sets as confidence sets, a Bernstein–von Mises theorem is needed. In case of the Dirichlet process we compare the posterior distribution with the empirical process ${\displaystyle \mathbb {P} _{n}={\frac {1}{n}}\sum _{i=1}^{n}\delta _{X_{i}}}$. Suppose ${\displaystyle {\mathcal {F}}}$ is a ${\displaystyle P_{0}}$-Donsker class, i.e.

${\displaystyle {\begin{aligned}{\sqrt {(}}n)\left(\mathbb {P} _{n}-P_{0}\right)\rightsquigarrow G_{P_{0}}\end{aligned}}}$

for some Brownian Bridge ${\displaystyle G_{P_{0}}}$. Suppose also that there exists a function ${\displaystyle F}$ such that ${\displaystyle F(x)\geq \sup _{f\in {\mathcal {F}}}f(x)}$ such that ${\displaystyle \int F^{2}\mathrm {d} H<\infty }$, then, ${\displaystyle P_{0}}$ almost surely

${\displaystyle {\sqrt {n}}\left(P-\mathbb {P} _{n}\right)|X_{1},\cdots ,X_{n}\rightsquigarrow G_{P_{0}}.}$

This implies that credible sets you construct are asymptotic confidence sets, and the Bayesian inference based on the Dirichlet process is asymptotically also valid frequentist inference.

For example, we might be interested in how people will vote on a number of questions in an upcoming election. A reasonable model for this situation might be to classify each voter as a liberal, a conservative or a moderate and then model the event that a voter says "Yes" to any particular question as a Bernoulli random variable with the probability dependent on which political cluster they belong to. By looking at how votes were cast in previous years on similar pieces of legislation one could fit a predictive model using a simple clustering algorithm such as k-means. That algorithm, however, requires knowing in advance the number of clusters that generated the data. In many situations, it is not possible to determine this ahead of time, and even when we can reasonably assume a number of clusters we would still like to be able to check this assumption. For example, in the voting example above the division into liberal, conservative and moderate might not be finely tuned enough; attributes such as a religion, class or race could also be critical for modelling voter behaviour, resulting in more clusters in the model.

As another example, we might be interested in modelling the velocities of galaxies using a simple model assuming that the velocities are clustered, for instance by assuming each velocity is distributed according to the normal distribution ${\displaystyle v_{i}\sim N(\mu _{k},\sigma ^{2})}$, where the ${\displaystyle i}$th observation belongs to the ${\displaystyle k}$th cluster of galaxies with common expected velocity. In this case it is far from obvious how to determine a priori how many clusters (of common velocities) there should be and any model for this would be highly suspect and should be checked against the data. By using a Dirichlet process prior for the distribution of cluster means we circumvent the need to explicitly specify ahead of time how many clusters there are, although the concentration parameter still controls it implicitly.

We consider this example in more detail. A first naive model is to presuppose that there are ${\displaystyle K}$ clusters of normally distributed velocities with common known fixed variance ${\displaystyle \sigma ^{2}}$. Denoting the event that the ${\displaystyle i}$th observation is in the ${\displaystyle k}$th cluster as ${\displaystyle z_{i}=k}$ we can write this model as:

${\displaystyle {\begin{aligned}(v_{i}\mid z_{i}=k,\mu _{k})&\sim N(\mu _{k},\sigma ^{2})\\\operatorname {P} (z_{i}=k)&=\pi _{k}\\({\boldsymbol {\pi }}\mid \alpha )&\sim \operatorname {Dir} \left({\frac {\alpha }{K}}\cdot \mathbf {1} _{K}\right)\\\mu _{k}&\sim H(\lambda )\end{aligned}}}$

That is, we assume that the data belongs to ${\displaystyle K}$ distinct clusters with means ${\displaystyle \mu _{k}}$ and that ${\displaystyle \pi _{k}}$ is the (unknown) prior probability of a data point belonging to the ${\displaystyle k}$th cluster. We assume that we have no initial information distinguishing the clusters, which is captured by the symmetric prior ${\displaystyle \operatorname {Dir} \left(\alpha /K\cdot \mathbf {1} _{K}\right)}$. Here ${\displaystyle \operatorname {Dir} }$ denotes the Dirichlet distribution and ${\displaystyle \mathbf {1} _{K}}$ denotes a vector of length ${\displaystyle K}$ where each element is 1. We further assign independent and identical prior distributions ${\displaystyle H(\lambda )}$ to each of the cluster means, where ${\displaystyle H}$ may be any parametric distribution with parameters denoted as ${\displaystyle \lambda }$. The hyper-parameters ${\displaystyle \alpha }$ and ${\displaystyle \lambda }$ are taken to be known fixed constants, chosen to reflect our prior beliefs about the system. To understand the connection to Dirichlet process priors we rewrite this model in an equivalent but more suggestive form:

${\displaystyle {\begin{aligned}(v_{i}\mid {\tilde {\mu }}_{i})&\sim N({\tilde {\mu }}_{i},\sigma ^{2})\\{\tilde {\mu }}_{i}&\sim G=\sum _{k=1}^{K}\pi _{k}\delta _{\mu _{k}}({\tilde {\mu }}_{i})\\({\boldsymbol {\pi }}\mid \alpha )&\sim \operatorname {Dir} \left({\frac {\alpha }{K}}\cdot \mathbf {1} _{K}\right)\\\mu _{k}&\sim H(\lambda )\end{aligned}}}$

Instead of imagining that each data point is first assigned a cluster and then drawn from the distribution associated to that cluster we now think of each observation being associated with parameter ${\displaystyle {\tilde {\mu }}_{i}}$ drawn from some discrete distribution ${\displaystyle G}$ with support on the ${\displaystyle K}$ means. That is, we are now treating the ${\displaystyle {\tilde {\mu }}_{i}}$ as being drawn from the random distribution ${\displaystyle G}$ and our prior information is incorporated into the model by the distribution over distributions ${\displaystyle G}$.

We would now like to extend this model to work without pre-specifying a fixed number of clusters ${\displaystyle K}$. Mathematically, this means we would like to select a random prior distribution ${\displaystyle G({\tilde {\mu }}_{i})=\sum _{k=1}^{\infty }\pi _{k}\delta _{\mu _{k}}({\tilde {\mu }}_{i})}$ where the values of the clusters means ${\displaystyle \mu _{k}}$ are again independently distributed according to ${\displaystyle H\left(\lambda \right)}$ and the distribution over ${\displaystyle \pi _{k}}$ is symmetric over the infinite set of clusters. This is exactly what is accomplished by the model:

${\displaystyle {\begin{aligned}(v_{i}\mid {\tilde {\mu }}_{i})&\sim N({\tilde {\mu }}_{i},\sigma ^{2})\\{\tilde {\mu }}_{i}&\sim G\\G&\sim \operatorname {DP} (H(\lambda ),\alpha )\end{aligned}}}$

With this in hand we can better understand the computational merits of the Dirichlet process. Suppose that we wanted to draw ${\displaystyle n}$ observations from the naive model with exactly ${\displaystyle K}$ clusters. A simple algorithm for doing this would be to draw ${\displaystyle K}$ values of ${\displaystyle \mu _{k}}$ from ${\displaystyle H(\lambda )}$, a distribution ${\displaystyle \pi }$ from ${\displaystyle \operatorname {Dir} \left(\alpha /K\cdot \mathbf {1} _{K}\right)}$ and then for each observation independently sample the cluster ${\displaystyle k}$ with probability ${\displaystyle \pi _{k}}$ and the value of the observation according to ${\displaystyle N\left(\mu _{k},\sigma ^{2}\right)}$. It is easy to see that this algorithm does not work in case where we allow infinite clusters because this would require sampling an infinite dimensional parameter ${\displaystyle {\boldsymbol {\pi }}}$. However, it is still possible to sample observations ${\displaystyle v_{i}}$. One can e.g. use the Chinese restaurant representation described below and calculate the probability for used clusters and a new cluster to be created. This avoids having to explicitly specify ${\displaystyle {\boldsymbol {\pi }}}$. Other solutions are based on a truncation of clusters: A (high) upper bound to the true number of clusters is introduced and cluster numbers higher than the lower bound are treated as one cluster.

Fitting the model described above based on observed data ${\displaystyle D}$ means finding the posterior distribution ${\displaystyle p\left({\boldsymbol {\pi }},{\boldsymbol {\mu }}\mid D\right)}$ over cluster probabilities and their associated means. In the infinite dimensional case it is obviously impossible to write down the posterior explicitly. It is, however, possible to draw samples from this posterior using a modified Gibbs sampler.[6] This is the critical fact that makes the Dirichlet process prior useful for inference.