Recall the last time you wanted to understand the distribution of given data. One alternative was to plot a histogram. However, it resulted in frustration due to the choice of the number of bins to use, which led to drastically different outcomes. Another alternative was kernel density estimation. Despite having a similar choice to make, it has the advantage of producing smooth estimates, which are more realistic for continuous quantities with regularities. However, kernel density estimation was unsatisfactory too: it did not aid in understanding the underlying structure of the data and, moreover, provided no means of quantifying the uncertainty associated with the results. In this article, we discuss a Bayesian approach to the estimation of data-generating distributions that addresses the aforementioned concerns.

The approach we shall discuss is based on the family of Dirichlet processes. How specifically such processes are constructed will be described in the next section; here, we focus on the big picture.

A Dirichlet process is a stochastic process, that is, an indexed sequence of random variables. Each realization of this process is a discrete probability distribution, which makes the process a distribution over distributions, similarly to a Dirichlet distribution. The process has only one parameter: a measure $$\nu: \mathcal{B} \to [0, \infty]$$ in a suitable finite measure space $$(\mathcal{X}, \mathcal{B}, \nu)$$ where $$\mathcal{X}$$ is a set, and $$\mathcal{B}$$ is a $$\sigma$$-algebra on $$\mathcal{X}$$. We shall adopt the following notation:

$P \sim \text{Dirichlet Process}(\nu)$

where $$P$$ is a random probability distribution that is distributed according to the Dirichlet process. Note that measure $$\nu$$ does not have to be a probability measure; that is, $$\nu(\mathcal{X}) = 1$$ is not required. To obtain a probability measure, one can divide $$\nu$$ by the total volume $$\lambda = \nu(\mathcal{X})$$:

$P_0(\cdot) = \frac{1}{\lambda} \nu(\cdot).$

Since this normalization is always possible, it is common and convenient to replace $$\nu$$ with $$\lambda P_0$$ and consider the process to be parametrized by two quantities instead of one:

$P \sim \text{Dirichlet Process}(\lambda P_0).$

Parameter $$\lambda$$ is referred to as the concentration parameter of the process.

There are two major alternatives of using the Dirichlet process for estimating distributions: as a direct prior for the data at hand and as a mixing prior. We begin with the former.

Direct prior

Given a data set of $$n$$ observations $$\{ x_i \}_{i = 1}^n$$, a Dirichlet process can be used as a prior:

\begin{align} x_i | P_x & \sim P_x, \text{ for } i = 1, \dots, n; \text{ and} \\ P_x & \sim \text{Dirichlet Process}(\lambda P_0). \tag{1} \end{align}

It is important to realize that the $$x_i$$’s are assumed to be distributed not according to the Dirichlet process but according to a distribution drawn from the Dirichlet process. Parameter $$\lambda$$ allows one to control the strength of the prior: the larger it is, the more shrinkage toward the prior is induced.

Inference

Due to the conjugacy property of the Dirichlet process in the above setting, the posterior is also a Dirichlet process and has the following simple form:

$P_x | \{ x_i \}_{i = 1}^n \sim \text{Dirichlet Process}\left( \lambda P_0 + \sum_{i = 1}^n \delta_{x_i} \right). \tag{2}$

That is, the total volume and normalized measure are updated as follows:

\begin{align} \lambda & := \lambda + n \quad \text{and} \\ P_0 & := \frac{\lambda}{\lambda + n} P_0 + \frac{1}{\lambda + n} \sum_{i = 1}^n \delta_{x_i}. \end{align}

Here, $$\delta_x(\cdot)$$ is the Dirac measure, meaning that $$\delta_x(X) = 1$$ if $$x \in X$$ for any $$X \subseteq \mathcal{X}$$, and otherwise, it is zero. It can be seen in Equation (2) that the base measure has simply been augmented with unit masses placed at the $$n$$ observed data points.

The main question now is, How to draw samples from a Dirichlet process given $$\lambda$$ and $$P_0$$?

As noted earlier, a draw from a Dirichlet process is a discrete probability distribution $$P_x$$. The probability measure of this distribution admits the following representation:

$P_x(\cdot) = \sum_{i = 1}^\infty p_i \delta_{x_i}(\cdot) \tag{3}$

where $$\{ p_i \}$$ is a set of probabilities that sum up to one, and $$\{ x_i \}$$ is a set of points in $$\mathcal{X}$$. Such a draw can be obtained using the so-called stick-breaking construction, which prescribes $$\{ p_i \}$$ and $$\{ x_i \}$$. To begin with, for practical computations, the infinite summation is truncated to retain the only first $$m$$ elements:

$P_x(\cdot) = \sum_{i = 1}^m p_i \delta_{x_i}(\cdot).$

Atoms $$\{ x_i \}_{i = 1}^m$$ are drawn independently from the normalized base measure $$P_0$$. The calculation of probabilities $$\{ p_i \}$$ is more elaborate, and this is where the construction and this article get their name, “stick breaking.” Imagine a stick of unit length, representing the total probability. The procedure is to keep breaking the stick into two parts where, for each iteration, the left part yields $$p_i$$, and the right one, the remainder, is carried over to the next iteration. How much to break off is decided on by drawing $$m$$ independent realizations from a carefully chosen beta distribution:

$q_i \sim \text{Beta}(1, \lambda), \text{ for } i = 1, \dots, m. \tag{4}$

All of them lie in the unit interval and are the proportions to break off of the remainder. When $$\lambda = 1$$, these proportions (of the reminder) are uniformly distributed. When $$\lambda < 1$$, the probability mass is shifted to the right, which means that there are likely to be a small number of large pieces, covering virtually the entire stick. When $$\lambda > 1$$, the probability mass is shifted to the left, which means that there are likely to be a large number of small pieces, struggling to reach the end of the stick.

Formally, the desired probabilities are given by the following expression:

$p_i = q_i \prod_{j = 1}^{i - 1} (1 - q_j), \text{ for } i = 1, \dots, m,$

which, as noted earlier, are the left parts of the remainder of the stick during each iteration. For instance, $$p_1 = q_1$$, $$p_2 = q_2 (1 - q_1)$$, and so on. Due to the truncation, the probabilities $$\{ p_i \}_{i = 1}^m$$ do not sum up to one, and it is common to set $$q_m := 1$$ so that $$p_m$$ takes up the remaining probability mass.

To recapitulate, a single draw from a Dirichlet process is obtained in two steps: prescribe atoms $$\{ x_i \}$$ via draws from the normalized base measure and prescribe the corresponding probabilities $$\{ p_i \}$$ via the stick-breaking construction. The two give a complete description of a discrete probability distribution. Recall that this distribution is still a single draw. By repeating this process many times, one obtains the distribution of this distribution, which can be used to, for instance, quantify uncertainty in the estimation.

Illustration

It is time to demonstrate how the Dirichlet process behaves as a direct prior. To this end, we shall use a data set containing velocities of “82 galaxies from 6 well-separated conic sections of an unfilled survey of the Corona Borealis region.” It was studied in Roeder (1990), which gives us a reference point.

For the curious reader, the source code of this notebook along with auxiliary scripts that are used for performing all the calculations presented below can be found on GitHub.

The empirical cumulative distribution function of the velocity is as follows:

Already here, it is apparent that the distribution is multimodal: there are two distinct regions, one to the left and one to the right, where the curve is flat, meaning there are no observations there. The proverbial histogram gives a confirmation:

It can be seen that there is a handful of galaxies moving relatively slowly and relatively fast compared to the majority located somewhere in the middle around twenty thousand kilometers per second. For completeness, kernel density estimation results in the following plot:

How many clusters of galaxies are there? What are their average velocities? How uncertain are these estimates? Our goal is to answer these questions by virtue of the Dirichlet process.

Now that the intention is to apply the presented theory in practice, we have to make all choices we have conveniently glanced over. Specifically, $$P_0$$ has to be chosen, and we shall use the following:

$P_0(\cdot) = \text{Gaussian}(\, \cdot \, | \mu_0, \sigma_0^2). \tag{5}$

In the above, $$\text{Gaussian}(\cdot)$$ refers to the probability measure of a Gaussian distribution with parameters $$\mu_0$$ and $$\sigma_0$$. In addition to these two, there is one more: $$\lambda$$. We shall set $$\mu_0$$ and $$\sigma_0$$ to 20 and 5, respectively—which correspond roughly to the mean and standard deviation of the data—and present results for different $$\lambda$$’s to investigate how the prior volume affects shrinkage toward the prior.

First, we do not condition on the data to get a better understanding of the prior itself, which corresponds to Equation (1). The following figure shows a single draw from four Dirichlet processes with different $$\lambda$$’s (the gray curves show the cumulative distribution function of the data as a reference):

It can be seen that the larger the prior volume, the smoother the curve. This is because larger $$\lambda$$’s “break” the stick into more pieces, allowing the normalized base measure to be extensively sampled, which, in the limit, converges to this very measure; see Equation (5).

Now, conditioning on the observed velocities of galaxies—that is, sampling as shown in Equation (2)—we obtain the following draws from the posterior Dirichlet distributions with different $$\lambda$$’s:

When the prior volume is small, virtually no data points come from $$P_0$$; instead, they are mostly uniform draws from the observed data set, leading to a curve that is nearly indistinguishable from the one of the data (the top curve). As $$\lambda$$ gets larger, the prior gets stronger, and the estimate gets shrunk toward it, up to a point where the observations appear to be entirely ignored (the bottom curve).

The above model has a serious limitation: it assumes a discrete probability distribution for the data-generating process, which can be seen in the prior and posterior given in Equation (1) and (2), respectively, and it is also apparent in the decomposition given in Equation (3). In some cases, it might be appropriate; however, there is arguably more situations where it is inadequate, including the running example.

Mixing prior

Instead of using a Dirichlet process as a direct prior for the given data, it can be used as a prior for mixing distributions from a given family. The resulting posterior will then naturally inherit the properties of the family, such as continuity. The general structure is as follows:

\begin{align} x_i | \theta_i & \sim P_x \left( \theta_i \right), \text{ for } i = 1, \dots, n; \tag{6} \\ \theta_i | P_\theta & \sim P_\theta, \text{ for } i = 1, \dots, n; \text{ and} \\ P_\theta & \sim \text{Dirichlet Process}(\lambda P_0). \\ \end{align}

The $$i$$th data point, $$x_i$$, is distributed according to distribution $$P_x$$ with parameters $$\theta_i$$. For instance, $$P_x$$ could refer to the Gaussian family with $$\theta_i = (\mu_i, \sigma_i)$$ identifying a particular member of the family by its mean and standard deviation. Parameters $$\{ \theta_i \}_{i = 1}^n$$ are unknown and distributed according to distribution $$P_\theta$$. Distribution $$P_\theta$$ is not known either and gets a Dirichlet process prior with measure $$\lambda P_0$$.

It can be seen in Equation (6) that each data point can potentially have its own unique set of parameters. However, this is not what usually happens in practice. If $$\lambda$$ is reasonably small, the vast majority of the stick—the one we explained how to break in the previous section—tends to be consumed by a small number of pieces. This makes many data points share the same parameters, which is akin to clustering. In fact, clustering is a prominent use case for the Dirichlet process.

Inference

Unlike the previous model, there is no conjugacy in this case, and hence the posterior is not a Dirichlet process. There is, however, a simple Markov chain Monte Carlo sampling strategy based on the stick-breaking construction. It belongs to the class of Gibbs samplers and is as follows.

Similarly to Equation (3), we have the following decomposition:

$P_m(\cdot) = \sum_{i = 1}^\infty p_i P_x(\cdot | \theta_i)$

where $$P_m$$ is the probability measure of the mixture. As before, the infinite decomposition has to be made finite to be usable in practice:

$P_m(\cdot) = \sum_{i = 1}^m p_i P_x(\cdot | \theta_i).$

Here, $$m$$ represents an upper limit on the number of mixture components. Each data point $$x_i$$, for $$i = 1, \dots, n$$, is mapped to one of the $$m$$ components, which we denote by $$k_i \in \{ 1, \dots, m \}$$. In other words, $$k_i$$ takes values from 1 to $$m$$ and gives the index of the component of the $$i$$th observation.

There are $$m + m \times |\theta| + n$$ parameters to be inferred where $$|\theta|$$ denotes the number of parameters of $$P_x$$. These parameters are $$\{ p_i \}_{i = 1}^m$$, $$\{ \theta_i \}_{i = 1}^m$$, and $$\{ k_i \}_{i = 1}^n$$. As usual in Gibbs sampling, the parameters assume arbitrary but compatible initial values. The sampler has the following three steps.

First, given $$\{ p_i \}$$, $$\{ \theta_i \}$$, and $$\{ x_i \}$$, the mapping of the observations to the mixture components, $$\{ k_i \}$$, is updated as follows:

$k_i \sim \text{Categorical}\left( m, \left\{ \frac{p_j P_x(x_i | \theta_j)}{\sum_{l = 1}^m p_l P_x(x_i | \theta_l)} \right\}_{j = 1}^m \right), \text{ for } i = 1, \dots, n.$

That is, $$k_i$$ is a draw from a categorical distribution with $$m$$ categories whose unnormalized probabilities are given by $$p_j P_x(x_i | \theta_j)$$, for $$j = 1, \dots, m$$.

Second, given $$\{ k_i \}$$, the probabilities of the mixture components, $$\{ p_i \}$$, are updated using the stick-breaking construction described earlier. This time, however, the beta distribution for sampling $$\{ q_i \}$$ in Equation (4) is replaced with the following:

$q_i \sim \text{Beta}\left( 1 + n_i, \lambda + \sum_{j = i + 1}^m n_j \right), \text{ for } i = 1, \dots, m,$

where

$n_i = \sum_{j = 1}^n I_{\{i\}}(k_j), \text{ for } i = 1, \dots, m,$

is the number of data points that are currently allocated to component $$i$$. Here, $$I_A$$ is the indicator function of a set $$A$$. As before, in order for the $$p_i$$’s to sum up to one, it is common to set $$q_m := 1$$.

Third, given $$\{ k_i \}$$ and $$\{ x_i \}$$, the parameters of the mixture components, $$\{ \theta_i \}$$, are updated. This is done by sampling from the posterior distribution of each component. In this case, the posterior is a prior of choice that is updated using the data points that are currently allocated to the corresponding component. To streamline this step, a conjugate prior for the data distribution, $$P_x$$, is commonly utilized, which we shall illustrate shortly.

To recapitulate, a single draw from the posterior is obtained in a number of steps where parameters or groups of parameters are updated in turn, while treating the other parameters as known. This Gibbs procedure is very flexible. Other parameters can be inferred too, instead of setting them to fixed values. An important example is the concentration parameter, $$\lambda$$. This parameter controls the formation of clusters, and one might let the data decide what the value should be, in which case a step similar to the third one is added to the procedure to update $$\lambda$$. This will be also illustrated below.

Illustration

We continue working with the galaxy data. For concreteness, consider the following choices:

\begin{align} \theta_i &= (\mu_i, \sigma_i), \text{ for } i = 1, \dots, n; \\ P_x (\theta_i) &= \text{Gaussian}(\mu_i, \sigma_i^2), \text{ for } i = 1, \dots, n; \text{ and} \\ P_0(\cdot) &= \text{Gaussian–Scaled-Inverse-}\chi^2(\, \cdot \, | \mu_0, \kappa_0, \nu_0, \sigma_0^2). \end{align} \tag{7}

In the above, $$\text{Gaussian–Scaled-Inverse-}\chi^2(\cdot)$$ refers to the probability measure of a bivariate distribution composed of a conditional Gaussian and an unconditional scaled inverse chi-squared distribution. Some intuition about this distribution can be built via the following decomposition:

\begin{align} \mu_i | \sigma_i^2 & \sim \text{Gaussian}\left(\mu_0, \frac{\sigma_i^2}{\kappa_0}\right) \text{ and} \\ \sigma_i^2 & \sim \text{Scaled-Inverse-}\chi^2(\nu_0, \sigma_0^2). \end{align} \tag{8}

This prior is a conjugate prior for a Gaussian data distribution with unknown mean and variance, which we assume here. This means that the posterior is also a Gaussian–scaled-inverse-chi-squared distribution. Given a data set with $$n$$ observations $$x_1, \dots, x_n$$, the four parameters of the prior are updated simultaneously (not sequentially) as follows:

\begin{align} \mu_0 & := \frac{\kappa_0}{\kappa_0 + n} \mu_0 + \frac{n}{\kappa_0 + n} \mu_x, \\ \kappa_0 & := \kappa_0 + n, \\ \nu_0 & := \nu_0 + n, \text{ and} \\ \sigma_0^2 & := \frac{1}{\nu_0 + n} \left( \nu_0 \sigma_0^2 + ss_x + \frac{\kappa_0 n}{\kappa_0 + n}(\mu_x - \mu_0)^2 \right) \end{align}

where $$\mu_x = \sum_{i = 1}^n x_i / n$$ and $$ss_x = \sum_{i = 1}^n (x_i - \mu_x)^2$$. It can be seen that $$\kappa_0$$ and $$\nu_0$$ act as counters of the number of observations; $$\mu_0$$ is a weighted sum of two means; and $$\nu_0 \sigma_0^2$$ is a sum of two sums of squares and a third term increasing the uncertainty due to the difference in the means. In the Gibbs sampler, each component (each cluster of galaxies) will have its own posterior based on the data points that are assigned to that component during each iteration of the process. Therefore, $$n$$, $$\mu_x$$, and $$ss_x$$ will generally be different for different components and, moreover, will vary from iteration to iteration.

We set $$\mu_0$$ to 20, which is roughly the mean of the data, and $$\nu_0$$ to 3, which is the smallest integer that allows the scaled chi-squared distribution to have a finite expectation. The choice of $$\kappa_0$$ and $$\sigma_0$$ is more subtle. Recall Equation (8). What we would like from the prior is to allow for free formation of clusters in a region generously covering the support of the data. To this end, the uncertainty in the mean, $$\mu_i$$, has to be high; however, it should not come from $$\sigma_i$$, since it would produce very diffuse clusters. We set $$\kappa_0$$ to 0.01 to magnify the variance of $$\mu_i$$ without affecting $$\sigma_i$$, and $$\sigma_0$$ to 1 to keep clusters compact.

Now, let us take a look at what the above choices entail. The following figure illustrates the prior for the mean of a component:

The negative part is unrealistic for velocity; however, it is rarely a problem in practice. What is important is that there is a generous coverage of the plausible values. The following figure shows the prior for the standard deviation of a component:

The bulk is below the standard deviation of the data; however, this is by choice: we expect more than one cluster of galaxies with similar velocities.

As mentioned earlier, we intend to include $$\lambda$$ in the inference. First, we put the following prior:

$\lambda \sim \text{Gamma}(\alpha_0, \beta_0). \tag{9}$

Note this is the rate parameterization of the Gamma family. Conditionally, this is a conjugate prior with the following update rule for the two parameters:

\begin{align} \alpha_0 & := \alpha_0 + m - 1 \quad \text{and} \\ \beta_0 & := \beta_0 - \sum_{i = 1}^{m - 1} \ln(1 - q_i) \end{align}

where $$\{ q_i \}$$ come from the stick-breaking construction. This is a fourth step in the Gibbs sampler. We set $$\alpha_0$$ and $$\beta_0$$ to 2 and 0.1, respectively, which entails the following prior assumption about $$\lambda$$:

The parameter is allowed to vary freely from small to large values, as desired.

Having chosen all priors and their hyperparameters, we are ready to investigate the behavior of the entire model; see Equations (6), (7), and (9). In what follows, we shall limit the number of mixture components to 25; that is, $$m = 25$$. Furthermore, we shall perform 2000 Gibbs iterations and discard the first half as a warm-up period. As before, we start without conditioning on the data to observe draws from the prior itself. The following figure shows two sample draws:

It can be seen that clusters of galaxies can appear anywhere in the region of interest and can be of various sizes. We conclude that the prior is adequate. When taking the observed velocities into account, we obtain a full posterior distribution in the form of 1000 draws. The following shows two random draws:

Indeed, mixture components have started to appear in the regions where there are observations.

Before we proceed to the final summary of results, it is prudent to inspect sample chains for a few parameters in order to ensure there are not problems with convergence to the stationary distribution. The following shows the number of occupied components among the 25 permitted:

The chain fluctuates around a fixed level without any prominent pattern, as it should. One can plot the actual marginal posterior distribution for the number of components; however, it is already clear that the distribution of the number of clusters of galaxies is mostly between 5 and 10 with a median of 7.

As for the concentration parameter, $$\lambda$$, the chain is as follows:

The behavior is uneventful, which is a good sign.

Let us now take a look at the posterior distributions of the first seven components highlighted earlier (note the different scales on the vertical axes):

The components clearly change roles, which can be seen by the multimodal nature of the distributions. Component 1 is most often at 10 (times $$10^6$$ m/s); however, it also peaks between 24 and 25 and even above 30. Components 2 and 3 are the most certain ones, which is due to a relatively large number of samples present in the corresponding region. They seem to exchanges roles and capture velocities of around 20 and 23. Components 4 and 5, on the other hand, appear to play the same role. Unlike Component 1, they are most likely to be found at around 33. Components 6 and 7 are similar too. They seem to be responsible for the small formation to the left and right next to the bulk in the middle (at 16); recall the histogram of the data. The small formation on the other side of the bulk at around 26 is captured as well, which is mostly done by Component 6.

Lastly, we summarize the inference using the following figure where the median distribution and a 95% uncertainty band—composed of distributions at the 0.025 and 0.975 quantiles—are plotted:

In this view, only five components are visible to the naked eye. The median curve matches well the findings in Roeder (1990). Judging by the width of the uncertainty band, there is a lot of plausible alternatives, and it is important to communicate this uncertainty to those who base decisions on the inference. The ability to quantify uncertainty with such ease is a prominent advantage of Bayesian inference.

Conclusion

In this article, the family of Dirichlet processes has been presented in the context of Bayesian inference. More specifically, it has been shown how a Dirichlet process can be utilized as a prior for an unknown discrete distribution and as a prior for mixing distributions from a given family. In both cases, it has been illustrated how to perform inference via a finite approximation and the stick-breaking construction.

Clearly, the overall procedure is more complicated than counting observations falling in a number of fixed bins, which is what a histogram does, or placing kernels all over the place, which is what a kernel density estimator does. However, “anything in life worth having is worth working for.” The advantages of the Bayesian approach include the ability to incorporate prior knowledge, which is crucial in situations with little data, and the ability to propagate and quantify uncertainty, which is a must.

Recall that the source code of this notebook along with auxiliary scripts that were used for performing the calculations presented above can be found on GitHub. Any feedback is welcome!

Acknowledgments

I would like to thank Mattias Villani for the insightful and informative graduate course in Bayesian statistics titled “Advanced Bayesian learning,” which was the inspiration behind writing this article, and for his guidance regarding the implementation.