Beta Process

The Beta process is a random measure

Definitions

As a Levy Process Definition

A beta process \(B \sim BP(c, B_0)\) is a positive Levy process whose Levy measure depends on 2 parameters:

1. \(c: \Omega \rightarrow \mathbf{R}\) is called the concentration function; if c is a constant, it is occasionally instead called the concentration parameter
2. \(B_0\) is a fixed measure on \(\Omega\) called the base measure

Properties

Conjugacy with the Bernoulli Process

If necessary, check out the Bernoulli process for a quick refresher. Akin to how the Beta and Bernoulli/Binomial distributions are conjugate, so too are the Beta and Bernoulli processes. Specifically, let \(B \sim BP(c, B_0)\) and \(X_n | B \sim BeP(B)\) for \(n = 1, ..., N\). Then

\[B | X_1, ..., X_N \sim BP(c+N, \frac{c}{c+n} B_0 + \frac{1}{c+n}\sum_{n=1}^N X_n)\]

or equivalently

\[B | X_1, ..., X_N \sim BP(c+N, \frac{c}{c+n} B_0 + \frac{1}{c+n}\sum_{j} m_{Nj} \delta_{\omega_j})\]

where \(m_{Nj}\) is the integer number of \(\{X_n\}_{n=1}^N\) with Dirac measure \(\delta_{\omega_j}\).

Marginalizing Out Beta Process Prior

See Bernoulli process for details.