API: Turing.RandomMeasures

ChineseRestaurantProcess(rpm, m)

The Chinese Restaurant Process for random probability measures rpm with counts m.

DirichletProcess(α)

The Dirichlet Process with concentration parameter α. Samples from the Dirichlet process can be constructed via the following representations.

Size-Biased Sampling Process

\[j_k \sim Beta(1, \alpha) * surplus\]

Stick-Breaking Process

\[v_k \sim Beta(1, \alpha)\]

Chinese Restaurant Process

\[p(z_n = k | z_{1:n-1}) \propto \begin{cases} \frac{m_k}{n-1+\alpha}, \text{if} m_k > 0\\ \frac{\alpha}{n-1+\alpha} \end{cases}\]

For more details see: https://www.stats.ox.ac.uk/~teh/research/npbayes/Teh2010a.pdf

PitmanYorProcess(d, θ, t)

The Pitman-Yor Process with discount d, concentration θ and t already drawn atoms. Samples from the Pitman-Yor Process can be constructed via the following representations.

Size-Biased Sampling Process

\[j_k \sim Beta(1-d, \theta + t*d) * surplus\]

Stick-Breaking Process

\[v_k \sim Beta(1-d, \theta + t*d)\]

Chinese Restaurant Process

\[p(z_n = k | z_{1:n-1}) \propto \begin{cases} \frac{m_k - d}{n+\theta}, \text{if} m_k > 0\\ \frac{\theta + d*t}{n+\theta} \end{cases}\]

For more details see: https://en.wikipedia.org/wiki/Pitman–Yor_process

SizeBiasedSamplingProcess(rpm, surplus)

The Size-Biased Sampling Process for random probability measures rpm with a surplus mass of surplus.

StickBreakingProcess(rpm)

The Stick-Breaking Process for random probability measures rpm.

_logpdf_table(d::AbstractRandomProbabilityMeasure, m::AbstractVector{Int})

Parameters:

• d: Random probability measure, e.g. DirichletProcess
• m: Cluster counts

Stick-breaking function.

This function accepts a vector (`v`) of length $K - 1$ where each element
is assumed to be in the unit interval, and returns a simplex of length
$K$.  If the supplied vector `v` is a vector of independent draws from
a Beta distribution (i.e., vⱼ | a ~ Beta(1, a), for j=1,...,K), then
returned simplex is generated via a stick-breaking process where
the first element of the stick is w₁ = v₁, the last element w_K =
∏ⱼ (1 - vⱼ), and the other elements are wₖ = vₖ ∏ⱼ₌₁ᵏ⁻¹(1 - vⱼ).
As $K$ goes to infinity, w is a draw from the Chinese Restaurant process
with mass parameter a.

Arguments

• v: A vector of length $K - 1$, where $K > 1$.

Return

• A simplex (w) of dimension $K$. Where ∑ₖ wₖ = 1, and each wₖ ≥ 0.