# Dependent Dirichlet Process

The Dependent Dirichlet Process (DDP) is a modified version of the Dirichlet Process
that essentially defines a Markov chain of DPs. The idea is that the paired traits and probabilities of the DP
can change over time: they can be born, move or die.

## Definition

Let \(D \sim DP(\mu)\) where \(\mu: \Omega \rightarrow \mathbb{R}_{+}\) is the base measure and
\(\alpha_{\mu} = \int_{\Omega} d\mu\) is the concentration parameter. We can think of \(D\) as an infinite
sum of traits and probabilities:

\[D = \sum_{k=1}^{\infty} \theta_k \pi_k \subset \Omega \times \mathbb{R}\]
The DDP is a Markov chain of DPs \((D_1, D_2, ...)\) where transitions are governed by 3 stochastic operations:

- Subsampling (death): Define \(q: \Omega \rightarrow [0, 1]\). Then, for each \((\theta, \pi) \in D_{t-1}\), sample
\(b_{\theta} \sim Bernoulli(q(\theta))\); in English, for each trait, flip a coin with probability
\(q(\theta_k)\). Keep only the traits which come up 1 (heads) and renormalize the random probability measure:

\[D_{t}^{subsample} \sim DP(q \mu_{t-1})\]
where

\[(q\mu)(A) = \int_A q(\theta) \mu(\theta)\]
- Transition (move): Define distribution \(T: \Omega \times \Omega \rightarrow \mathbb{R}_+\). For
each \((\theta, \pi)\) in \(D_{t}^{subsample}\), sample \(\theta^{\prime} \sim T(\theta^{\prime} \lvert
\theta)\) and set

\[D_{t}^{transition} \sim DP(T q \mu_{t-1})\]
where

\[(t \mu)( A ) = \int_A \int_{\Omega} T(\theta^{\prime} \lvert \theta) \mu(d\theta)\]
Intuitively, this just says that the locations of the DPs transition according to \(T\).

- Superposition (birth): Sample \(F \sim DP(\nu)\) and sample \((c_D, c_F) \sim
Dir(T q \mu_{t-1}(\Omega), v(Omega))\). Then set \(D_t\) as the union of \((\theta, c_D \pi)\)
for all \(D_t^{transition}\) and \((\theta, c_F \pi)\) for all \((\theta, \pi) \in F\). More succinctly,

\[D_t \sim DP(T q \mu_{t-1} + \nu)\]
After these three steps have been taken, we have the next DP in the Markov chain of DPs
defined as the Dependent Dirichlet Process.

### Low Variance Asymptotics