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\textbf{Reflection - Reversible jump MCMC - Green}\\
\textbf{<NAME> \hfill \today}

Given a set of parameters (${\theta_i}$ for $i = 1, 2, \hdots n$) that describe an unknown distribution, Gibbs sampling systematically visits subsets of these parameters and updates them based on the \textit{full conditionals}. The \textit{full conditionals} are the conditional distributions of a subset $T \subset \{ 1, 2, \hdots n \}$ of the parameters given all other paramters not in $T$. In other words, we use the conditional distributions $P(\theta_T \mid \theta_{-T})$ where $\theta_{-T}$ denotes all the parameters that are not in T. Note that the paper introduces the concept of `jumping' between subspaces of differing dimensionality, in this case, we are only concerned with one of those subspaces. An example of a full conditional is given in 6.1 where $\theta_i \mid \hdots \sim Beta(\alpha,\beta)$ where $\hdots$ refers to all other parameters associated with that dimension. A (not full) conditional is any other conditional distribution that is not a full conditional (an example can be seen in section 3 where the probability of parameters $\theta^{(1)}$ associated with subspace 1 are conditioned on that subspace being chosen ($P(\theta^{(1)} \mid k = 1)$).

This technique is called `reversible jump' as the sampler needs to evaluate proposals that include parameters that might exist in a different subspace. The sampler must therefore allow `jumps' to the parameters associated with a different subspace (the jump includes a change in parameter dimension). The jumps must be `reversible' to maintain detailed balance between subspaces.

% The posterior we are approximating is $p(k, \theta^{(k)} \ mid y)$
We could not directly use Variational Bayes to solve the problem as it is formulated here. In theory, we could ask for a family of approximating functions $Q$ and select $q^* \in Q$ that minimizes the $KL$ divergence to the joint posterior $P(k, \theta^{(k)} \mid y)$. This is minimization objective would change with different $k$ and $\theta^{(k)}$ (due to changes in dimension of $\theta^{(k)}$) making this technique invalid.

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