Figure - PMC

Skip to main content

View full-text article in PMC

. Author manuscript; available in PMC: 2019 Mar 1.

Published in final edited form as: IEEE/ACM Trans Comput Biol Bioinform. 2018 Mar-Apr;15(2):537–550. doi: 10.1109/TCBB.2015.2440244

Fig. 2 — Illustration of sequential resolution sampling. (a) Initially, we utilize the standard MCMC algorithm to produce the posterior distribution of the selection indicators in at resolution 1, i.e. P₁(· | S, X, y), which is then used to guide the construction of the proposal function in the SRS-MCMC algorithm to produce P₂(· | S, X, y) for at resolution 2. This procedure is performed sequentially until resolution K to generate the posterior distribution P_K(· | S, X, y) for our target model . (b) Decomposition of the proposal function T(· → · | •) (red) includes two steps for drawing a proposed sample. Step 1 (green): draw { ${\tilde{c}}_{*}^{(k - 1)}, {\tilde{γ}}_{*}^{(k - 1)}$ } from the posterior distribution P_k₋₁(· | ·) under the model at resolution k−1. Step 2 (blue): sample { $c_{*}^{(k - 1)}, γ_{*}^{(k - 1)}$ } given { ${\tilde{c}}_{*}^{(k - 1)}, {\tilde{γ}}_{*}^{(k - 1)}$ } in step 1 and the current state of the Markov chain using H(· | ·). (c) A binary tree represents the sampling scheme for $c_{g, *}^{(k)}$ based on the probability mass function h(· | ·). It is determined by $c_{g^{'}, *}^{(k)}$ and ${\tilde{c}}_{g^{'}, o}^{(k)}$ for g′ satisfying ${\tilde{b}}_{g g^{'}}^{(k)} = 1$ , and $c_{g, o}^{(k)}$ .

Inline graphic — Illustration of sequential resolution sampling. (a) Initially, we utilize the standard MCMC algorithm to produce the posterior distribution of the selection indicators in at resolution 1, i.e. P₁(· | S, X, y), which is then used to guide the construction of the proposal function in the SRS-MCMC algorithm to produce P₂(· | S, X, y) for at resolution 2. This procedure is performed sequentially until resolution K to generate the posterior distribution P_K(· | S, X, y) for our target model . (b) Decomposition of the proposal function T(· → · | •) (red) includes two steps for drawing a proposed sample. Step 1 (green): draw { ${\tilde{c}}_{*}^{(k - 1)}, {\tilde{γ}}_{*}^{(k - 1)}$ } from the posterior distribution P_k₋₁(· | ·) under the model at resolution k−1. Step 2 (blue): sample { $c_{*}^{(k - 1)}, γ_{*}^{(k - 1)}$ } given { ${\tilde{c}}_{*}^{(k - 1)}, {\tilde{γ}}_{*}^{(k - 1)}$ } in step 1 and the current state of the Markov chain using H(· | ·). (c) A binary tree represents the sampling scheme for $c_{g, *}^{(k)}$ based on the probability mass function h(· | ·). It is determined by $c_{g^{'}, *}^{(k)}$ and ${\tilde{c}}_{g^{'}, o}^{(k)}$ for g′ satisfying ${\tilde{b}}_{g g^{'}}^{(k)} = 1$ , and $c_{g, o}^{(k)}$ .