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MCMC Model Composition Search Strategy in a Hierarchical Model

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### MCMC Model Composition Search Strategy in a Hierarchical Model

by Susan J. Simmons

University of North Carolina Wilmington

Variable selection

- In many hierarchical settings, it is of interest to be able to identify important features or variables
- We propose using an MC3 search strategy to identify these important features
- We denote the response variable by yij where i=1,…,L (number of clusters or groups) and j=1,…,ni (number of observations within group i).

yij ~ N(qi, si2) and qi ~ N(Xi´b,t2)

Where X is P x L matrix of explanatory variables (P=#variables and L=# obsn)

Hyperparameters

- We set prior distributions on each of the parameters:

bj ~ N(0,100) t2 ~ Inv-c2(1)

si2 ~ Inv-c2(1)

Combining the data and prior distributions give us an implicit posterior distribution, but the full conditional posterior distributions have a nice form

where

Samples from Full Posterior distribution

- With nice parametric full conditional distributions, we can use the Gibbs sampler to get posterior samples from joint posterior distribution p(q,s2,t2,b |y).
- Using these samples, we can estimate

The Search (MC3)

- The model selection vector is a binary vector that indicates which features are in the model. The vector is of length PFor example, say there are 5 variables, then one possibility is [1 0 0 1 0].
- (0) The search begins by randomly selecting the first model selection vector, and using this model, calculate P(D|M(1)).
- (1) A position along the model selection vector is randomly chosen, and then switched (0 to 1, or 1 to 0).
- (2) The probability of the data given this model is calculated (P(D|M(2)).
- (3) A transition probability aij is calculated as min (1, P(D|M(j))/ P(D|M(i))).
- (4) This transition probability is the success probability in a randomly chosen Bernoulli trial.
- (5) If the value of the Bernoulli trial is 1, the chain moves to model j and this becomes the new model selection vector. If the value is 0, the old model selection vector is maintained.

Repeat (1) – (5).

More on the search algorithm

- To ensure that the chain does not get stuck in a local minimum, we generated 10 chains (each of length 2000). Total of 20,000 models investigated.
- Once all the chains have completed, we can calculate the activation probability for each feature as
- P(bj ≠0|D) = ∑P(bj ≠0|D,M(k))P(M(k)|D)

Where P(bj ≠0|D,M(k)) = 0 if feature j is not in the model and 1 if feature jis in the model, and P(M(k)|D) is calculated as

Plant QTL experiments

Line 1

Clones

Line 2

There are 165 different lines (or clusters) and in this simulation, ni=10 for i=1,…,165. We generated 60 different simulations scenarios.

Simulations

- The 60 different simulations had effect sizes (1 through 9) and with two different error distributions. (gamma(0.5,1) and gamma(3,1)). In addition, we simulated models between 1 and 6 QTL.
- The model used to simulate the data was yij=m+∑ajxij + eij

where aj is the effect of marker j, m is the overall mean and eij is the error (gamma)

Conclusions

- Results so far are promising.
- Try more complex models.
- Need to make model flexible to deal with p>>n (restricted parameter search).

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