MCMC in structure space

MCMC in order space

Current work with Marco Grzegorczyk

• MCMC in structure rather than order space.

• Design new proposal moves that achieve faster mixing and convergence.

First idea

Propose new parents from the distribution:

•Identify those new parents that are involved in the formation of directed cycles.

•Orphan them, and sample new parents for them subject to the acyclicity constraint.

Problem: This move is not reversible

Design a complementary backward move, which proposes “illegal” strucures.

•Select a node X.

•Select a subset of its parents, propose new parents for these parents such that you get directed cycles that involve node X.

•Orphan node X, then select new parents subject to the acylicity constraint.

Move reversible, but maths complicated

Devise a simpler move with similar mixing and convergence

•Identify a pair of nodes X Y

•Orphan both nodes.

•Sample new parents from the Boltzmann distribution subject to the acyclicity constraint such the inverse edge Y X is included.

This move is reversible!

Acceptance probability

• Does the new method avoid the bias intrinsic to order MCMC?

• How do convergence and mixing compare to structure and order MCMC?

• What is the effect on the network reconstruction accuracy?

• Does the new method avoid the bias intrinsic to order MCMC?

• How do convergence and mixing compare to structure and order MCMC?

• What is the effect on the network reconstruction accuracy?

Estimating the bias of the method

• Consider a small network with only five nodes.

• A complete enumeration of structures is possible to compute the correct posterior distribution.

• Compute the difference between the predicted and the true marginal posterior probability, for all edges

• Does the new method avoid the bias intrinsic to order MCMC?

• How do convergence and mixing compare to structure and order MCMC?

• What is the effect on the network reconstruction accuracy?

Alarm network

Devised by Beinlich et al., 1989N=37 nodes

46 directed edges

We generated data sets with m=25,50,100,250,500,750,1000

instances

• Does the new method avoid the bias intrinsic to order MCMC?

• How do convergence and mixing compare to structure and order MCMC?

• What is the effect on the network reconstruction accuracy?

• Does the new method avoid the bias intrinsic to order MCMC?

• How do convergence and mixing compare to structure and order MCMC?

• What is the effect on the network reconstruction accuracy?

• The new method avoids the bias intrinsic to order MCMC.

• Its convergence and mixing are similar to order MCMC; both methods outperform structure MCMC.

• Its network reconstruction accuracy is similar to order MCMC; both methods outperform structure MCMC.

• We expect to get an improvement over order MCMC when using explicit prior knowledge.

Conclusions

Thank you