Bayesian Networks in the Presence of Temporal Information - Bayesian Networks in R: With Applications in Systems Biology - page 71

Initial Model

v w

b d

Phase Update

χ ρ

ζ

CP
Birth

CP
Death

CP
Shift

Edge
Birth

Edge
Death

Parameter
Update

RJ-
MCMC

RJ-
MCMC

RJ-
MCMC

RJ-
MCMC

sample

MH

Markov Chain Update

Fig. 3.4 The ARTIVA algorithm for learning Auto-Regressive TIme VArying networks. The birth,
the death, and the shift of a changepoint (CP) are proposed with probabilities b , d ,and v , respec-
tively. Updating the regression model describing interactions for a gene within a phase is proposed
with probability w . Varying the number of CPs or the number of arcs changes the dimension of
the state-space and requires RJ-MCMC. Proposed shifts in changepoint positions are accepted ac-
cording to a standard Metropolis-Hastings step. The probabilities of choosing each modification
satisfy b + d + v + w = 1and χ + ζ + ρ = 1

Each nonzero value in A h indicates a relationship between the expression levels of
two variables X i and X j and is therefore a good indicator of a putative biological
interaction between those variables. Each of these interactions will be represented
with an arc going from a parent X j at time t 1toatarget X i at time point t ,for

all t in phase h . Note that the regulation coefficients are specific to each temporal
phase. Finally, for each target X i , the vector of CPs delimiting homogeneous phases
is denoted by

−

i 0 i 0 i

1.
In order to learn the auto-regressive time-varying network models, ARTIVA uses
the Reversible Jump Markov chain Monte Carlo (RJ-MCMC) approach from Green
( 1995 ). RJ-MCMC starts with a randomly generated initial model. At each iteration
of the algorithm, a modification of the model is proposed (Fig. 3.4 ) and can be ac-
cepted or rejected with a specific probability, which is computed from the data. The
resulting reversible Markov chain sampler can jump between parameter spaces of
different dimensions and converges to its stationary distribution after a sufficiently
large number of burn-in iterations. After the burn-in, RJ-MCMC provides a good
approximation of the probability of each time-varying network model.

ξ =( ξ ,..., ξ

k + 1 )

,where ξ = 1 + d and ξ

k + 1 =

n +