  We consider a problem of inferring contact network from nodal states observed
during an epidemiological process. In a black-box Bayesian optimisation
framework this problem reduces to a discrete likelihood optimisation over the
set of possible networks. The high dimensionality of this set, which grows
quadratically with the number of network nodes, makes this optimisation
computationally challenging. Moreover, the computation of the likelihood of a
network requires estimating probabilities of the observed data to realise
during the evolution of the epidemiological process on this network. A
stochastic simulation algorithm struggles to estimate rare events of observing
the data (corresponding to the ground truth network) during the evolution with
a significantly different network, and hence prevents optimisation of the
likelihood. We replace the stochastic simulation with solving the chemical
master equation for the probabilities of all network states. This equation also
suffers from the curse of dimensionality due to the exponentially large number
of network states. We overcome this by approximating the probability of states
in the tensor-train decomposition. This enables fast and accurate computation
of small probabilities and likelihoods. Numerical simulations demonstrate
efficient black-box Bayesian inference of the network.
