Parameter Estimation in Sparse Linear-Gaussian State-Space Models via Reversible Jump Markov Chain Monte Carlo

State-space models are ubiquitous for modelling complex systems that evolve over time. In such models, key parameters are usually unknown and must be estimated. In particular, linear systems are parametrised by the transition matrix that encodes the dependencies among state dimensions. Due to physical and computational constraints, it is desirable to estimate this matrix by promoting sparsity in its components, in such a way that the interactions between elements of the state-space are reduced. In this work, we propose a novel Bayesian methodology to estimate model parameters and promote sparsity. The method, called SpaRJ, is based on reversible jump Markov chain Monte Carlo, and allows for the exploration of the space of sparse transition matrices in an efficient manner via the adaptation of the implicit model dimension. The methodology has strong theoretical guarantees and exhibits good performance in numerical examples.