We propose a generic approach to inference in the non-linear, non-Gaussian state space model. This approach builds on recent developments in precision-based algorithms to estimating general state space models with multivariate observations and states. The baseline algorithm approximates the conditional distribution of the states by a multivariate t density, which is then used for integrated likelihood estimation via importance sampling or for posterior simulation using Markov chain Monte Carlo (MCMC). We build further upon this baseline approach to consider more sophisticated algorithms such as accept-reject Metropolis-Hasting and variational approximation. To illustrate the proposed approach, we estimate the risk of a liquidity trap in the US under a time-varying parameter vector autoregressive (TVP-VAR) model with stochastic volatility.