State space models work on two different layers of noise: a noise infused process model and additional measurement noise. A noise infused process model may track the annual population size of salmon, where the noise in this layer may be used to account for un-modelable environmental fluctuations or random perturbations to migratory routes. Subsequently, the population size is observed via noisy measurements, where this may be due to challenges in accurately counting the size of the population of salmon. As a result, estimating parameters through these two layers of noise requires dealing with considerable uncertainty. The widely adopted Integrated Nested Laplace Approximation (INLA) is designed to approximately integrate out some parts of the model, accelerating and simplifying the process of estimating parameters. The INLA approximation lies in the assumption that performing the integral is equivalent to integrating a Gaussian. The alternative to using INLA, and also checking validity of the INLA assumption, typically requires high dimensional and slow but very accurate Monte Carlo integration. This forces the practitioner to chose between the extremes of quick and rough or slow and precise. In this work we devise an INLA diagnostic /alternative model integration approach allowing the user to decide where to stand in a continuous variant of the previously binary speed vs accuracy tradeoff. Additionally, the proposed approach outputs a measure of confidence in the applied approximate integral. The method is based on probabilistic numerics, a new area of research bringing together numerical analysis, applied mathematics, statistics, and computer science. This is joint work with Charlie Zhou (Simon Fraser University) and Oksana Chkrebtii (the Ohio State University).