In section 1.6 Constrained interval unscented Kalman filter Markov chain Monte Carlo (CIUKF-MCMC) of the Materials and Methods section, there is an error in Theorem 1. Specifically, in several of the equations in the theorem, some of the indices on x and y in the exponents are incorrect. Please find the correct theorem below;

Theorem 1 (Marginal likelihood (Theorem 1 of [26] and 12.1 of [67])) Let y_k denote the set of all observations up to time t_k as defined in Section 1.2. Let the initial condition be uncertain with distribution p (x₀|θ). Then the marginal likelihood is defined recursively in three stages:

for k = 1,2,…

1. Predict the new state from previous data
2. update the prediction with the current data
3. and marginalize out uncertainty in the states