08 August 2022
Given a linear Gaussian state space model (LGSSM), we have previously derived the optimal proposals for filtering SMC, \(p(z_1 \given x_1)\) and \(\{p(z_t \given z_{t - 1}, x_t)\}_{t = 2}^T\). For smoothing, we need the optimal SMC proposals to be conditioned on the future: \(p(z_1 \given x_{1:T})\) and \(\{p(z_t \given z_{t - 1}, x_{1:T})\}_{t = 2}^T\). How can we compute these proposals?
The first proposal is easy since it’s just the marginal smoothing distribution which can be computed like this.
For \(p(z_t \given z_{t - 1}, x_{1:T})\), we reuse \(p(z_{t - 1} \given z_t, x_{1:T})\) from the derivation of Kalman smoothing in equation 19 of this note: \begin{align} p(z_t \given z_{t - 1}, x_{1:T}) = \frac{p(z_{t - 1} \given z_t, x_{1:T}) p(z_t \given x_{1:T})}{p(z_{t - 1} \given x_{1:T})}. \end{align} Since \(p(z_t \given x_{1:T})\) is Gaussian, obtained from smoothing, and \(p(z_{t - 1} \given z_t, x_{1:T})\) is Gaussian whose mean is a linear function of \(z_t\), we can compute left hand side as a conjugate posterior for the “prior” \(p(z_t \given x_{1:T})\) and “likelihood” \(p(z_{t - 1} \given z_t, x_{1:T})\).
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