SMC Samplers - Optimal Backward Kernel Explanation

Question

In Sequential Monte Carlo Samplers of Del Moral (2006) we see that the optimal backward kernel is $$ L_{n-1}^{\text{opt}} (x_{n-1} \mid x_n) = \frac{\eta_{n-1}(x_{n-1}) K_n(x_n \mid x_{n-1})}{\eta_n(x_n)} $$ I am very confused by the notation used in the paper. It seems like sometimes $L$ and $K$ are kernels, sometimes they are densities and same for $\eta$ and $\pi$.

• What does the numerator mean? Is it $K_n$ operating on the left on the measure $\eta_{n-1}$ $$ \eta_{n-1} K_n(A) = \int_E \eta_{n-1}(d x_{n-1}) K_n(x_{n-1}, A) \qquad \qquad A\in\mathcal{E} $$ or is it just a multiplication of their densities? $$ \frac{d \eta_{n-1}}{d \lambda} \cdot \frac{d K_n(x_{n-1}, \cdot)}{d\lambda} $$
• What does the denominator mean? I have never seen a ratio of measures like that. Doesn't seem to make any sense? Surely $L_{n-1}: E\times\mathcal{E}\to [0,1]$ is a Markov Kernel whereas both $\eta_{n-1} K_n$ and $\eta_n$ are measures? $$ L_{n-1}^{\text{opt}}(x_{n}, ) = \frac{\eta_{n-1} K_n}{\eta_n} $$

This notation is used thoughout the whole paper and it's very confusing for me!

Answer 1

In the setting where everything has densities, I think that the interpretation is fairly standard, so I will not focus on that case here.

At the measure-theoretic level, the numerator

$$L_{n-1}^{\text{opt}} (x_{n-1} \mid x_n) = \frac{\eta_{n-1}(x_{n-1}) K_n(x_n \mid x_{n-1})}{\eta_n(x_n)}$$

should be interpreted as follows: the measure $\eta_{n-1}(x_{n-1}) K_n(x_n \mid x_{n-1})$ can be given an unambiguous meaning as a probability measure on the space $E_{n-1} \times E_n$ (one could emphasise this by writing it as e.g. $\eta_{n-1}(dx_{n-1}) K_n(dx_n \mid x_{n-1})$). The optimal $L$-kernel is then obtained by taking regular conditional probabilities of this joint measure with respect to $x_n$.