# SMC Samplers - Optimal Backward Kernel Explanation

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!

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