in Bayesian parameter inference in nonlinear state space models (SSM) of the form $$\begin{aligned}
    x_{t}|x_{t-1}  \sim  f_{\theta}(x_{t}|x_{t-1}), \quad
    y_{t}|x_t      \sim  g_{\theta}(y_{t}|x_t),
\label{eq:SSM}\end{aligned}$$ where the latent states and the measurements are denoted by $\mathbf{x} = x_{0:T} \triangleq \{x_t\}_{t=0}^T$ and $\mathbf{y} = \measurementsdef$, respectively. Here, $f_{\theta}(\cdot)$ and $g_{\theta}(\cdot)$ denote the transition and observation kernels, respectively, parametrised by the unknown static parameter vector $\theta \in \Theta \subset \mathbb{R}^d$. The initial state is distributed according to some distribution $\mu(x_0)$ which, for notational simplicity, is assumed to be independent of $\theta$.

The aim of Bayesian parameter inference (in SSMs) is to compute the *parameter posterior distribution* $$\begin{aligned}
    p(\theta | \mathbf{y}) = \frac{ p_{\theta}(\mathbf{y}) p(\theta)}{p(\mathbf{y})},
    \label{eq:parameterPosterior}\end{aligned}$$ where $p(\theta)$ denotes the prior of $\theta$ and $p_{\theta}(\mathbf{y})$ denotes the likelihood, which for an SSM can be expressed as $$\begin{aligned}
    p_{\theta}(\mathbf{y}) = p_{\theta}(y_1) \prod_{t=2}^T p_{\theta}(y_t|y_{1:t-1}).
    \label{eq:likeFunc}\end{aligned}$$ The one-step ahead predictor $p_{\theta}(y_t|y_{1:t-1})$, and thus also the likelihood function, is in general not analytically tractable. However, unbiased estimators of the likelihood can be constructed using sequential Monte Carlo (SMC) [@DoucetJohansen2011; @DelMoral2004] and these can be used as *plug-in estimators*. This is especially useful in the Metropolis-Hastings (MH) algorithm that can be used for estimating the parameter posterior in .

This combination 