\Bigg].\end{aligned}$$ From this, we see that can be written as an additive functional of the form $$\begin{aligned}
\sum_{t=1}^T 
\mathbb{E}_{\theta} \left[ 
(\xi_{\theta}(x_t,x_{t-1}))^2 
+ 
\big(( \xi_{\theta}(x_t,x_{t-1}),\alpha_{\theta}(x_{t-1}) \big)^{\dagger} 
\Big| \mathbf{y} 
\right],\end{aligned}$$ which can be estimated using the FL smoother as before. However, for this we need to compute the quantities $\alpha_{\theta}(x_{t-1})$. One option is to make use of a type of fixed-lag approximation for $\alpha_{\theta}(x_{t-1})$, by assuming that $x_s$ and $x_t$ are conditionally independent given $y_{1:\kappa_t}$, whenever $|s-t| > \Delta$. This approach has previously been used by @DoucetJacobRubenthaler2013. Alternatively, we can use a filter approximation according to $$\begin{aligned}
    \widehat{\alpha}_{\theta} \Big( x_{t}\pIdx{i} \Big) 
    = 
    \widehat{\alpha}_{\theta} \Big( x_{t-1}^{a_{t}\pIdx{i}} \Big) + 
    \xi_{\theta} \Big( x_t\pIdx{i},x_{t-1}^{a_{t}\pIdx{i}} \Big),
    \label{eq:alphaEst}\end{aligned}$$ for $i=1, \ldots, N$. Note that this approach suffers from the same particle degeneracy as the APF. However, this only affects a small number of terms and in our experience this approximation works sufficiently well to give estimates with reasonable variance. The resulting estimator using is $$\begin{aligned}
    \widehat{\mathsf{I}}^{(2)}_{T}(\theta|u)
    &= \sum_{t=1}^T \sum_{i=1}^N 
    w_{\kappa_t}\pIdx{i} \eta_{\theta} \Big( \tilde{x}_{\kappa_t,t}\pIdx{i}, \tilde{x}_{\kappa_t,t-1}\pIdx{i} \Big), \label{eq:LouisIdentityEstTerm2Final} \text{ where} \\
\eta_{\theta}(x_t, x_{t-1})
&=
\xi_{\theta}(x_t, x_{t-1})^2 + \big( \xi_{\theta}(x_t, x_{t-1}),\widehat{\alpha}_{\theta}(x_{t-1}) \big)^{\dagger}. \nonumber\end{aligned}$$ Hence, the Hessian can be estimated using by inserting the estimators from , and .

Regularisation of the 