for a vector $v$. From this, we can construct an estimator of using the estimator of the gradient in , of the form $$\begin{aligned}
    \widehat{\mathsf{I}}_T(\theta|u) 
    =
    -\DDtheta \log p(\theta)
    + \widehat{\mathsf{S}}_T(\theta|u)^2
    - \widehat{\mathsf{I}}^{(1)}_T(\theta|u)
    - \widehat{\mathsf{I}}^{(2)}_T(\theta|u),
    \label{eq:LouisIdentityEst}\end{aligned}$$ where we introduce $\mathsf{I}^{(1)}_T(\theta)= \mathbb{E}_{\theta} \left[ \DDtheta \log p_{\theta}(\mathbf{x},\mathbf{y}) | \mathbf{y} \right]$ and ${\mathsf{I}^{(2)}_T(\theta)= \mathbb{E}_{\theta} \left[ \Dtheta \log p_{\theta}(\mathbf{x},\mathbf{y})^2| \mathbf{y} \right]}$. We obtain the estimator of the first term analogously to as $$\begin{aligned}
    \widehat{\mathsf{I}}^{(1)}_{T}(\theta|u)
    &= \sum_{t=1}^T \sum_{i=1}^N 
    w_{\kappa_t}\pIdx{i}  \zeta_{\theta} \Big( \tilde{x}_{\kappa_t,t}\pIdx{i}, \tilde{x}_{\kappa_t,t-1}\pIdx{i} \Big), \text{ where} \label{eq:LouisIdentityEstTerm1Final} \\
    \zeta_{\theta}(x_t, x_{t-1})
    &=
    \DDtheta \log f_{\theta}(x_t|x_{t-1}) + \DDtheta \log g_{\theta}(y_t|x_{t}). \nonumber\end{aligned}$$ The estimator of the second term needs a bit more work and we start by rewriting the last term in as $$\begin{aligned}
&\sum_{t=1}^T \sum_{s=1}^T \mathbb{E}_{\theta} \left[ \xi_{\theta}(x_t,x_{t-1}) \xi_{\theta}(x_s,x_{s-1})^{\top} \Big| \mathbf{y} \right] \nonumber \\
&=
\sum_{t=1}^T 
\bigg\{ 
\mathbb{E}_{\theta} \left[ \xi_{\theta}(x_t,x_{t-1})^2 \Big| \mathbf{y} \right] \nonumber \\
&+ \sum_{s=1}^{t-1} 
\mathbb{E}_{\theta} \left[ \big( \xi_{\theta}(x_t,x_{t-1}),\xi_{\theta}(x_s,x_{s-1}) \big)^{\dagger} \Big| \mathbf{y} \right]
\bigg\}, \label{eq:LouisIdentityEstTerm2}\end{aligned}$$ where we have introduced the operator $(a,b)^{\dagger}=ab^{\top}+ba^{\top}$ for brevity. Consider the last term appearing in this expression, we can rewrite it as $$\begin{aligned}
&\sum_{s=1}^{t-1} 
\mathbb{E}_{\theta} \left[ \xi_{\theta}(x_t,x_{t-1}) \xi_{\theta}(x_s,x_{s-1})^{\top} \Big| \mathbf{y} \right] \\
&=
\mathbb{E}_{\theta} \Bigg[ \xi_{\theta}(x_t,x_{t-1}) 
\underbrace{\left\{ \sum_{s=1}^{t-1} 
\mathbb{E}_{\theta} \left[ \xi_{\theta}(x_s,x_{s-1}) 
\big| x_{t-1}, y_{1:t-1} \right] \right\}^{\top}}
_{\triangleq \alpha_{\theta}(x_{t-1})^{\top}}
 \Big| \mathbf{y} 