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\begin{equation*}
\langle O \rangle_\text{thermal} = \int d\mathbf{R}_0\dots d\mathbf{R}_{M-1} \ 
\underbrace{\frac{1}{Z}\rho(\mathbf{R}_0,\mathbf{R}_1;\tau)
\rho(\mathbf{R}_1,\mathbf{R}_2;\tau) \dots \rho(\mathbf{R}_{M-1},\mathbf{R}_0;\tau)}_\text{sampling probability} \ 
\underbrace{\frac{(\hat{\mathcal{O}} \rho)(\mathbf{R}_0,\mathbf{R}_1;\tau)}{\rho(\mathbf{R}_0,\mathbf{R}_1;\tau)}}_\text{value to average}
\end{equation*}
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