\begin{algorithm}[h]
\caption{Coupled MCMC}
\label{alg:cmcmc}
\begin{enumerate}
\item{Input: data $y_{1:T}$, level $l\in\mathbb{N}$, particle number $N\in\mathbb{N}$, iteration number $M\in\mathbb{N}$ and proposal $q_l$.}
\item{Initialize: Sample $\theta_0^{l}$ from the prior and then run Algorithm \ref{alg:dpf} with parameter $\theta_0^{l}$ to give $(\overline{w}_{0,\Delta_l:T}^l(l),\overline{w}_{0,\Delta_{l-1}:T}^{l-1}(l))$ , denoting the normalizing constant estimate $\hat{\tilde{p}}_{\theta_0^{l}}^N(y_{1:T})$. Set $k=1$.}
\item{Iterate:
Sample $\theta'|\theta_{k-1}^{l}$ from the proposal $q_l(\cdot|\theta_{k-1}^{l})$ and then run Algorithm \ref{alg:dpf} with parameter $\theta'$, denoting the normalizing constant estimate $\hat{\tilde{p}}_{\theta'}^N(y_{1:T})$ and proposed paths $({\overline{w}_{\theta',\Delta_l:T}^l},\overline{w}_{\theta',\Delta_{l-1}:T}^{l-1})$.
Set $\theta_k^{l}=\theta'$, $(\overline{w}_{k,\Delta_l:T}^l(l),\overline{w}_{k,\Delta_{l-1}:T}^{l-1}(l))=({\overline{w}_{\theta',\Delta_l:T}^l},\overline{w}_{\theta',\Delta_{l-1}:T}^{l-1})$ and $\hat{\tilde{p}}_{\theta_k^{l}}^N(y_{1:T})=\hat{\tilde{p}}_{\theta'}^N(y_{1:T})$ with probability
$$
\min\left\{1,\frac{\hat{\tilde{p}}_{\theta'}^N(y_{1:T})\pi(\theta')q_l(\theta_k^{l}|\theta')}{\hat{\tilde{p}}_{\theta_k^{l}}^N(y_{1:T})\pi(\theta_k^{l})q_l(\theta'|\theta_k^{l})}\right\}
$$
otherwise set $\theta_k^{l}=\theta_{k-1}^l$, $(\overline{w}_{k,\Delta_l:T}^l(l),\overline{w}_{k,\Delta_{l-1}:T}^{l-1}(l))=(\overline{w}_{k-1,\Delta_l:T}^l(l),\overline{w}_{k-1,\Delta_{l-1}:T}^{l-1}(l))$ and $\hat{\tilde{p}}_{\theta_k^{l}}^N(y_{1:T})=\hat{\tilde{p}}_{\theta_{k-1}^l}^N(y_{1:T})$. Set $k=k+1$ and if $k=M+1$ go to step 4, otherwise go to the start to step 3.}
\item{Output: $(\theta_{0:M}^{l},\overline{w}_{0:M,\Delta_l:T}^l(l),\overline{w}_{0:M,\Delta_{l-1}:T}^{l-1}(l))$.}
\end{enumerate}
\end{algorithm}

\begin{algorithm}
[h]
\caption{Coupled MCMC}
\begin{enumerate}
\item{Input: data $y_{1:T}$, level $l\in\mathbb{N}$, particle number $N\in\mathbb{N}$, iteration number $M\in\mathbb{N}$ and proposal $q_l$.}
\item{Initialize: Sample $\theta_0^{l}$ from the prior and then run Algorithm \ref{alg:dpf} with parameter $\theta_0^{l}$ to give $(\overline{w}_{0,\Delta_l:T}^l(l),\overline{w}_{0,\Delta_{l-1}:T}^{l-1}(l))$ , denoting the normalizing constant estimate $\hat{\tilde{p}}_{\theta_0^{l}}^N(y_{1:T})$. Set $k=1$.}
\item{Iterate:
Sample $\theta'|\theta_{k-1}^{l}$ from the proposal $q_l(\cdot|\theta_{k-1}^{l})$ and then run Algorithm \ref{alg:dpf} with parameter $\theta'$, denoting the normalizing constant estimate $\hat{\tilde{p}}_{\theta'}^N(y_{1:T})$ and proposed paths $({\overline{w}_{\theta',\Delta_l:T}^l},\overline{w}_{\theta',\Delta_{l-1}:T}^{l-1})$.
Set $\theta_k^{l}=\theta'$, $(\overline{w}_{k,\Delta_l:T}^l(l),\overline{w}_{k,\Delta_{l-1}:T}^{l-1}(l))=({\overline{w}_{\theta',\Delta_l:T}^l},\overline{w}_{\theta',\Delta_{l-1}:T}^{l-1})$ and $\hat{\tilde{p}}_{\theta_k^{l}}^N(y_{1:T})=\hat{\tilde{p}}_{\theta'}^N(y_{1:T})$ with probability
$$
\min\left\{1,\frac{\hat{\tilde{p}}_{\theta'}^N(y_{1:T})\pi(\theta')q_l(\theta_k^{l}|\theta')}{\hat{\tilde{p}}_{\theta_k^{l}}^N(y_{1:T})\pi(\theta_k^{l})q_l(\theta'|\theta_k^{l})}\right\}
$$
otherwise set $\theta_k^{l}=\theta_{k-1}^l$, $(\overline{w}_{k,\Delta_l:T}^l(l),\overline{w}_{k,\Delta_{l-1}:T}^{l-1}(l))=(\overline{w}_{k-1,\Delta_l:T}^l(l),\overline{w}_{k-1,\Delta_{l-1}:T}^{l-1}(l))$ and $\hat{\tilde{p}}_{\theta_k^{l}}^N(y_{1:T})=\hat{\tilde{p}}_{\theta_{k-1}^l}^N(y_{1:T})$. Set $k=k+1$ and if $k=M+1$ go to step 4, otherwise go to the start to step 3.}
\item{Output: $(\theta_{0:M}^{l},\overline{w}_{0:M,\Delta_l:T}^l(l),\overline{w}_{0:M,\Delta_{l-1}:T}^{l-1}(l))$.}
\end{enumerate}
\end{algorithm}
{
    "title": "bayesian parameter inference for partially observed stochastic volterra   equations",
    "id": "2310.03114",
    "abstract": "in this article we consider bayesian parameter inference for a type of partially observed stochastic volterra equation (sve). sves are found in many areas such as physics and mathematical finance. in the latter field they can be used to represent long memory in unobserved volatility processes. in many cases of practical interest, sves must be time-discretized and then parameter inference is based upon the posterior associated to this time-discretized process. based upon recent studies on time-discretization of sves (e.g. richard et al. 2021), we use euler-maruyama methods for the afore-mentioned discretization. we then show how multilevel markov chain monte carlo (mcmc) methods (jasra et al. 2018) can be applied in this context. in the examples we study, we give a proof that shows that the cost to achieve a mean square error (mse) of $\\mathcal{o}(\\epsilon^2)$, $\\epsilon>0$, is {$\\mathcal{o}(\\epsilon^{-\\tfrac{4}{2h+1}})$, where $h$ is the hurst parameter. if one uses a single level mcmc method then the cost is $\\mathcal{o}(\\epsilon^{-\\tfrac{2(2h+3)}{2h+1}})$} to achieve the same mse. we illustrate these results in the context of state-space and stochastic volatility models, with the latter applied to real data.",
    "categories": "stat.co stat.me",
    "doi": "",
    "created": "2023-10-04",
    "updated": "2024-02-19",
    "authors": [
        "ajay jasra",
        "hamza ruzayqat",
        "amin wu"
    ],
    "affiliation": [],
    "url": "https://arxiv.org/abs/2310.03114"
}
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