Jacquier-Johannes-Polson (2007) MCMC Maximum Likelihood for Latent State Models

Summary

Develops a Markov chain Monte Carlo (MCMC) algorithm for computing the frequentist maximum likelihood estimator (MLE) in latent state models without gradient methods. $J$ independent copies of the latent state path are augmented into the Gibbs joint target; as $J$ grows, the marginal for $\theta$ concentrates at the MLE — the same mechanism as simulated annealing, but applicable when $L(\theta)$ is itself an integral over latent variables. The algorithm also provides the optimal smoothed distribution of latent states at the MLE. Demonstrated on a stochastic volatility (SV) model and a multivariate Merton jump-diffusion model.

Key Claims

• Augmented joint density $p_J^m(\theta, \tilde{X}^J) \propto \bigl[\prod_{j=1}^J p(Y|\theta,X^j)p(X^j|\theta)\bigr]\cdot m(\theta)$; marginal $p_J(\theta) \propto L(\theta)^J \cdot m(\theta)/m_J$ concentrates at MLE as $J\to\infty$. $J=1$ with prior $m(\theta)=p(\theta)$ recovers standard Bayesian MCMC.
• Convergence theorem: $\sqrt{J}(\theta^{(g)} - \hat\theta) \to \mathcal{N}(0,\, \ell''(\hat\theta)^{-1})$ as $J\to\infty$. Consequence (a): $\mathrm{Var}(\sqrt{J}\text{ draws})$ estimates the observed Fisher information matrix. Consequence (b): normality of scaled draws (Jarque-Bera / quantile plots) is the convergence diagnostic — does not require knowing $\hat\theta$.
• Dominating measure $m(\theta)$: needed when $L(\theta)^J$ fails to integrate for any $J$ (e.g., jump models with $\sigma_v\to 0$ degeneracy). Often $m(\theta)\propto 1$ suffices for $J\geq 2$.
• SV model ($\log V_t = a + d \cdot \log V_{t-1} + \sigma_v v_t$): $J=20$, $G=25{,}000$, ${\sim}20$ min on SUN workstation. Smoothed volatility path nearly identical to Bayesian smoother; draw variance of $d$ collapses by $\sqrt{J}$ as predicted.
• Multivariate Merton jump-diffusion ($K=3$ assets): $G=5{,}000$, $J=20$; variance of $\lambda$ draws reduces by factor 4.4 (vs. predicted $\sqrt{20}=4.47$); some slight finite-$T$ bias (e.g., mean of $m_Z^1 \approx -2.7$ vs. true $-3.0$).
• CPU time is linear in $J$ and $G$ — no asymptotic overhead compared to Bayesian MCMC at same $J$.

Concepts Introduced or Extended

• MCMC Maximum Likelihood
• Stochastic Volatility
• Gibbs Sampler

Entities Mentioned

• Eric Jacquier
• Michael Johannes
• Nicholas Polson

Quotes

"As we increase $J$, $p_J^m(\theta)$ collapses onto the maximum of $\log L(\theta)$, the finite-sample MLE."

"The algorithm also provides the optimal smoothing distribution of the latent variable... especially important in non-linear or non-normal latent variable models for which the Kalman filter is misspecified."

My Take

A clean and practically important bridge between Bayesian and frequentist MCMC. The $J$-dial unifies both approaches in a single framework, and the convergence theorem provides both standard errors and a diagnostic without extra computation. Main limitation: CPU cost grows linearly in $J$, so large $J$ may be expensive for complex models. The "dominating measure" discussion is model-specific and provides limited general guidance. The applications convincingly demonstrate $J=10\text{--}20$ suffices in practice, making the approach competitive with Bayesian MCMC.