Add support for additional approaches for adaptive tuning of proposal parameters

[matt-graham]

The tuning parameters of the Barker proposal (at a minimum a scalar step size and potentially a diagonal or dense preconditioner) would ideally be adaptively tuned in a initial warm-up stage of the chain using an adaptive Markov chain Monte Carlo approach.

Livingstone and Zanella (2022) suggest using a scheme from Andrieu and Thoms (2008), corresponding to

$$ \begin{aligned} \log(\sigmat) &= \log(\sigma{t-1}) + \gamma_t (\alpha(X_t, Y_t) - \alpha^*),\ \mut &= \mu{t-1} + \gamma_t (Xt - \mu{t-1}) \ \Sigma_t &= \Sigma_t + \gamma_t ((X_t - \mu_t)(X_t - \mut)^T - \Sigma{t-1}) \ \end{aligned} $$

where

• $t$ is the Markov chain sample index,
• $\sigma_t$ is the scalar step size at sample $t$,
• $\mu_t$ is the estimated mean of the target distribution at sample $t$,
• $\Sigma_t$ is the estimated covariance of the target distribution at sample $t$,
• $X_t$ is the chain state at sample $t$,
• $Y_t$ is the proposed move at sample $t$,
• $\alpha(X_t, Y_t)$ is the Metropolis-Hastings acceptance probability at sample $t$,
• $\alpha^*$ is a target acceptance probability,
• $\gamma_t = t^{-\kappa}$ is the learning rate at sample $t$ with $\kappa \in (0.5, 1)$.

Tasks

• [ ] Add dual averaging scheme from Hoffman and Gelman (2014)
• [x] Add robust shape adaptation scheme from Vihola (2012) / ramcmc package

[matt-graham]

In addition to the above adaptation scheme for the step size we may also wish to consider implementing a dual averaging based scheme as described in Algorithm 5 in Hoffman and Gelman (2014) (based on Nesterov (2009)) and implemented in for example Stan.

For the covariance / preconditioner adaptation, we may also want to consider the robust scheme decribed in Vihola (2011) and implemented in the ramcmc package.