Tempering

Tempering(
    betas: Tensor | None = None,
    ess_tol: Tensor | float = 0.5,
    ess_tol_init: Tensor | float = 0.5,
    beta_factor: Tensor | float = 1.05,
    min_beta: Tensor | float = 0.0001,
    max_layers: int = 20,
)

Likelihood tempering.

The intermediate densities, \(\{\pi_{k}(\theta)\}_{k=1}^{N}\), generated using this approach take the form \[\pi_{k}(\theta) \propto (Q_{\sharp}\rho(\theta))^{1-\beta_{k}}\pi(\theta)^{\beta_{k}},\] where \(Q_{\sharp}\rho(\cdot)\) denotes the pushforward of the reference density, \(\rho(\cdot)\), under the preconditioner, \(Q(\cdot)\), \(\pi(\cdot)\) denotes the target density, and \(0 \leq \beta_{1} < \cdots < \beta_{N} = 1\).

It is possible to provide this class with a set of \(\beta\) values to use. If these are not provided, they will be determined automatically by finding the largest possible \(\beta\), at each iteration, such that the ESS of a reweighted set of samples distributed according to (a TT approximation to) the previous bridging density does not fall below a given value.

Parameters

betas : Tensor | None = None: A set of \(\beta\) values to use for the intermediate distributions. If not specified, these will be determined automatically.
ess_tol : Tensor | float = 0.5: If selecting the \(\beta\) values adaptively, the minimum allowable ESS of the samples (distributed according to an approximation of the previous bridging density) when selecting the next bridging density.
ess_tol_init : Tensor | float = 0.5: If selecting the \(\beta\) values adaptively, the minimum allowable ESS of the samples when selecting the initial bridging density.
beta_factor : Tensor | float = 1.05: If selecting the \(\beta\) values adaptively, the factor by which to increase the current \(\beta\) value by prior to checking whether the ESS of the reweighted samples is sufficiently high.
min_beta : Tensor | float = 0.0001: If selecting the \(\beta\) values adaptively, the minimum allowable \(\beta\) value.
max_layers : int = 20: If selecting the \(\beta\) values adaptively, the maximum number of layers to construct. Note that, if the maximum number of layers is reached, the final bridging density may not be the target density.