SigmoidSmoothing
SigmoidSmoothing(
self,
gammas: Sequence | Tensor | float,
betas: Sequence | Tensor | float = 1.0,
)Uses a sigmoid function in place of an indicator function.
This bridge must be used with a RareEventFunc as the target function.
Parameters
gammas : Sequence | Tensor | float-
A sequence of values, \(\{\gamma_{k}\}_{k=1}^{N}\), which define the sigmoid functions.
betas : Sequence | Tensor | float = 1.0-
A sequence of values, \(\{\beta_{k}\}_{k=1}^{N}\), to use to temper the density of the parameter. If these are not provided, a value of \(\beta_{k}=1\) will be used when defining all intermediate densities.
Notes
This bridge is used in rare event estimation problems to approximate the optimal biasing density, which takes the form \[ \pi^{*}(\theta) \propto \pi(\theta)\mathbb{I}_{\mathcal{F}}(\theta), \qquad \textrm{where } \mathcal{F} := \{\theta : F(\theta) \geq z\}. \] In the above, \(\theta\) denotes a set of parameters with density \(\pi(\cdot)\), \(F(\cdot)\) denotes the system response function, and \(z\) denotes a (scalar–valued) rare event threshold.
The intermediate densities generated using this approach take the form (Cui, Dolgov, and Zahm 2023) \[ \pi_{k}(\theta) \propto (Q_{\sharp}\rho(\theta))^{1-\beta_{k}} \pi(\theta)^{\beta_{k}}g_{\gamma_{k}}(z). \] In the above, \(Q_{\sharp}\rho(\cdot)\) denotes the pushforward of the reference density, \(\rho(\cdot)\), under the preconditioner, \(Q(\cdot)\), and \(g_{\gamma_{k}}(\cdot)\) denotes the sigmoid function, which is defined as \[ g_{\gamma_{k}}(z) := (1 + \exp(\gamma_{k}(F(\theta) - z)))^{-1}. \] The sequences \(\{\beta_{k}\}_{k=1}^{N}\) and \(\{\gamma_{k}\}_{k=1}^{N}\) must satisfy \(0 \leq \gamma_{1} \leq \cdots \leq \gamma_{N}\) and \(0 \leq \beta_{1} \leq \cdots \leq \beta_{N} = 1\).