CF: Control functionals (CF)

Description Usage Arguments Value Warning On the choice of σ, the kernel and the Stein order Author(s) References See Also

View source: R/kernel_methods.R

Description

This function performs control functionals as described in Oates et al (2017). To choose between different kernels using cross-validation, use CF_crossval.

Usage

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CF(
  integrands,
  samples,
  derivatives,
  steinOrder = NULL,
  kernel_function = NULL,
  sigma = NULL,
  K0 = NULL,
  est_inds = NULL,
  one_in_denom = FALSE,
  diagnostics = FALSE
)

Arguments

integrands

An N by k matrix of integrands (evaluations of the function of interest)

samples

An N by d matrix of samples from the target

derivatives

An N by d matrix of derivatives of the log target with respect to the parameters

steinOrder

(optional) This is the order of the Stein operator. The default is 1 in the control functionals paper (Oates et al, 2017) and 2 in the semi-exact control functionals paper (South et al, 2020). The following values are currently available: 1 for all kernels and 2 for "gaussian", "matern" and "RQ". See below for further details.

kernel_function

(optional) Choose between "gaussian", "matern", "RQ", "product" or "prodsim". See below for further details.

sigma

(optional) The tuning parameters of the specified kernel. This involves a single length-scale parameter in "gaussian" and "RQ", a length-scale and a smoothness parameter in "matern" and two parameters in "product" and "prodsim". See below for further details.

K0

(optional) The kernel matrix. One can specify either this or all of sigma, steinOrder and kernel_function. The former involves pre-computing the kernel matrix using K0_fn and is more efficient when using multiple estimators out of CF, SECF and aSECF or when using the cross-validation functions.

est_inds

(optional) A vector of indices for the estimation-only samples. The default when est_inds is missing or NULL is to perform both estimation of the control variates and evaluation of the integral using all samples. Otherwise, the samples from est_inds are used in estimating the control variates and the remainder are used in evaluating the integral. Splitting the indices in this way can be used to reduce bias from adaption and to make computation feasible for very large sample sizes (small est_inds is faster), but in general in will increase the variance of the estimator.

one_in_denom

(optional) Whether or not to include a 1 + in the denominator of the control functionals estimator, as in equation 2 on p703 of Oates et al (2017). The 1 + in the denominator is an arbitrary choice so we set it to zero by default.

diagnostics

(optional) A flag for whether to return the necessary outputs for plotting or estimating using the fitted model. The default is false since this requires some additional computation when est_inds is NULL.

Value

A list with the following elements:

Warning

Solving the linear system in CF has O(N^3) complexity and is therefore not suited to large N. Using est_inds will instead have an O(N_0^3) cost in solving the linear system and an O((N-N_0)^2) cost in handling the remaining samples, where N_0 is the length of est_inds. This can be much cheaper for large N.

On the choice of σ, the kernel and the Stein order

The kernel in Stein-based kernel methods is L_x L_y k(x,y) where L_x is a first or second order Stein operator in x and k(x,y) is some generic kernel to be specified.

The Stein operators for distribution p(x) are defined as:

Here \nabla_x is the first order derivative wrt x and Δ_x = \nabla_x^T \nabla_x is the Laplacian operator.

The generic kernels which are implemented in this package are listed below. Note that the input parameter sigma defines the kernel parameters σ.

In the above equations, z(x) = ∑_j x[j]^2 and z(x,y) = ∑_j (x[j] - y[j])^2. For the last two kernels, the code only has implementations for steinOrder=1. Each combination of steinOrder and kernel_function above is currently hard-coded but it may be possible to extend this to other kernels in future versions using autodiff. The calculations for the first three kernels above are detailed in South et al (2020).

Author(s)

Leah F. South

References

Oates, C. J., Girolami, M. & Chopin, N. (2017). Control functionals for Monte Carlo integration. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 79(3), 695-718.

South, L. F., Karvonen, T., Nemeth, C., Girolami, M. and Oates, C. J. (2020). Semi-Exact Control Functionals From Sard's Method. https://arxiv.org/abs/2002.00033

See Also

See ZVCV for examples and related functions. See CF_crossval for a function to choose between different kernels for this estimator.


ZVCV documentation built on July 2, 2020, 2:38 a.m.

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