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
This function performs semi-exact control functionals as described in South et al (2020).
To choose between different kernels using cross-validation, use SECF_crossval
.
1 2 3 4 5 6 7 8 9 10 11 12 13 |
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 |
polyorder |
(optional) The order of the polynomial to be used in the parametric component, with a default of 1. We recommend keeping this value low (e.g. only 1-2). |
steinOrder |
(optional) This is the order of the Stein operator. The default is |
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 |
est_inds |
(optional) A vector of indices for the estimation-only samples. The default when |
apriori |
(optional) A vector containing the subset of parameter indices to use in the polynomial. Typically this argument would only be used if the dimension of the problem is very large or if prior information about parameter dependencies is known. The default is to use all parameters 1:d where d is the dimension of the target. |
diagnostics |
(optional) A flag for whether to return the necessary outputs for plotting or estimating using the fitted model. The default is |
A list with the following elements:
expectation
: The estimate(s) of the (k) expectation(s).
f_true
: (Only if est_inds
is not NULL
) The integrands for the evaluation set. This should be the same as integrands[setdiff(1:N,est_inds),].
f_hat
: (Only if est_inds
is not NULL
) The fitted values for the integrands in the evaluation set. This can be used to help assess the performance of the Gaussian process model.
a
: (Only if diagnostics
= TRUE
) The value of a as described in South et al (2020), where predictions are of the form f_hat = K0*a + Phi*b for heldout K0 and Phi matrices and estimators using heldout samples are of the form mean(f - f_hat) + b[1].
b
: (Only if diagnostics
= TRUE
) The value of b as described in South et al (2020), where predictions are of the form f_hat = K0*a + Phi*b for heldout K0 and Phi matrices and estimators using heldout samples are of the form mean(f - f_hat) + b[1].
ksd
: (Only if diagnostics
= TRUE
) An estimated kernel Stein discrepancy based on the fitted model that can be used for diagnostic purposes. See South et al (2020) for further details.
bound_const
: (Only if diagnostics
= TRUE
and est_inds
=NULL
) This is such that the absolute error for the estimator should be less than ksd \times bound_const.
Solving the linear system in SECF has O(N^3+Q^3) complexity where N is the sample size and Q is the number of terms in the polynomial.
Standard SECF is therefore not suited to large N. The method aSECF is designed for larger N and details can be found at aSECF
and in South et al (2020).
An alternative would be to use est_inds which has O(N_0^3 + Q^3) complexity in solving the linear system and O((N-N_0)^2) complexity in
handling the remaining samples, where N_0 is the length of est_inds. This can be much cheaper for small N_0 but the estimation of the
Gaussian process model is only done using N_0 samples and the evaluation of the integral only uses N-N_0 samples.
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:
steinOrder=1
: L_x g(x) = \nabla_x^T g(x) + \nabla_x \log p(x)^T g(x) (see e.g. Oates el al (2017))
steinOrder=2
: L_x g(x) = Δ_x g(x) + \nabla_x log p(x)^T \nabla_x g(x) (see e.g. South el al (2020))
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.
"gaussian"
: A Gaussian kernel,
k(x,y) = exp(-z(x,y)/σ^2)
"matern"
: A Matern kernel with σ = (λ,ν),
k(x,y) = bc^{ν}z(x,y)^{ν/2}K_{ν}(c z(x,y)^{0.5})
where b=2^{1-ν}(Γ(ν))^{-1}, c=(2ν)^{0.5}λ^{-1} and K_{ν}(x) is the modified Bessel function of the second kind. Note that λ is the length-scale parameter and ν is the smoothness parameter (which defaults to 2.5 for steinOrder=1 and 4.5 for steinOrder=2).
"RQ"
: A rational quadratic kernel,
k(x,y) = (1+σ^{-2}z(x,y))^{-1}
"product"
: The product kernel that appears in Oates et al (2017) with σ = (a,b)
k(x,y) = (1+a z(x) + a z(y))^{-1} exp(-0.5 b^{-2} z(x,y))
"prodsim"
: A slightly different product kernel with σ = (a,b) (see e.g. https://www.imperial.ac.uk/inference-group/projects/monte-carlo-methods/control-functionals/),
k(x,y) = (1+a z(x))^{-1}(1 + a z(y))^{-1} exp(-0.5 b^{-2} z(x,y))
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).
Leah F. South
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 ZVCV for examples and related functions. See SECF_crossval
for a function to choose between different kernels for this estimator.
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