Kriging::logMargPostFun

Description

Compute the Log-Marginal Posterior Density of a Kriging Model Object for a given Vector \(\boldsymbol{\theta}\) of Correlation Ranges

Usage

  • Python

    # k = Kriging(...)
    k.logMargPostFun(theta, grad = FALSE)
    
  • R

    # k = Kriging(...)
    k$logMargPostFun(theta, grad = FALSE)
    
  • Matlab/Octave

    % k = Kriging(...)
    k.logMargPostFun(theta, grad = FALSE)
    

Arguments

Argument

Description

theta

Numeric vector of correlation range parameters at which the function is to be evaluated.

grad

Logical. Should the function return the gradient (w.r.t theta)?

Details

The log-marginal posterior density relates to the jointly robust prior \(\pi_{\texttt{JR}}(\boldsymbol{\theta},\, \sigma^2, \, \boldsymbol{\beta}) \propto \pi(\boldsymbol{\theta}) \, \sigma^{-2}\). The marginal (or integrated) posterior is the function \(\boldsymbol{\theta}\) obtained by marginalizing out the GP variance \(\sigma^2\) and the vector \(\boldsymbol{\beta}\) of trend coefficients. Due to the form of the prior, the marginalization can be done on the likelihood \(p_{\texttt{marg}}(\boldsymbol{\theta}\,\vert \,\mathbf{y}) \propto \pi(\boldsymbol{\theta}) \times L_{\texttt{marg}}(\boldsymbol{\theta};\,\mathbf{y})\).

Value

The value of the log-marginal posterior density \(\log p_{\texttt{marg}}(\boldsymbol{\theta} \,|\, \mathbf{y})\). By maximizing this function we should get the estimate of \(\boldsymbol{\theta}\) obtained when using objective = "LMP" in the fit.Kriging method.

Examples

f <- function(x) 1 - 1 / 2 * (sin(12 * x) / (1 + x) + 2 * cos(7 * x) * x^5 + 0.7)
set.seed(123)
X <- as.matrix(runif(10))
y <- f(X)

k <- Kriging(y, X, "matern3_2", objective="LMP")
print(k)

lmp <- function(theta) k$logMargPostFun(theta)$logMargPost

t <- seq(from = 0.01, to = 2, length.out = 101)
plot(t, lmp(t), type = "l")
abline(v = k$theta(), col = "blue")

Results

* data: 10x[0.0455565,0.940467] -> 10x[0.194057,1.00912]
* trend constant (est.): 0.388566
* variance (est.): 0.158896
* covariance:
  * kernel: matern3_2
  * range (est.): 0.313364
  * fit:
    * objective: LMP
    * optim: BFGS