E.theta.toy: Expectation and variance with respect to theta
In calibrator: Bayesian Calibration of Complex Computer Codes

Description Usage Arguments Note Author(s) References See Also Examples

Function E.theta.toy returns expectation of H_1(D) with respect to theta; Edash.theta.toy returns expectation with respect to Edash. Function E.theta.toy also returns information about nonlinear behaviour of h1(x,theta).

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E.theta.toy(D2=NULL,  H1=NULL, x1=NULL, x2=NULL, phi, give.mean=TRUE)
Edash.theta.toy(x, t.vec, k,  H1, fast.but.opaque=FALSE, a=NULL, b=NULL,
phi=NULL)

`D2`	Observation points
`H1`	Regression function for D1
`phi`	hyperparameters. Default value of `NULL` only to be used in `Edash.theta.toy()` when `fast.but.opaque` is `TRUE`
`x`	lat/long point (for `Edash.theta.toy`)
`t.vec`	Matrix whose rows are parameter values (for `Edash.theta.toy`)
`k`	Integer specifying column (for `Edash.theta.toy`)
`give.mean`	In `E.theta.toy()`, Boolean, with default `TRUE` meaning to return the mean (expectation), and `FALSE` meaning to return the “variance”
`fast.but.opaque`	In `Edash.theta.toy()`, Boolean, with default `FALSE` meaning to use a slow but clear method. If `TRUE`, use faster code but parameters `a` and `b` must then be specified
`a`	Constant term, needed if `fast.but.opaque` is `TRUE`: (V_theta^{-1}+2Omega_t)^{-1}V_theta^{-1}m_theta. Specifying `a` in advance saves execution time
`b`	Linear term, needed if `fast.but.opaque` is `TRUE`: 2(V_theta^{-1}+2Omega_t)^{-1}Omega_t (multiplied by `t[k,]` in `Edash.theta.toy()`).
`x1`	In `E.theta.toy(g=F,...)`, the value of `x` in h_1(x,theta). The default value is `NULL` because in simple cases such as that implemented here, the output is independent of `x1` and `x2`
`x2`	In `E.theta.toy(g=F,...)`, the value of `x` in h_1(x,theta)

A terse discussion follows; see the calex.pdf vignette and the 1D case study in directory inst/doc/one/dim/ for more details and examples.

Function E.theta.toy(give.mean=FALSE,...) does not return the variance! The matrix returned is a different size from the variance matrix!

It returns the thing that must be added to crossprod(E_theta(h1(x,theta)),t(E_theta(h1(x,theta)))) to give E_theta(h1(x,theta).t(h1(x,theta))).

In other words, it returns E_theta(h1(x,theta).t(h1(x,theta)))- crossprod(E_theta(h1(x,theta)),t(E_theta(h1(x,theta)))).

If the terms of h1() are of the form c(o,theta) (where o is a vector that is a function of x alone, and independent of theta), then the function will include the variance matrix, in the lower right corner (zeroes elsewhere).

Function E.theta() must be updated if h1.toy() changes: unlike E.theta() and Edash.theta(), it does not “know” where the elements that vary with theta are, nor their (possibly x-dependent) coefficients.

This form of the function requires x1 and x2 arguments, for good form's sake, even though the returned value is independent of x in the toy example. To see why it is necessary to include x, consider a simple case with h1(x,theta)=(1,x*theta)'. Now E_theta(h(x,theta)) is just (1,x*mean(theta))' but

E_theta(h_1(x,theta)h1(x,theta)^T)

is a 2-by-2 matrix (M, say) with E_theta(M)=h1(x,thetabar)h_1(x,thetabar)^T + variance terms.

ommitted: see dvi

All three functions here are intimately connected to the form of h1.toy() and changing it (or indeed H1.toy()) will usually require rewriting all three functions documented here. Look at the definition of E.theta.toy(give=F), and you will see that even changing the meat of h1.toy() from c(1,x) to c(x,1) would require a redefinition of E.theta.toy(g=F).

The only place that E.theta.toy(g=F) is used is internally in hh.fun().

Robin K. S. Hankin

M. C. Kennedy and A. O'Hagan 2001. Bayesian calibration of computer models. Journal of the Royal Statistical Society B, 63(3) pp425-464
M. C. Kennedy and A. O'Hagan 2001. Supplementary details on Bayesian calibration of computer models, Internal report, University of Sheffield. Available at http://www.tonyohagan.co.uk/academic/ps/calsup.ps
R. K. S. Hankin 2005. Introducing BACCO, an R bundle for Bayesian analysis of computer code output, Journal of Statistical Software, 14(16)

toys

data(toys)
E.theta.toy(D2=D2.toy,      H1=H1.toy,phi=phi.toy)
E.theta.toy(D2=D2.toy[1,],  H1=H1.toy,phi=phi.toy)
E.theta.toy(D2=x.toy,       H1=H1.toy,phi=phi.toy)
Edash.theta.toy(x=x.toy,t.vec=t.vec.toy,k=1, H1=H1.toy,phi=phi.toy)

Loading required package: emulator
Loading required package: mvtnorm
      h1.const   x   y   A B   C
obs.1        1 0.5 0.7 0.8 0 0.6
obs.2        1 0.9 0.3 0.8 0 0.6
obs.3        1 0.3 0.1 0.8 0 0.6
obs.4        1 0.1 0.9 0.8 0 0.6
obs.5        1 0.7 0.5 0.8 0 0.6
       h1.const   x   y   A B   C
code.1        1 0.5 0.7 0.8 0 0.6
       h1.const   x   y   A B   C
code.1        1 0.5 0.6 0.8 0 0.6
       h1.const   x   y          A         B          C
code.1        1 0.5 0.6 0.06559159 0.1862606 0.06428571