# prob.psi1

0th

Percentile

##### A priori probability of psi1, psi2, and theta

Function to determine the a-priori probability of $\psi_1$ and $\psi_2$ of the hyperparameters, and $\theta$, given the apriori means and standard deviations.

Function sample.theta() samples $\theta$ from its prior distribution.

Keywords
array
##### Usage
prob.psi1(phi,lognormally.distributed=TRUE)
prob.psi2(phi,lognormally.distributed=TRUE)
prob.theta(theta,phi,lognormally.distributed=FALSE)
sample.theta(n=1,phi)
##### Arguments
phi

Hyperparameters

theta

Parameters

lognormally.distributed

Boolean variable with FALSE meaning to assume a Gaussian distribution and TRUE meaning to use a lognormal distribution.

n

In function sample.theta(), the number of observations to take

##### Details

These functions use package mvtnorm to calculate the probability density under the assumption that the PDF is lognormal. One implication would be that phi$psi2.apriori$mean and phi$psi1.apriori$mean are the means of the logarithms of the elements of psi1 and psi2 (which are thus assumed to be positive). The sigma matrix is the covariance matrix of the logarithms as well.

In these functions, interpretation of argument phi depends on the value of Boolean argument lognormally.distributed. Take prob.theta() as an example. If lognormally.distributed is TRUE, then log(theta) is normally distributed with mean phi$theta.aprior$mean and variance phi$theta.apriori$sigma. If FALSE, theta is normally distributed with mean phi$theta.aprior$mean and variance phi$theta.apriori$sigma.

Interpretation of phi$theta.aprior$mean depends on the value of lognormally.distributed: if TRUE it is the expected value of log(theta); if FALSE, it is the expectation of theta.

The reason that prob.theta has a different default value for lognormally.distributed is that some elements of theta might be negative, contraindicating a lognormal distribution

##### References

• 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)

p.eqn4.supp, stage1, p.eqn8.supp

• prob.psi1
• prob.psi2
• prob.theta
• sample.theta
##### Examples
# NOT RUN {
data(toys)
prob.psi1(phi=phi.toy)
prob.psi2(phi=phi.toy)

prob.theta(theta=theta.toy,phi=phi.toy)

sample.theta(n=4,phi=phi.toy)

# }

Documentation reproduced from package calibrator, version 1.2-8, License: GPL-2

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