T =

1
Output Y(s)
??
T
Input (s)
s3
s1
s2
sP
n Number of Input Points (3) T Number of
Outputs at each input, i.e., Replications (5)
2
Special Cases Id like to be able to deal with

• When T 1 (i.e., 1 Replication) and assuming
  normality it should reduce to a GP.
• When n 1 (i.e., 1 input point) it should reduce
  to CDF estimation as with a univariate DP prior.

3
Gelfand, Kottas, and MacEachernJASA vol. 100,
No. 471 (2005)
Output Z(x)
Each color is a different replication
Input (x)
x3
x1
x2
4
Gelfand, Kottas, and MacEachernJASA vol. 100,
No. 471 (2005)
5
How to break the association of outputs across
inputs?

• Could add a (uniform?) prior on Replication
  membership.
• This could in principle be dealt with by
  inserting a Metropolis step within the Gibbs
  sampler .
• Im unsure whether this Gelfand et al. technique
  is helpful for estimating the CDF when there is
  only one input point.

6
Another Idea

• Where CH(f)(F1,,Fn) is a copula, either
• Elliptically contoured (e.g., Gaussian)
• Fairlie-Gumbel-Morgenstern

7
Copulae

• Gaussian
• F-G-M