1. n 3 = 7 Input (s) Output X(s) s1s1 s2s2 s3s3 sPsP m = Number of Input Points (=3) n i = Number of Outputs at Input s i X (i) = Set of Outputs X j (i) at Input s i X (p) = Predicted value at Input s p X (1) X (2) X (3) X (p)

2. Special Cases Id like to include in a model for this data When n i = 1 for all i (i.e., 1 output/input) and assuming normality it should reduce to a GP. When m = 1 (i.e., 1 input point) it should reduce to CDF estimation as with a univariate DP prior.

3. Partially Exchangeable A partially exchangeable sequence consists of a number of subsequences; data are exchangeable within a subsequence but not between subsequences. Exchangeable means that all permutations of the data yield the same likelihood. For our purposes, we structure the data as: Subsequence mSubsequence 2Subsequence 1

4. The de Finetti Theorem A partially exchangeable sequence of random variables with m subsequences has joint CDF: The measure (F 1,…, F m ) is called de Finettis measure. If the subsequence CDFs are independent then:

5. Dependent Subsequences How can we model dependence? Method 1: Latent Variable (models correlations indirectly) Method 2: Copulae (models correlations directly) Other Methods: Multivariate Beta?

6. Dirichlet Distribution Factoids If F follows a Dirichlet Distribution then F(t) has a beta distribution and therefore has moments 1 0 t F0F0 CDF envelope

7. Copulae Gaussian F-G-M Copulae are functions that tie together univariate CDFs in order to build a multivariate CDF with a desired correlation structure.

8. Integrating out the DPs When both the data and the subsequence CDFs are correlated over the input space the overall CDF can take the form: i in 1,…,m, j in 1,…,n

9. Inference (I hope!) Given the data, we want to do inference about the shape of the input distributions and prediction at a new point in the input space (s p ):