1. Multivariate Regression

2. Bayesian Multivariate Regression

3. Hedged Prediction Technology Vovk, Gammerman, Shafer(2005) The VC-dimension based error bounds are ridiculously pessimistic in practice- but tight in the ‘distributionindependent’ framework Hedged prediction technology gives individual error estimates (confidences) on predictions that are in practice much better

4. Hedged Prediction Technology Vovk, Gammerman, Shafer(2005) Based on Kolmogorov complexity, make a prediction that makes the current history + prediction as ‘random’ as possible Based on non-conformance measure, predict continuation of (x1,y1), (x2,y2), … (xk,yk), (x(k+1), Y) that makes (x(k+1),Y) as ‘conforming’ as possible.

5. Hedged Prediction Technology Vovk, Gammerman, Shafer(2005) LN 2.9

6. Hedged Prediction Technology Vovk, Gammerman, Shafer(2005)

7. LN 2.9

8. P-values from Lagrange Multipliers Lagrange multipliers measure ‘force’ between point and constraint -- ideal as non-conformance measure 3% wrongly classified-- confidence of classifier is 97% for point outside margin NOTE: Under exchangeability hypothesis

9. P-values from Lagrange Multipliers Lagrange multipliers measure ‘force’ between point and constraint -- ideal as non-conformance measure 7% support vectors: Giving ‘corridor’ as prediction set has confidence 93%. NOTE: Under exchangeability hypothesis, ie not for time series.