Pattern Recognition and Machine Learning

C. PROPERTIES OF MATRICES 701 These last two equations can also be written in the form A = ∑M i=1 λiuiuTi (C.45) A−^1 = ∑M i=1 1 ...

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Appendix D. Calculus of Variations We can think of a functiony(x)as being an operator that, for any input valuex, returns an out ...

704 D. CALCULUS OF VARIATIONS Figure D.1 A functional derivative can be defined by considering how the value of a functional F[y ...

D. CALCULUS OF VARIATIONS 705 from which we can read off the functional derivative by comparison with (D.3). Requiring that the ...

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Appendix E Lagrange Multipliers Lagrange multipliers, also sometimes calledundetermined multipliers, are used to find the statio ...

708 E. LAGRANGE MULTIPLIERS Figure E.1 A geometrical picture of the technique of La- grange multipliers in which we seek to maxi ...

E. LAGRANGE MULTIPLIERS 709 Figure E.2 A simple example of the use of Lagrange multipli- ers in which the aim is to maximizef(x ...

710 E. LAGRANGE MULTIPLIERS problem of maximizingf(x)subject tog(x)  0 is obtained by optimizing the Lagrange function (E.4) wi ...

REFERENCES 711 References Abramowitz, M. and I. A. Stegun (1965).Handbook of Mathematical Functions. Dover. Adler, S. L. (1981). ...

712 REFERENCES Uncertainty in Artificial Intelligence: Proceed- ings of the Fifth Conference, pp. 21–30. Morgan Kaufmann. Bach, ...

REFERENCES 713 Bishop, C. M. (1991). A fast procedure for retraining the multilayer perceptron.International Journal of Neural S ...

714 REFERENCES Bishop, C. M. and J. Winn (2000). Non-linear Bayesian image modelling. InProceedings Sixth European Conference on ...

REFERENCES 715 Choudrey, R. A. and S. J. Roberts (2003). Variational mixture of Bayesian independent component an- alyzers.Neura ...

716 REFERENCES Duda, R. O., P. E. Hart, and D. G. Stork (2001).Pat- tern Classification(Second ed.). Wiley. Durbin, R., S. Eddy, ...

REFERENCES 717 Gamerman, D. (1997).Markov Chain Monte Carlo: Stochastic Simulation for Bayesian Inference. Chapman and Hall. Gel ...

718 REFERENCES Hassibi, B. and D. G. Stork (1993). Second order derivatives for network pruning: optimal brain surgeon. In S. J. ...

REFERENCES 719 Jaakkola, T. and M. I. Jordan (2000). Bayesian parameter estimation via variational methods. Statistics and Compu ...

720 REFERENCES Kindermann, R. and J. L. Snell (1980).Markov Ran- dom Fields and Their Applications. American Mathematical Societ ...