## Generalized Mellin Transform for the Analysis of Learning MachinesDepartment of Electrical Engineering, Stanford University., 1965 - Artificial intelligence - 134 pages This work introduces a generalization of the Mellin integral transform, and then applies it as a new technique for the analysis of statistical learning machines. Specifically, this transform technique is demonstrated for Bayes' estimation and binary detection of communication signals in the framework of statistical decision theory. It is shown that the use of the Mellin transform markedly simplifies the analysis of sequential estimation of an unknown parameter in a statistical population, whether or not an optimum (admissible) decision rule is used for processing the observations. The loss function, which need not be error-squared, or the likelihood ratio is treated as a random variable, and is described by a transform expression derived from the basic Bayes' learning or estimation equation. This transform expression for the loss is the Mellin transform pair to the probability density of the loss, and contains, as unspecified parameters, the number of observations, the a priori assumptions about the unknown parameter that is to be estimated, and the decision (data processing) rule. The transform expression for the loss is shown to be a convenient and appealing measure of the performance of the learning machine. Convergence, bias, and the rate of convergence of the estimation procedure are readily displayed through the transform associated with the loss function. The average cost of errors for a binary detector, previously adapted through the use of sequential estimation, is derived through the convolution integral and Parseval's relationships for the Mellin transform. (Author). |

### What people are saying - Write a review

We haven't found any reviews in the usual places.

### Contents

lNTRODUCTlON | 1 |

ANALYSlS OF LEARNlNG MACHlNES | 10 |

GENERALlZATlON OF THE MELLlN lNTEGRAL TRANSFORM | 18 |

5 other sections not shown

### Common terms and phrases

adaptive machines adaptive period adaptive procedure adaptive system amplitude analysis of learning Appendix arbitrary Attn Bayes Calif characteristic function CLs|e,zn code words cost of detector CRef denotes detection detector errors discussed distribution function encoding error-squared loss estimation problem estimation procedure estimation rule estimation theory example func functional operation gamma Gaussian noise Gaussian population given hypergeometric functions integral transform introduced inverse Kotlarski learning machine problems likelihood ratio lognormal loss function Mellin Mellin transform minimizes observations obtained optimum rule output P.O. Box posteriori density priori assumptions prob probability density function probability distribution processing rule random variable rate of convergence Re{s region of convergence rule for processing sequential estimation Stanford Elect statistical decision theory statistical learning machines statistical population statistically independent Tech test signals theorem tion trans transform expression transform method transform pairs transform technique transmission unknown parameter variance Wright-Patterson AFB zero