Law of total variance

Summary

In probability theory, the law of total variance or variance decomposition formula or conditional variance formulas or law of iterated variances also known as Eve's law, states that if X and Y are random variables on the same probability space, and the variance of Y is finite, then \operatorname{Var}(Y) = \operatorname{E}[\operatorname{Var}(Y \mid X)] + \operatorname{Var}(\operatorname{E}[Y \mid X]). In language perhaps better known to statisticians than to probability theorists, the two terms are the "unexplained" and the "explained" components of the variance respectively (cf. fraction of variance unexplained, explained variation). In actuarial science, specifically credibility theory, the first component is called the expected value of the process variance (EVPV) and the second is called the variance of the hypothetical means (VHM). These two components are also the source of the term "Eve's law", from the initia

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Weight Initialization for High Order and Multilayer Perceptrons

Proper weight initialization is one of the most important prerequisites for fast convergence of feed-forward neural networks like high order and multilayer perceptrons. In order to determine the optimal value of the initial weight variance (or range), which is a important parameter of random weight initialization methods for high order perceptrons, a wide range of experiments (more than

200,000

simulations) was performed, using seven different data sets, three weight distributions, three activation functions, and several network orders. The results of these experiments are compared to weight initialization techniques for multilayer perceptrons, which leads to the proposal of a suitable weight initialization method for high order perceptrons. Experiments over a large range of initial weight variances are performed (more than

20,000

simulations) for multilayer perceptrons and compared to weight initialization methods proposed by other authors. The results of this comparison are justified by sufficiently small confidence intervals.

1994