# Mathematical Perspectives on Neural Networks

By Paul Smolensky; Michael C. Mozer et al. | Go to book overview

10 Statistical Analysis of Neural Networks

L. F. Abbott Brandeis University

INTRODUCTION

Artificial neural networks are large, complex dynamic systems often constructed to perform a function of practical value. The behavior of a network is determined by rules prescribing how each unit evolves with time under the influence of other network elements and external inputs. On the basis of these rules, computer simulations can be used to determine how well a network performs a desired task. If we want to develop an analytic approach to evaluating and understanding network performance, however, a detailed microscopic description involving every network element may be too complex to be of much value. Instead, we need a macroscopic description using a small number of variables that are more directly related to the task being evaluated. By analogy, a person designing a system to handle gases does not want to deal with individual molecular positions and collisions, but rather needs to know about density and pressure. The statistical approach discussed here is a method for deriving, from the microscopic rules governing a network, a macroscopic description that is both simpler and more directly applicable to the evaluation of network function.

The statistical approach to the study of neural networks involves averaging over individual elements to develop a more compact description of network activity. This averaging allows us to characterize the network with a small number of macroscopic variables instead of using a separate variable for each network element. If we wish to describe how the state of the network changes with time, we also need to recast the dynamic equations of the network in terms of these same macroscopic variables. Because it involves averaging, the resulting description of the network is statistical in nature, that is, probabilistic. In the case of large networks, however,

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