By Herbert K. H. Lee
Bayesian Nonparametrics through Neural Networks is the 1st booklet to target neural networks within the context of nonparametric regression and type, operating in the Bayesian paradigm. Its target is to demystify neural networks, placing them firmly in a statistical context instead of treating them as a black field. This strategy is not like latest books, which are likely to deal with neural networks as a laptop studying set of rules rather than a statistical version. as soon as this underlying statistical version is well-known, different normal statistical strategies may be utilized to enhance the version.
The Bayesian technique permits larger accounting for uncertainty. This publication covers uncertainty in version selection and techniques to house this factor, exploring a few principles from records and computing device studying. an in depth dialogue at the selection of past and new noninformative priors is integrated, besides a considerable literature evaluation. Written for statisticians utilizing statistical terminology, Bayesian Nonparametrics through Neural Networks will lead statisticians to an elevated figuring out of the neural community version and its applicability to real-world difficulties.
To illustrate the foremost mathematical ideas, the writer makes use of examples in the course of the e-book: one on ozone toxins and the opposite on credits functions. The technique confirmed is proper for regression and classification-type difficulties and is of curiosity as a result common power functions of the methodologies defined within the ebook.
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Extra info for Baysian Nonparametrics via Neural Networks (ASA-SIAM Series on Statistics and Applied Probability)
Example text
Probability of acceptance now increases smoothly and monotonically with age, resulting in a legal model. This model also fits almost as well, misclassifying only four more cases than the six basis function model. In many cases, there are statistical gains to be made from using a simpler model. In this case, there is also a legal gain. Statistical issues of choosing the number of basis functions will be addressed in Chapter 4. 26 Chapter 2. 6. Predicted probability of loan acceptance from a 6-node network using only the age of the applicant.
The outputs of each layer are taken as the inputs of the next layer. As was proved by several groups, a single layer is all that is necessary to span most spaces of interest, so there is no additional flexibility to be gained by using multiple layers (Cybenko (1989); Funahashi (1989); Hornik, Stinchcombe, and White (1989)). However, sometimes adding layers will give a more compact representation, whereby a complex function can be approximated by a smaller total number of nodes in multiple layers than the number of nodes necessary if only a single layer is used.
Thus, by requiring that |Z'Z| > 0, we can guarantee linear independence. The same logic applies to datasets with more inputs and to networks with more hidden nodes. These sorts of restrictions are now common in the mixture model literature (Diebolt and Robert (1994); Wasserman (2000)). For indicator function basis functions, the requirement on the determinant also prevents impropriety in the posterior. Consider the case of trying to fit an indicator function where the parameter is the threshold.
