By Herbert K. H. Lee
Bayesian Nonparametrics through Neural Networks is the 1st publication to target neural networks within the context of nonparametric regression and class, operating in the Bayesian paradigm. Its objective is to demystify neural networks, placing them firmly in a statistical context instead of treating them as a black field. This technique is unlike current books, which are likely to deal with neural networks as a computing device studying set of rules rather than a statistical version. as soon as this underlying statistical version is well-known, different common statistical options will be utilized to enhance the version.
The Bayesian strategy permits larger accounting for uncertainty. This booklet covers uncertainty in version selection and techniques to house this factor, exploring a few rules from records and computer studying. an in depth dialogue at the selection of earlier and new noninformative priors is incorporated, besides a considerable literature evaluate. 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 innovations, the writer makes use of examples through the booklet: one on ozone toxins and the opposite on credits functions. The method validated is suitable for regression and classification-type difficulties and is of curiosity due to the frequent capability functions of the methodologies defined within the publication.
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Extra info for Baysian Nonparametrics via Neural Networks (ASA-SIAM Series on Statistics and Applied Probability)
2. For this reason, we also need to bound the individual Yjh parameters, \Yjh\ < D. It can also be helpful to bound the ftj parameters for reasons of numerical stability during computations. 7. The logistic functions allow values between zero and one so that two columns of Zf Z could be very similar but not identical. This condition is known as multicollinearity in the context of regression. 3. Noninformative Priors 41 estimates. It is computationally desirable to avoid this case, which we can do by requiring the determinant to be larger than some small positive number C rather than merely requiring it to be nonzero.
Maximum likelihood fit for a two-hidden node network. 3. 2. original parameters) will be more diffuse, more closely matching the lack of information we really have about the parameters themselves. This approach lets the data have more influence on the posterior. Let us now take a look at several proposed hierarchical priors. Muller and Rios Insua (1998) proposed a three-stage hierarchical model with a relatively simple structure, although many parameters are multivariate. 2. 4. DAG for the Muller and Rios Insua model.
In many problems, the Jeffreys prior is intuitively reasonable and leads to a proper posterior. However, there are some known situations where the prior seems unreasonable or fails to produce a reasonable or even proper posterior (see, for example, Jeffreys (1961), Berger and Bernardo (1992), Berger, De Oliveira, and Sanso (2001), or Schervish (1995, pp. 122-123)). We will see that we have problems with posterior impropriety with the Jeffreys prior for a neural network. Jeffreys (1961) made arguments that it is often better to treat classes of parameters as independent and compute the priors independently (treating parameters from other classes as fixed during the computation).