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Erschienen in:
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2014 | OriginalPaper | Buchkapitel

4. Generalized Linear Models

verfasst von : Jean-Michel Marin, Christian P. Robert

Erschienen in: Bayesian Essentials with R

Verlag: Springer New York

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Abstract

Generalized linear models are extensions of the linear regression model described in the previous chapter. In particular, they avoid the selection of a single transformation of the data that must achieve the possibly conflicting goals of normality and linearity imposed by the linear regression model, which is for instance impossible for binary or count responses. The trick that allows both a feasible processing and an extension of linear regression is first to turn the covariates into a real number by a linear projection and then to transform this value so that it fits the support of the response. We focus here on the Bayesian analysis of probit and logit models for binary data and of log-linear models for contingency tables.

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Vorheriges Kapitel Regression and Variable Selection
Nächstes Kapitel Capture–Recapture Experiments
Fußnoten
1
This upper indexing allows for the distinction between x i , the ith component of the covariate vector, and \({\mathbf{x}}^{i}\), the ith vector of covariates in the sample.
 
2
This algorithm had been used by particle physicists, including Metropolis, since the late 1940s, but, as is often the case, the connection with statistics was not made until much later!
 
3
Guaranteed convergence as in accept–reject algorithms is sometimes achievable with MCMC methods using techniques such as perfect sampling or renewal. But such techniques require a much more advanced study of the target distribution and the transition kernel of the algorithm. These conditions are not met very often in practice (see Robert and Casella 2004, Chap. 13).
 
4
In R, this estimation can be conducted using the acf function.
 
5
We stress that we do not resort to an MH algorithm for the purpose of simulating exactly from the corresponding conditional since this would require an infinite number of iterations but rather that we use a single iteration of the MH algorithm as a substitute for the simulation from the conditional since the resulting MCMC algorithm is still associated with the same stationary distribution.
 
6
The use of the is.matrix test ensures that the function can be computed at one point as well as on multiple points and thus allows for calls from plot and other graphical functions.
 
7
A choice of parameters that depend on the data for the Metropolis–Hastings proposal is completely valid, both from an MCMC point of view (meaning that this is not a self-tuning algorithm) and from a Bayesian point of view (since the parameters of the proposal are not those of the prior).
 
8
We do not include the graphs for the other values of τ, but the curious reader can check that there is indeed a clear difference with the case τ = 1.
 
9
Note that the matrix \({\mathbf{X}}^{\mathsf{T}}\mathbf{X}\) is not the Fisher information matrix outside of the normal model. However, the (genuine) Fisher information matrix usually involves a function of \(\boldsymbol{\beta }\) that prevents its use as a prior (inverse) covariance matrix on \(\boldsymbol{\beta }\).
 
10
The factor 2 in the covariance matrix allows some amount of overdispersion, which is always welcomed in importance sampling settings, if only for variance finiteness purposes.
 
11
The theoretical motivation for setting the number of covariate vectors equal to the dimension of \(\boldsymbol{\beta }\) will be made clear below.
 
12
This technique is called the device of imaginary observations and was proposed by the Italian statistician Bruno de Finetti for prior elicitation.
 
13
Note that the fact that the g j ’s do not take their values in {0, 1} but rather in (0, 1) does not create any difficulty in the implementation of Algorithm 4.7.
 
Literatur
Agresti, A. (1996). An Introduction to Categorical Data Analysis. John Wiley, New York.
Chambers, J., Cleveland, W., Kleiner, B., and Tukey, P. (1983). Graphical Methods for Data Analysis. Chapman and Hall, New York.
Flury, B. and Riedwyl, H. (1988). Multivariate Statistics, A Practical Approach. Cambridge University Press, Cambridge.
Gelman, A., Carlin, J., Stern, H., and Rubin, D. (2013). Bayesian Data Analysis. Chapman and Hall, New York, second edition.
McCullagh, P. and Nelder, J. (1989). Generalized Linear Models. Chapman and Hall, New York.
Robert, C. and Casella, G. (2004). Monte Carlo Statistical Methods. Springer-Verlag, New York, second edition.
Robert, C. and Casella, G. (2009). Introducing Monte Carlo Methods with R. Use R! Springer-Verlag, New York.
Whittaker, J. (1990). Graphical Models in Applied Multivariate Statistics. John Wiley, Chichester.
Metadaten
Titel
Generalized Linear Models
verfasst von
Jean-Michel Marin
Christian P. Robert
Copyright-Jahr
2014
Verlag
Springer New York
Buch
Bayesian Essentials with R

Print ISBN: 978-1-4614-8686-2

Electronic ISBN: 978-1-4614-8687-9

Copyright-Jahr: 2014

DOI
https://doi.org/10.1007/978-1-4614-8687-9_4

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