Introduction to Bayesian Scientific Computing

by Calvetti, Daniela; Somersalo, Erkki

• ISBN13: 9780387733937
• ISBN10: 0387733930
• Format: Paperback
• Copyright: 2007-12-01
• Publisher: Springer Verlag

Summary

This book has been written for undergraduate and graduate students in various areas of mathematics and its applications. It is for students who are willing to get acquainted with Bayesian approach to computational science but not necessarily to go through the full immerson into the statistical analysis. It has also been written for researchers working in areas where mathematical and statistical modeling are of central importance, such as biology and engineering.

Table of Contents

Inverse problems and subjective computing	p. 1
What do we talk about when we talk about random variables?	p. 2
Through the formal theory, lightly	p. 5
How normal is it to be normal?	p. 16
Basic problem of statistical inference	p. 21
On averaging	p. 22
Maximum Likelihood, as frequentists like it	p. 31
The praise of ignorance: randomness as lack of information	p. 39
Construction of Likelihood	p. 41
Enter, Subject: Construction of Priors	p. 48
Posterior Densities as Solutions of Statistical Inverse Problems	p. 55
Basic problem in numerical linear algebra	p. 61
What is a solution?	p. 61
Direct linear system solvers	p. 63
Iterative linear system solvers	p. 67
Ill-conditioning and errors in the data	p. 77
Sampling: first encounter	p. 91
Sampling from Gaussian distributions	p. 92
Random draws from non-Gaussian densities	p. 99
Rejection sampling: prelude to Metropolis-Hastings	p. 102
Statistically inspired preconditioners	p. 107
Priorconditioners: specially chosen preconditioners	p. 108
Sample-based preconditioners and PCA model reduction	p. 118
Conditional Gaussian densities and predictive envelopes	p. 127
Gaussian conditional densities	p. 128
Interpolation, splines and conditional densities	p. 134
Envelopes, white swans and dark matter	p. 144
More applications of the Gaussian conditioning	p. 147
Linear inverse problems	p. 147
Aristotelian boundary conditions	p. 151
Sampling: the real thing	p. 161
Metropolis-Hastings algorithm	p. 168
Wrapping up: hypermodels, dynamic priorconditioners and Bayesian learning	p. 183
MAP estimation or marginalization?	p. 189
Bayesian hypermodels and priorconditioners	p. 193
References	p. 197
Index	p. 199
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