This book has been written for undergraduate and graduate students in various disciplines of mathematics. The authors, internationally recognized experts in their field, have developed a superior teaching and learning tool that makes it easy to grasp new concepts and apply them in practice. The book’s highly accessible approach makes it particularly ideal if you want to become acquainted with the Bayesian approach to computational science, but do not need to be fully immersed in detailed statistical analysis.
Author(s): Daniela Calvetti, Erkki Somersalo
Publisher: Springer
Year: 2007
Language: English
Pages: 211
Preface......Page 6
Contents......Page 11
Inverse problems and subjective computing......Page 13
What do we talk about when we talk about random variables?......Page 14
Through the formal theory, lightly......Page 17
How normal is it to be normal?......Page 28
Basic problem of statistical inference......Page 33
On averaging......Page 34
Maximum Likelihood, as frequentists like it......Page 43
The praise of ignorance: randomness as lack of information......Page 51
Construction of Likelihood......Page 53
Enter, Subject: Construction of Priors......Page 60
Posterior Densities as Solutions of Statistical Inverse Problems......Page 67
What is a solution?......Page 72
Direct linear system solvers......Page 74
Iterative linear system solvers......Page 78
Ill-conditioning and errors in the data......Page 88
Sampling: first encounter......Page 102
Sampling from Gaussian distributions......Page 103
Random draws from non-Gaussian densities......Page 110
Rejection sampling: prelude to Metropolis-Hastings......Page 113
Statistically inspired preconditioners......Page 118
Priorconditioners: specially chosen preconditioners......Page 119
Sample-based preconditioners and PCA model reduction......Page 129
Conditional Gaussian densities and predictive envelopes......Page 138
Gaussian conditional densities......Page 139
Interpolation, splines and conditional densities......Page 145
Envelopes, white swans and dark matter......Page 155
Linear inverse problems......Page 158
Aristotelian boundary conditions......Page 162
Sampling: the real thing......Page 172
Metropolis--Hastings algorithm......Page 179
Wrapping up: hypermodels, dynamic priorconditioners and Bayesian learning......Page 194
MAP estimation or marginalization?......Page 200
Bayesian hypermodels and priorconditioners......Page 204
References......Page 207
Index......Page 208
