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Confidence in nonparametric credible sets?
Published on Aug 22, 201210927 Views
In nonparametric statistics the posterior distribution is used in exactly the same way as in any Bayesian analysis. It supposedly gives us the likelihood of various parameter values given the data. A
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Chapter list
Confidence in Nonparametric Credible Sets?00:00
Contents02:10
Co-authors04:01
Disclaimer04:22
1. Nonparametric Bayes04:58
Bayesian nonparametric inference (1)05:01
Bayesian nonparametric inference (2)06:12
Bayesian nonparametric inference (3)06:49
Gaussian priors07:47
Example: Logistic regression08:02
Example: heat equation11:04
Example: genomics13:20
Example: earth science14:10
Notation: the Bayesian machine14:50
Notation: the Bayesian machine - asymptotics in n16:06
Frequentist Bayes (1)16:35
Frequentist Bayes (2)19:01
Frequentist Bayes (3)19:45
Frequentist Bayes (4)19:54
Frequentist Bayes (5)20:24
What do the frequentists say? - rates20:38
2. Gaussian Process Priors23:05
Gaussian process23:06
Integrated Brownian motion23:52
Stationary processes24:54
Other Gaussian processes25:45
Posterior contraction rates for Gaussian priors (1)26:06
Posterior contraction rates for Gaussian priors (2)28:25
Settings (1)28:35
Settings (2)29:54
Posterior contraction rates for Gaussian priors (3)30:10
Posterior contraction rates for Gaussian priors (4)32:12
Brownian Motion32:23
Integrated Brownian Motion35:35
Stationary processes36:30
Stationary processes - radial basis36:45
Stationary processes - Matérn38:02
Time-scaling Gaussian processes (1)38:27
Time-scaling Gaussian processes (2)38:46
Time-scaling integrated Brownian motion (1)39:10
Time-scaling integrated Brownian motion (2)40:02
Time-scaling smooth stationary process40:15
Adaptation (1)40:47
Adaptation by random scaling - example42:19
Recovery: summary43:42
3. Credible Sets44:03
Notation: the Bayesian machine44:06
Frequentist Bayes44:24
Uncertainty quantification: an early answer44:38
Linear Gaussian inverse problems (1)46:13
Linear Gaussian inverse problems (2)47:05
Linear Gaussian inverse problems (3)47:32
Sobolev models and priors (1)48:34
Sobolev models and priors (2)49:46
Linear Gaussian inverse problem - rate of contraction49:58
Example: reconstruct derivative (1)50:35
Example: reconstruct derivative (2)51:17
Example: reconstruct derivative (n=100)51:23
Example: reconstruct derivative (n=100 000)53:05
Linear Gaussian inverse problem - credible balls (1)53:37
Linear Gaussian inverse problem - credible balls (2)55:38
Linear Gaussian inverse problem - scaling the prior56:03
Example: reconstruct derivative (n=1000)56:21
Credible sets: first summary56:30
Example: heat equation (n=10 000, n=100 000 000)57:53
4. Adaptive Credible Sets58:16
Adaptation (2)58:23
Example: genomics58:44
Linear Gaussian inverse problem - random prior smoothness (1)58:45
Linear Gaussian inverse problem - random prior smoothness (2)01:00:11
Example: reconstructing a derivative01:00:42
What do the frequentists say? - Honesty (1)01:02:02
What do the frequentists say? - Honesty (2)01:02:43
What do the frequentists say? - Honesty (3)01:03:23
What do the frequentists say? - Discrepancy between estimation and uncertainty quantification (1)01:03:27
What do the frequentists say? - Discrepancy between estimation and uncertainty quantification (2)01:04:22
What do the frequentists say? - Self-similarity01:04:54
Linear Gaussian inverse problems - Credible sets are honest over self-similar functions01:05:50
Example: reconstruct derivative (n=1000)01:06:04
Credible sets are honest over prior sets? (1)01:06:19
Credible sets are honest over prior sets? (2)01:06:28
Credible sets are honest over prior sets? (3)01:06:38
Example: reconstructing a derivative01:07:00
Conclusions and Conjectures (1)01:07:42
Conclusions and Conjectures (2)01:07:48
Conclusions and Conjectures (3)01:07:53
Conclusions and Conjectures (4)01:08:00
Conclusions and Conjectures (5)01:08:09
Conclusions and Conjectures (6)01:08:14
Conclusions and Conjectures (7)01:08:16
Conclusions and Conjectures (8)01:08:23