ip, ist, josé bioucas, 2007 1 probability the mathematical language to quantify uncertainty ...

1 P, IST, José Bioucas, 2007 Probability mathematical language to quantify uncertain Observation mechanism: Priors: Parameters Role in inverse problems

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1IP, IST, José Bioucas, 2007

Probability

The mathematical language to quantify uncertainty

Observation mechanism:

Priors:

Parameters

Role in inverse problems

2IP, IST, José Bioucas, 2007

Overview of Probability

Definition; Properties

Independency; Conditional probability; Bayes theorem

Random variables; Cumulative distribution function

Examples of random variables

Bivariate distributions; Marginal distribution;Conditional distribution

Multivariate distributions; Marginal distribution;Conditional distributions

Expectation of a random variable; Variance; Covariance

Conditional expectation of a random variable; Variance; Covariance

Weak law of large numbers

Ref. Larry Wasserman, All of Statistics. A Concise Course in Statistical Inference, Springer, 2004

3IP, IST, José Bioucas, 2007

Definition

Frequencist interpretation:

Number of occurenciesof A

Number of repetitions

Bayesain interpretation: Measures an observer’s strength of belief that A is true

In probability the interpretation does not matter

sample space

event

4IP, IST, José Bioucas, 2007

Independency; Conditional probability

5IP, IST, José Bioucas, 2007

Random variables

6IP, IST, José Bioucas, 2007

Random variables

7IP, IST, José Bioucas, 2007

Some discrete random variables

From Wikipedia

8IP, IST, José Bioucas, 2007

Some discrete random variables

From Wikipedia

9IP, IST, José Bioucas, 2007

Some continuous random variables

10IP, IST, José Bioucas, 2007

Some continuous random variables

11IP, IST, José Bioucas, 2007

Some continuous random variables

12IP, IST, José Bioucas, 2007

Bivariate distributions

13IP, IST, José Bioucas, 2007

Bivariate distributions

14IP, IST, José Bioucas, 2007

Bivariate distributions

15IP, IST, José Bioucas, 2007

Expectation

16IP, IST, José Bioucas, 2007

Expectation

17IP, IST, José Bioucas, 2007

Expectation

18IP, IST, José Bioucas, 2007

Expectation

19IP, IST, José Bioucas, 2007

Expectation

20IP, IST, José Bioucas, 2007

Expectation

21IP, IST, José Bioucas, 2007

Expectation

22IP, IST, José Bioucas, 2007

Multivariate Normal

23IP, IST, José Bioucas, 2007

Multivariate Normal

24IP, IST, José Bioucas, 2007

Inequalities

25IP, IST, José Bioucas, 2007

Laws of large numbers

26IP, IST, José Bioucas, 2007

Central Limit Theorem

Geodesic Saliency Using Background Priors

Shmoo Quantify

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THE PARISH OF SALFORD PRIORS · THE PARISH OF SALFORD PRIORS Incorporating Abbots’ Salford, Dunnington, Iron ross, Pitchill, Rushford, Salford Priors and The evingtons. Introduction

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Priors, Normal Models , Computing Posteriors

Compression under uncertain priors

Salford Priors Village Design Statement Priors.pdf · Salford Priors Village Design Statement Created Date: 20140108114054Z

a catalog of noninformative priors

Turner Priors

Wing-Kin Ma, José M. Bioucas-Dias, Tsung-Han Chan, Nicolas …bioucas/files/ieee_spm_2014_HU.pdf · 2014-09-23 · Wing-Kin Ma, José M. Bioucas-Dias, Tsung-Han Chan, Nicolas Gillis,

Abstractdivided into internal priors and external priors. The internal image priors are directly learned on the input test image itself, while the external image priors need to be

Face Super-Resolution Guided by 3D Facial Priors...Face Super-Resolution Guided by 3D Facial Priors 3 branch. Speci cally, the 3D facial priors contain rich hierarchical features,

SALFORD PRIORS PARISH OUNIL NEWSLETTER

Data Dependent Priors for Stable Learning - … · Background PAC-Bayes Analysis Linear Classiﬁers Data distribution dependent priors Data Dependent Priors for Stable Learning John

MVPA with SpaceNet: sparse structured priors

Almeida, Miguel; Bioucas-Dias, J.; Vigário, R. Separation ... · Almeida, Miguel; Bioucas-Dias, J.; Vigário, R. Separation of phase-locked sources in pseudo-real MEG data Published

Weakly informative priors - Columbia Universitygelman/presentations/weakpriorstalk.pdf · Weakly informative priors Static sensitivity analysis Conservatism of Bayesian inference

Salford Priors, Warwickshire 1 - Cotswold Archaeologyreports.cotswoldarchaeology.co.uk/.../0490-Salford-Priors-Warks-PAR...Salford Priors, Warwickshire ... shallow banana-shaped gullies

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IMPROPER PRIORS FIDUCIAL INFERENCE - Statistics · “Improper priors are just limits of proper priors ... ” We present ingredients in a mathematical theory for statistics which

ICIC Data Analysis Workshop 2016 · 2020. 8. 5. · A word on priors in Theory or physics driven priors –, Mass > 0 Data driven priors & combination of experiments – Prior = old

Smoothing/Priors/ Regularization for Maxent Models

Gibbs Sampling, Conjugate Priors and Coupling

ip, ist, josé bioucas, 2007 1 probability the mathematical language to quantify uncertainty ...

Documents

jos bioucas

expectation slide

wikipedia slide

bivariate distributions

multivariate normal

continuous random variables

discrete random variables

sample space event slide