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IP, IST, José Bioucas, 2007 1 Probability The mathematical language to quantify uncertainty  Observation mechanism:  Priors:  Parameters Role in inverse.

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Presentation on theme: "IP, IST, José Bioucas, 2007 1 Probability The mathematical language to quantify uncertainty  Observation mechanism:  Priors:  Parameters Role in inverse."— Presentation transcript:

1 IP, IST, José Bioucas, Probability The mathematical language to quantify uncertainty  Observation mechanism:  Priors:  Parameters Role in inverse problems

2 IP, IST, José Bioucas, 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

3 IP, IST, José Bioucas, Definition Frequencist interpretation: Number of occurencies of 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

4 IP, IST, José Bioucas, Independency; Conditional probability    

5 IP, IST, José Bioucas, Random variables  

6 IP, IST, José Bioucas, Random variables   

7 IP, IST, José Bioucas, Some discrete random variables From Wikipedia

8 IP, IST, José Bioucas, Some discrete random variables From Wikipedia

9 IP, IST, José Bioucas, Some continuous random variables

10 IP, IST, José Bioucas, Some continuous random variables

11 IP, IST, José Bioucas, Some continuous random variables

12 IP, IST, José Bioucas, Bivariate distributions

13 IP, IST, José Bioucas, Bivariate distributions

14 IP, IST, José Bioucas, Bivariate distributions

15 IP, IST, José Bioucas, Expectation

16 IP, IST, José Bioucas, Expectation

17 IP, IST, José Bioucas, Expectation

18 IP, IST, José Bioucas, Expectation

19 IP, IST, José Bioucas, Expectation

20 IP, IST, José Bioucas, Expectation

21 IP, IST, José Bioucas, Expectation

22 IP, IST, José Bioucas, Multivariate Normal

23 IP, IST, José Bioucas, Multivariate Normal

24 IP, IST, José Bioucas, Inequalities

25 IP, IST, José Bioucas, Laws of large numbers

26 IP, IST, José Bioucas, Central Limit Theorem


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