IP, IST, José Bioucas, 2007 1 Probability The mathematical language to quantify uncertainty ...
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![Page 1: IP, IST, José Bioucas, 2007 1 Probability The mathematical language to quantify uncertainty Observation mechanism: Priors: Parameters Role in inverse.](https://reader036.fdocuments.in/reader036/viewer/2022062621/551c06e15503469e4f8b4efd/html5/thumbnails/1.jpg)
1IP, IST, José Bioucas, 2007
Probability
The mathematical language to quantify uncertainty
Observation mechanism:
Priors:
Parameters
Role in inverse problems
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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
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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
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4IP, IST, José Bioucas, 2007
Independency; Conditional probability
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5IP, IST, José Bioucas, 2007
Random variables
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6IP, IST, José Bioucas, 2007
Random variables
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7IP, IST, José Bioucas, 2007
Some discrete random variables
From Wikipedia
![Page 8: IP, IST, José Bioucas, 2007 1 Probability The mathematical language to quantify uncertainty Observation mechanism: Priors: Parameters Role in inverse.](https://reader036.fdocuments.in/reader036/viewer/2022062621/551c06e15503469e4f8b4efd/html5/thumbnails/8.jpg)
8IP, IST, José Bioucas, 2007
Some discrete random variables
From Wikipedia
![Page 9: IP, IST, José Bioucas, 2007 1 Probability The mathematical language to quantify uncertainty Observation mechanism: Priors: Parameters Role in inverse.](https://reader036.fdocuments.in/reader036/viewer/2022062621/551c06e15503469e4f8b4efd/html5/thumbnails/9.jpg)
9IP, IST, José Bioucas, 2007
Some continuous random variables
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10IP, IST, José Bioucas, 2007
Some continuous random variables
![Page 11: IP, IST, José Bioucas, 2007 1 Probability The mathematical language to quantify uncertainty Observation mechanism: Priors: Parameters Role in inverse.](https://reader036.fdocuments.in/reader036/viewer/2022062621/551c06e15503469e4f8b4efd/html5/thumbnails/11.jpg)
11IP, IST, José Bioucas, 2007
Some continuous random variables
![Page 12: IP, IST, José Bioucas, 2007 1 Probability The mathematical language to quantify uncertainty Observation mechanism: Priors: Parameters Role in inverse.](https://reader036.fdocuments.in/reader036/viewer/2022062621/551c06e15503469e4f8b4efd/html5/thumbnails/12.jpg)
12IP, IST, José Bioucas, 2007
Bivariate distributions
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13IP, IST, José Bioucas, 2007
Bivariate distributions
![Page 14: IP, IST, José Bioucas, 2007 1 Probability The mathematical language to quantify uncertainty Observation mechanism: Priors: Parameters Role in inverse.](https://reader036.fdocuments.in/reader036/viewer/2022062621/551c06e15503469e4f8b4efd/html5/thumbnails/14.jpg)
14IP, IST, José Bioucas, 2007
Bivariate distributions
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15IP, IST, José Bioucas, 2007
Expectation
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16IP, IST, José Bioucas, 2007
Expectation
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17IP, IST, José Bioucas, 2007
Expectation
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18IP, IST, José Bioucas, 2007
Expectation
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19IP, IST, José Bioucas, 2007
Expectation
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20IP, IST, José Bioucas, 2007
Expectation
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21IP, IST, José Bioucas, 2007
Expectation
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22IP, IST, José Bioucas, 2007
Multivariate Normal
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23IP, IST, José Bioucas, 2007
Multivariate Normal
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24IP, IST, José Bioucas, 2007
Inequalities
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25IP, IST, José Bioucas, 2007
Laws of large numbers
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26IP, IST, José Bioucas, 2007
Central Limit Theorem