CHAPTER 2:INTRODUCTION TOPROBABILITYCOMPUTER VISION: MODELS, LEARNING ANDINFERENCELukas Tencer
Random variables2       A random variable x denotes a quantity that is        uncertain       May be result of experimen...
Discrete Random Variables3                                                                   Ordered                      ...
Continuous Random Variable4    - Probability density function (PDF) of a distribution, integrates to 1                    ...
Joint Probability5       Consider two random variables x and y       If we observe multiple paired instances, then      ...
Joint Probability6            Computer vision: models, learning and inference. ©2011                                      ...
Marginalization7    We can recover probability distribution of any variable in a joint      distribution by integrating (o...
Marginalization8    We can recover probability distribution of any variable in a joint      distribution by integrating (o...
Marginalization9    We can recover probability distribution of any variable in a joint      distribution by integrating (o...
Marginalization10     We can recover probability distribution of any variable in a joint       distribution by integrating...
Conditional Probability11        Conditional probability of x given that y=y1 is relative         propensity of variable ...
Conditional Probability12        Conditional probability can be extracted from joint probability        Extract appropri...
Conditional Probability13        More usually written in compact form     • Can be re-arranged to give                  C...
Conditional Probability14        This idea can be extended to more than two         variables                 Computer vi...
Bayes’ Rule15     From before:     Combining:     Re-arranging:                  Computer vision: models, learning and inf...
Bayes’ Rule Terminology16           Likelihood – propensity                  Prior – what we know           for observing ...
Independence17        If two variables x and y are independent then variable x         tells us nothing about variable y ...
Independence18        If two variables x and y are independent then variable x         tells us nothing about variable y ...
Independence19        When variables are independent, the joint factorizes into         a product of the marginals:      ...
Expectation20     Expectation tell us the expected or average value of     some function f [x] taking into account the dis...
Expectation21     Expectation tell us the expected or average value of     some function f [x] taking into account the dis...
Expectation: Common Cases22          Computer vision: models, learning and inference. ©2011                               ...
Expectation: Rules23     Rule 1:     Expected value of a constant is the constant                 Computer vision: models,...
Expectation: Rules24     Rule 2:     Expected value of constant times function is constant     times expected value of fun...
Expectation: Rules25     Rule 3:     Expectation of sum of functions is sum of expectation of     functions               ...
Expectation: Rules26     Rule 4:     Expectation of product of functions in variables x and y     is product of expectatio...
Conclusions27     • Rules of probability are compact and simple     • Concepts of marginalization, joint and conditional  ...
28            Thank You            for you attentionBased on:Computer vision: models, learning and inference. ©2011 Simon ...
Additional29        Calculating k-th moment
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Introduction to Probability

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Introduction to Probability

  1. 1. CHAPTER 2:INTRODUCTION TOPROBABILITYCOMPUTER VISION: MODELS, LEARNING ANDINFERENCELukas Tencer
  2. 2. Random variables2  A random variable x denotes a quantity that is uncertain  May be result of experiment (flipping a coin) or a real world measurements (measuring temperature)  If observe several instances of x we get different values  Some values occur more than others and this information is captured by a probability distribution Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
  3. 3. Discrete Random Variables3 Ordered - histogram Unordered - hintongram Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
  4. 4. Continuous Random Variable4 - Probability density function (PDF) of a distribution, integrates to 1 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
  5. 5. Joint Probability5  Consider two random variables x and y  If we observe multiple paired instances, then some combinations of outcomes are more likely than others  This is captured in the joint probability distribution  Written as Pr(x,y)  Can read Pr(x,y) as “probability of x and y” Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
  6. 6. Joint Probability6 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
  7. 7. Marginalization7 We can recover probability distribution of any variable in a joint distribution by integrating (or summing) over the other variables Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
  8. 8. Marginalization8 We can recover probability distribution of any variable in a joint distribution by integrating (or summing) over the other variables Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
  9. 9. Marginalization9 We can recover probability distribution of any variable in a joint distribution by integrating (or summing) over the other variables Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
  10. 10. Marginalization10 We can recover probability distribution of any variable in a joint distribution by integrating (or summing) over the other variables Works in higher dimensions as well – leaves joint distribution between whatever variables are left Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
  11. 11. Conditional Probability11  Conditional probability of x given that y=y1 is relative propensity of variable x to take different outcomes given that y is fixed to be equal to y1.  Written as Pr(x|y=y1) Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
  12. 12. Conditional Probability12  Conditional probability can be extracted from joint probability  Extract appropriate slice and normalize (because dos not sums to 1) Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
  13. 13. Conditional Probability13  More usually written in compact form • Can be re-arranged to give Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
  14. 14. Conditional Probability14  This idea can be extended to more than two variables Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
  15. 15. Bayes’ Rule15 From before: Combining: Re-arranging: Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
  16. 16. Bayes’ Rule Terminology16 Likelihood – propensity Prior – what we know for observing a certain about y before seeing value of x given a certain x value of y Posterior – what we Evidence –a constant to know about y after ensure that the left hand seeing x side is a valid distribution Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
  17. 17. Independence17  If two variables x and y are independent then variable x tells us nothing about variable y (and vice-versa) Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
  18. 18. Independence18  If two variables x and y are independent then variable x tells us nothing about variable y (and vice-versa) Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
  19. 19. Independence19  When variables are independent, the joint factorizes into a product of the marginals: Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
  20. 20. Expectation20 Expectation tell us the expected or average value of some function f [x] taking into account the distribution of x Definition: Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
  21. 21. Expectation21 Expectation tell us the expected or average value of some function f [x] taking into account the distribution of x Definition in two dimensions: Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
  22. 22. Expectation: Common Cases22 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
  23. 23. Expectation: Rules23 Rule 1: Expected value of a constant is the constant Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
  24. 24. Expectation: Rules24 Rule 2: Expected value of constant times function is constant times expected value of function Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
  25. 25. Expectation: Rules25 Rule 3: Expectation of sum of functions is sum of expectation of functions Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
  26. 26. Expectation: Rules26 Rule 4: Expectation of product of functions in variables x and y is product of expectations of functions if x and y are independen Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
  27. 27. Conclusions27 • Rules of probability are compact and simple • Concepts of marginalization, joint and conditional probability, Bayes rule and expectation underpin all of the models in this book • One remaining concept – conditional expectation – discussed later Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
  28. 28. 28 Thank You for you attentionBased on:Computer vision: models, learning and inference. ©2011 Simon J.D. Princehttp://www.computervisionmodels.com/
  29. 29. Additional29  Calculating k-th moment

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