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Basic Probability

Basic Probability. Introduction. Probability is the study of randomness and uncertainty. In the early days, probability was associated with games of chance (gambling). Simple Games Involving Probability.

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Basic Probability

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  1. Basic Probability Pattern Classification, Chapter 1

  2. Introduction • Probability is the study of randomness and uncertainty. • In the early days, probability was associated with games of chance (gambling). Pattern Classification, Chapter 1

  3. Simple Games Involving Probability Game: A fair die is rolled. If the result is 2, 3, or 4, you win $1; if it is 5, you win $2; but if it is 1 or 6, you lose $3. Should you play this game? Pattern Classification, Chapter 1

  4. Random Experiment • a random experiment is a process whose outcome is uncertain. • Examples: • Tossing a coin once or several times • Picking a card or cards from a deck • Measuring temperature of patients • ... Pattern Classification, Chapter 1

  5. Events & Sample Spaces Sample Space The sample space is the set of all possible outcomes. Event An event is any collection of one or more simple events Simple Events The individual outcomes are called simple events. Pattern Classification, Chapter 1

  6. Example Experiment: Toss a coin 3 times. • Sample space  •  = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}. • Examples of events include • A = {HHH, HHT,HTH, THH} = {at least two heads} • B = {HTT, THT,TTH} = {exactly two tails.} Pattern Classification, Chapter 1

  7. Basic Concepts (from Set Theory) • The union of two events A and B, AB, is the event consisting of all outcomes that are either in Aor in Bor in both events. • The complement of an event A, Ac, is the set of all outcomes in that are not in A. • The intersection of two events A and B, AB, is the event consisting of all outcomes that are in both events. • When two events A and B have no outcomes in common, they are said to be mutually exclusive, or disjoint, events. Pattern Classification, Chapter 1

  8. Example Experiment: toss a coin 10 times and the number of heads is observed. • Let A = { 0, 2, 4, 6, 8, 10}. • B = { 1, 3, 5, 7, 9}, C = {0, 1, 2, 3, 4, 5}. • AB= {0, 1, …, 10} = . • AB contains no outcomes. So A and B are mutually exclusive. • Cc = {6, 7, 8, 9, 10}, A  C = {0, 2, 4}. Pattern Classification, Chapter 1

  9. Rules • Commutative Laws: • AB = B A, AB = B A • Associative Laws: • (AB) C = A (B C ) • (AB)  C = A (B  C) . • Distributive Laws: • (AB)C = (A  C)  (B  C) • (AB)C = (A  C) (B  C) • DeMorgan’s Laws: Pattern Classification, Chapter 1

  10. Venn Diagram  A B A∩B Pattern Classification, Chapter 1

  11. Probability • A Probability is a number assigned to each subset (events) of a sample space . • Probability distributions satisfy the followingrules: Pattern Classification, Chapter 1

  12. Axioms of Probability • For any event A, 0  P(A)  1. • P() =1. • If A1, A2, … An is a partition of A, then • P(A) = P(A1)+ P(A2)+...+ P(An) • (A1, A2, … An is called a partition of A if A1A2 …An = A and A1, A2, … An are mutually exclusive.) Pattern Classification, Chapter 1

  13. Properties of Probability • For any event A, P(Ac) = 1 - P(A). • If AB, then P(A)  P(B). • For any two events A and B, • P(AB) = P(A) + P(B) - P(AB). • For three events, A, B, and C, • P(ABC) = P(A) + P(B) + P(C) - • P(AB) - P(AC) - P(BC) + P(AB C). Pattern Classification, Chapter 1

  14. Example • In a certain population, 10% of the people are rich, 5% are famous, and 3% are both rich and famous. A person is randomly selected from this population. What is the chance that the person is • not rich? • rich but not famous? • either rich or famous? Pattern Classification, Chapter 1

  15. Intuitive Development (agrees with axioms) • Intuitively, the probability of an event a could be defined as: Where N(a) is the number that event a happens in n trials Pattern Classification, Chapter 1

  16. Here We Go Again: Not So Basic Probability

  17. More Formal: • W is the Sample Space: • Contains all possible outcomes of an experiment • w inW is a single outcome • A inW is a set of outcomes of interest Pattern Classification, Chapter 1

  18. Independence • The probability of independent events A, B and C is given by: P(A,B,C) = P(A)P(B)P(C) A and B are independent, if knowing that A has happened does not say anything about B happening Pattern Classification, Chapter 1

  19. Bayes Theorem • Provides a way to convert a-priori probabilities to a-posteriori probabilities: Pattern Classification, Chapter 1

  20. Conditional Probability • One of the most useful concepts! W A B Pattern Classification, Chapter 1

  21. Bayes Theorem • Provides a way to convert a-priori probabilities to a-posteriori probabilities: Pattern Classification, Chapter 1

  22. Using Partitions: • If events Ai are mutually exclusive and partition W Pattern Classification, Chapter 1

  23. Random Variables • A (scalar) random variable X is a function that maps the outcome of a random event into real scalar values W X(w) w Pattern Classification, Chapter 1

  24. Random Variables Distributions • Cumulative Probability Distribution (CDF): • Probability Density Function (PDF): Pattern Classification, Chapter 1

  25. Random Distributions: • From the two previous equations: Pattern Classification, Chapter 1

  26. Uniform Distribution • A R.V. X that is uniformly distributed between x1 and x2 has density function: X1 X2 Pattern Classification, Chapter 1

  27. Gaussian (Normal) Distribution • A R.V. X that is normally distributed has density function: m Pattern Classification, Chapter 1

  28. Statistical Characterizations • Expectation (Mean Value, First Moment): • Second Moment: Pattern Classification, Chapter 1

  29. Statistical Characterizations • Variance of X: • Standard Deviation of X: Pattern Classification, Chapter 1

  30. Mean Estimation from Samples • Given a set of N samples from a distribution, we can estimate the mean of the distribution by: Pattern Classification, Chapter 1

  31. Variance Estimation from Samples • Given a set of N samples from a distribution, we can estimate the variance of the distribution by: Pattern Classification, Chapter 1

  32. Pattern Classification

  33. Chapter 1: Introduction to Pattern Recognition (Sections 1.1-1.6) Machine Perception An Example Pattern Recognition Systems The Design Cycle Learning and Adaptation Conclusion

  34. Machine Perception • Build a machine that can recognize patterns: • Speech recognition • Fingerprint identification • OCR (Optical Character Recognition) • DNA sequence identification Pattern Classification, Chapter 1

  35. An Example • “Sorting incoming Fish on a conveyor according to species using optical sensing” Sea bass Species Salmon Pattern Classification, Chapter 1

  36. Problem Analysis • Set up a camera and take some sample images to extract features • Length • Lightness • Width • Number and shape of fins • Position of the mouth, etc… • This is the set of all suggested features to explore for use in our classifier! Pattern Classification, Chapter 1

  37. Preprocessing • Use a segmentation operation to isolate fishes from one another and from the background • Information from a single fish is sent to a feature extractor whose purpose is to reduce the data by measuring certain features • The features are passed to a classifier Pattern Classification, Chapter 1

  38. Pattern Classification, Chapter 1

  39. Classification • Select the length of the fish as a possible feature for discrimination Pattern Classification, Chapter 1

  40. Pattern Classification, Chapter 1

  41. The length is a poor feature alone! Select the lightness as a possible feature. Pattern Classification, Chapter 1

  42. Pattern Classification, Chapter 1

  43. Threshold decision boundary and cost relationship • Move our decision boundary toward smaller values of lightness in order to minimize the cost (reduce the number of sea bass that are classified salmon!) Task of decision theory Pattern Classification, Chapter 1

  44. Adopt the lightness and add the width of the fish Fish xT = [x1, x2] Lightness Width Pattern Classification, Chapter 1

  45. Pattern Classification, Chapter 1

  46. We might add other features that are not correlated with the ones we already have. A precaution should be taken not to reduce the performance by adding “noisy features” • Ideally, the best decision boundary should be the one which provides an optimal performance such as in the following figure: Pattern Classification, Chapter 1

  47. Pattern Classification, Chapter 1

  48. However, our satisfaction is premature because the central aim of designing a classifier is to correctly classify novel input Issue of generalization! Pattern Classification, Chapter 1

  49. Pattern Classification, Chapter 1

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