Advanced Probabilistic Modeling in R Day 1

Transcription

1 Advanced Probabilistic Modeling in R Day 1 Roger Levy University of California, San Diego July 20, /24

2 Today s content Quick review of probability: axioms, joint & conditional probabilities, Bayes Rule, conditional independence Bayes Nets (a.k.a. directed acyclic graphical models, DAGs) The Gaussian distribution Example: human phoneme categorization Maximum likelihood estimation Bayesian parameter estimation Frequentist hypothesis testing Bayesian hypothesis testing 2/24

3 Probability spaces Traditionally, probability spaces are defined in terms of sets. An event E is a subset of a sample space Ω: E Ω. 3/24

4 Probability spaces Traditionally, probability spaces are defined in terms of sets. An event E is a subset of a sample space Ω: E Ω. A probability space P on a sample space Ω is a function from events E in Ω to real numbers such that the following three axioms hold: 1. P(E) 0 for all E Ω (non-negativity). 2. If E 1 and E 2 are disjoint, then P(E 1 E 2 ) = P(E 1 )+P(E 2 ) (disjoint union). 3. P(Ω) = 1 (properness). 3/24

5 Joint, conditional, and marginal probabilities Given the joint distribution P(X,Y) over two random variables X and Y, the conditional distribution P(Y X) is defined as P(Y X) P(X,Y) P(X) 4/24

6 Joint, conditional, and marginal probabilities Given the joint distribution P(X,Y) over two random variables X and Y, the conditional distribution P(Y X) is defined as P(Y X) P(X,Y) P(X) The marginal probability distribution P(X) is P(X = x) = y P(X = x,y = y) These concepts can be extended to arbitrary numbers of random variables. 4/24

7 The chain rule A joint probability can be rewritten as the product of marginal and conditional probabilities: P(E 1,E 2 ) = P(E 2 E 1 )P(E 1 ) 5/24

8 The chain rule A joint probability can be rewritten as the product of marginal and conditional probabilities: P(E 1,E 2 ) = P(E 2 E 1 )P(E 1 ) And this generalizes to more than two variables: 5/24

9 The chain rule A joint probability can be rewritten as the product of marginal and conditional probabilities: P(E 1,E 2 ) = P(E 2 E 1 )P(E 1 ) And this generalizes to more than two variables: P(E 1,E 2 ) = P(E 2 E 1 )P(E 1 )

10 The chain rule A joint probability can be rewritten as the product of marginal and conditional probabilities: P(E 1,E 2 ) = P(E 2 E 1 )P(E 1 ) And this generalizes to more than two variables: P(E 1,E 2 ) = P(E 2 E 1 )P(E 1 ) P(E 1,E 2,E 3 ) = P(E 3 E 1,E 2 )P(E 2 E 1 )P(E 1 )

11 The chain rule A joint probability can be rewritten as the product of marginal and conditional probabilities: P(E 1,E 2 ) = P(E 2 E 1 )P(E 1 ) And this generalizes to more than two variables: P(E 1,E 2 ) = P(E 2 E 1 )P(E 1 ) P(E 1,E 2,E 3 ) = P(E 3 E 1,E 2 )P(E 2 E 1 )P(E 1 )..

12 The chain rule A joint probability can be rewritten as the product of marginal and conditional probabilities: P(E 1,E 2 ) = P(E 2 E 1 )P(E 1 ) And this generalizes to more than two variables: P(E 1,E 2 ) = P(E 2 E 1 )P(E 1 ) P(E 1,E 2,E 3 ) = P(E 3 E 1,E 2 )P(E 2 E 1 )P(E 1 )..

13 The chain rule A joint probability can be rewritten as the product of marginal and conditional probabilities: P(E 1,E 2 ) = P(E 2 E 1 )P(E 1 ) And this generalizes to more than two variables: P(E 1,E 2 ) = P(E 2 E 1 )P(E 1 ) P(E 1,E 2,E 3 ) = P(E 3 E 1,E 2 )P(E 2 E 1 )P(E 1 ).. P(E 1,E 2,...,E n ) = P(E n E 1,E 2,...,E n 1 )...P(E 2 E 1 )P(E 1 ) 5/24

14 The chain rule A joint probability can be rewritten as the product of marginal and conditional probabilities: P(E 1,E 2 ) = P(E 2 E 1 )P(E 1 ) And this generalizes to more than two variables: P(E 1,E 2 ) = P(E 2 E 1 )P(E 1 ) P(E 1,E 2,E 3 ) = P(E 3 E 1,E 2 )P(E 2 E 1 )P(E 1 ).. P(E 1,E 2,...,E n ) = P(E n E 1,E 2,...,E n 1 )...P(E 2 E 1 )P(E 1 ) Breaking a joint probability down into the product a marginal probability and several joint probabilities this way is called chain rule decomposition. 5/24

15 Bayes Rule (Bayes Theorem) P(A B) = P(B A)P(A) P(B) 6/24

16 Bayes Rule (Bayes Theorem) P(A B) = P(B A)P(A) P(B) With extra background random variables I: P(A B,I) = P(B A,I)P(A I) P(B I) 6/24

17 Bayes Rule (Bayes Theorem) P(A B) = P(B A)P(A) P(B) With extra background random variables I: P(A B,I) = P(B A,I)P(A I) P(B I) This theorem follows directly from def n of conditional probability: P(A, B) = P(B A)P(A) 6/24

18 Bayes Rule (Bayes Theorem) P(A B) = P(B A)P(A) P(B) With extra background random variables I: P(A B,I) = P(B A,I)P(A I) P(B I) This theorem follows directly from def n of conditional probability: P(A, B) = P(B A)P(A) P(A, B) = P(A B)P(B) 6/24

19 Bayes Rule (Bayes Theorem) P(A B) = P(B A)P(A) P(B) With extra background random variables I: P(A B,I) = P(B A,I)P(A I) P(B I) This theorem follows directly from def n of conditional probability: P(A, B) = P(B A)P(A) P(A, B) = P(A B)P(B) So P(A B)P(B) = P(B A)P(A) 6/24

20 Bayes Rule (Bayes Theorem) P(A B) = P(B A)P(A) P(B) With extra background random variables I: P(A B,I) = P(B A,I)P(A I) P(B I) This theorem follows directly from def n of conditional probability: So P(A, B) = P(B A)P(A) P(A, B) = P(A B)P(B) P(A B)P(B) = P(B A)P(A) P(A B)P(B) P(B) = P(B A)P(A) P(B) 6/24

21 Bayes Rule (Bayes Theorem) P(A B) = P(B A)P(A) P(B) With extra background random variables I: P(A B,I) = P(B A,I)P(A I) P(B I) This theorem follows directly from def n of conditional probability: So P(A, B) = P(B A)P(A) P(A, B) = P(A B)P(B) P(A B)P(B) = P(B A)P(A) P(A B)P(B) P(B) = P(B A)P(A) P(B) 6/24

22 Other ways of writing Bayes Rule Likelihood Prior {}}{{}}{ P(B A) P(A) P(A B) = P(B) }{{} Normalizing constant The hardest part of using Bayes Rule was calculating the normalizing constant (a.k.a. the partition function) 7/24

23 Other ways of writing Bayes Rule Likelihood Prior {}}{{}}{ P(B A) P(A) P(A B) = P(B) }{{} Normalizing constant The hardest part of using Bayes Rule was calculating the normalizing constant (a.k.a. the partition function) Hence there are often two other ways we write Bayes Rule: 7/24

24 Other ways of writing Bayes Rule Likelihood Prior {}}{{}}{ P(B A) P(A) P(A B) = P(B) }{{} Normalizing constant The hardest part of using Bayes Rule was calculating the normalizing constant (a.k.a. the partition function) Hence there are often two other ways we write Bayes Rule: 1. Emphasizing explicit marginalization: P(A B) = P(B A)P(A) P(A = a,b) a 7/24

25 Other ways of writing Bayes Rule Likelihood Prior {}}{{}}{ P(B A) P(A) P(A B) = P(B) }{{} Normalizing constant The hardest part of using Bayes Rule was calculating the normalizing constant (a.k.a. the partition function) Hence there are often two other ways we write Bayes Rule: 1. Emphasizing explicit marginalization: 2. Ignoring the partition function: P(A B) = P(B A)P(A) P(A = a,b) a P(A B) P(B A)P(A) 7/24

26 (Conditional) Independence Events A and B are said to be Conditionally Independent given information C if P(A,B C) = P(A C)P(B C) Conditional independence of A and B given C is often expressed as A B C 8/24

27 Directed graphical models A lot of the interesting joint probability distributions in the study of language involve conditional independencies among the variables 9/24

28 Directed graphical models A lot of the interesting joint probability distributions in the study of language involve conditional independencies among the variables So next we ll introduce you to a general framework for specifying conditional independencies among collections of random variables 9/24

29 Directed graphical models A lot of the interesting joint probability distributions in the study of language involve conditional independencies among the variables So next we ll introduce you to a general framework for specifying conditional independencies among collections of random variables It won t allow us to express all possible independencies that may hold, but it goes a long way 9/24

30 Directed graphical models A lot of the interesting joint probability distributions in the study of language involve conditional independencies among the variables So next we ll introduce you to a general framework for specifying conditional independencies among collections of random variables It won t allow us to express all possible independencies that may hold, but it goes a long way And I hope that you ll agree that the framework is intuitive too! 9/24

31 A non-linguistic example Imagine a factory that produces three types of coins in equal volumes: 10/24

32 A non-linguistic example Imagine a factory that produces three types of coins in equal volumes: Fair coins; 10/24

33 A non-linguistic example Imagine a factory that produces three types of coins in equal volumes: Fair coins; 2-headed coins; 10/24

34 A non-linguistic example Imagine a factory that produces three types of coins in equal volumes: Fair coins; 2-headed coins; 2-tailed coins. 10/24

35 A non-linguistic example Imagine a factory that produces three types of coins in equal volumes: Fair coins; 2-headed coins; 2-tailed coins. Generative process: 10/24

36 A non-linguistic example Imagine a factory that produces three types of coins in equal volumes: Fair coins; 2-headed coins; 2-tailed coins. Generative process: The factory produces a coin of type X and sends it to you; 10/24

37 A non-linguistic example Imagine a factory that produces three types of coins in equal volumes: Fair coins; 2-headed coins; 2-tailed coins. Generative process: The factory produces a coin of type X and sends it to you; You receive the coin and flip it twice, with H(eads)/T(ails) outcomes Y 1 and Y 2 10/24

38 A non-linguistic example Imagine a factory that produces three types of coins in equal volumes: Fair coins; 2-headed coins; 2-tailed coins. Generative process: The factory produces a coin of type X and sends it to you; You receive the coin and flip it twice, with H(eads)/T(ails) outcomes Y 1 and Y 2 Receiving a coin from the factory and flipping it twice is sampling (or taking a sample) from the joint distribution P(X,Y 1,Y 2 ) 10/24

39 This generative process a Bayes Net The directed acyclic graphical model (DAG), or Bayes net: X Y 1 Y 2 Semantics of a Bayes net: the joint distribution can be expressed as the product of the conditional distributions of each variable given only its parents 11/24

40 This generative process a Bayes Net The directed acyclic graphical model (DAG), or Bayes net: X Y 1 Y 2 Semantics of a Bayes net: the joint distribution can be expressed as the product of the conditional distributions of each variable given only its parents In this DAG, P(X,Y 1,Y 2 ) = P(X)P(Y 1 X)P(Y 2 X) 11/24

41 This generative process a Bayes Net The directed acyclic graphical model (DAG), or Bayes net: X Y 1 Y 2 Semantics of a Bayes net: the joint distribution can be expressed as the product of the conditional distributions of each variable given only its parents In this DAG, P(X,Y 1,Y 2 ) = P(X)P(Y 1 X)P(Y 2 X) X P(X) 1 Fair 3 2-H T 3 11/24

42 This generative process a Bayes Net The directed acyclic graphical model (DAG), or Bayes net: X Y 1 Y 2 Semantics of a Bayes net: the joint distribution can be expressed as the product of the conditional distributions of each variable given only its parents In this DAG, P(X,Y 1,Y 2 ) = P(X)P(Y 1 X)P(Y 2 X) X P(X) 1 Fair 3 2-H T 3 X P(Y 1 = H X) P(Y 1 = T X) 1 1 Fair H T /24

43 This generative process a Bayes Net The directed acyclic graphical model (DAG), or Bayes net: X Y 1 Y 2 Semantics of a Bayes net: the joint distribution can be expressed as the product of the conditional distributions of each variable given only its parents In this DAG, P(X,Y 1,Y 2 ) = P(X)P(Y 1 X)P(Y 2 X) X P(X) 1 Fair 3 2-H T 3 X P(Y 1 = H X) P(Y 1 = T X) 1 1 Fair H T 0 1 X P(Y 2 = H X) P(Y 2 = T X) 1 1 Fair H T /24

44 Conditional independence in Bayes nets X P(X) 1 Fair 3 2-H T 3 X P(Y 1 = H X) P(Y 1 = T X) 1 1 Fair H T 0 1 X P(Y 2 = H X) P(Y 2 = T X) 1 1 Fair H T 0 1 Question: Conditioned on not having any further information, are the two coin flips Y 1 and Y 2 in this generative process independent? 12/24

45 Conditional independence in Bayes nets X P(X) 1 Fair 3 2-H T 3 X P(Y 1 = H X) P(Y 1 = T X) 1 1 Fair H T 0 1 X P(Y 2 = H X) P(Y 2 = T X) 1 1 Fair H T 0 1 Question: Conditioned on not having any further information, are the two coin flips Y 1 and Y 2 in this generative process independent? That is, if C = {}, is it the case that A B C? 12/24

46 Conditional independence in Bayes nets X P(X) 1 Fair 3 2-H T 3 X P(Y 1 = H X) P(Y 1 = T X) 1 1 Fair H T 0 1 X P(Y 2 = H X) P(Y 2 = T X) 1 1 Fair H T 0 1 Question: Conditioned on not having any further information, are the two coin flips Y 1 and Y 2 in this generative process independent? That is, if C = {}, is it the case that A B C? No! 12/24

47 Conditional independence in Bayes nets X P(X) 1 Fair 3 2-H T 3 X P(Y 1 = H X) P(Y 1 = T X) 1 1 Fair H T 0 1 X P(Y 2 = H X) P(Y 2 = T X) 1 1 Fair H T 0 1 Question: Conditioned on not having any further information, are the two coin flips Y 1 and Y 2 in this generative process independent? That is, if C = {}, is it the case that A B C? No! P(Y2 = H) = 1 2 (you can see this by symmetry) 12/24

48 Conditional independence in Bayes nets X P(X) 1 Fair 3 2-H T 3 X P(Y 1 = H X) P(Y 1 = T X) 1 1 Fair H T 0 1 X P(Y 2 = H X) P(Y 2 = T X) 1 1 Fair H T 0 1 Question: Conditioned on not having any further information, are the two coin flips Y 1 and Y 2 in this generative process independent? That is, if C = {}, is it the case that A B C? No! P(Y2 = H) = 1 2 (you can see this by symmetry) Coin was fair Coin was 2-H {}}{ 1 But P(Y2 = H Y 1 = H) = 3 1 {}}{ = /24

49 Formally assessing conditional independence in Bayes Nets The comprehensive criterion for assessing conditional independence is known as D-separation. 13/24

50 Formally assessing conditional independence in Bayes Nets The comprehensive criterion for assessing conditional independence is known as D-separation. A path between two disjoint node sets A and B is a sequence of edges connecting some node in A with some node in B 13/24

51 Formally assessing conditional independence in Bayes Nets The comprehensive criterion for assessing conditional independence is known as D-separation. A path between two disjoint node sets A and B is a sequence of edges connecting some node in A with some node in B Any node on a given path has converging arrows if two edges on the path connect to it and point to it. 13/24

52 Formally assessing conditional independence in Bayes Nets The comprehensive criterion for assessing conditional independence is known as D-separation. A path between two disjoint node sets A and B is a sequence of edges connecting some node in A with some node in B Any node on a given path has converging arrows if two edges on the path connect to it and point to it. A node on the path has non-converging arrows if two edges on the path connect to it, but at least one does not point to it. 13/24

53 Formally assessing conditional independence in Bayes Nets The comprehensive criterion for assessing conditional independence is known as D-separation. A path between two disjoint node sets A and B is a sequence of edges connecting some node in A with some node in B Any node on a given path has converging arrows if two edges on the path connect to it and point to it. A node on the path has non-converging arrows if two edges on the path connect to it, but at least one does not point to it. A third disjoint node set C d-separates A and B if for every path between A and B, either: 1. there is some node on the path with converging arrows which is not in C; or 2. there is some node on the path whose arrows do not converge and which is in C. 13/24

54 Major types of d-separation A node set C d-separates A and B if for every path between A and B, either: 1. there is some node on the path with converging arrows which is not in C; or 2. there is some node on the path whose arrows do not converge and which is in C. Commoncause d- separation (from knowing Z) Z Intervening d- separation (from knowing Y) X X Explaining away: knowing Z prevents d- separation Y D- separation in the absence of knowledge of Z X Y Y X Y Z Z Z 14/24

55 Back to our example X Y 1 Y 2 15/24

56 Back to our example X Y 1 Y 2 Without looking at the coin before flipping it, the outcome Y 1 of the first flip gives me information about the type of coin, and affects my beliefs about the outcome of Y 2 15/24

57 Back to our example X Y 1 Y 2 Without looking at the coin before flipping it, the outcome Y 1 of the first flip gives me information about the type of coin, and affects my beliefs about the outcome of Y 2 But if I look at the coin before flipping it, Y 1 and Y 2 are rendered independent 15/24

58 An example of explaining away I saw an exhibition about the, uh... 16/24

59 An example of explaining away I saw an exhibition about the, uh... There are several causes of disfluency, including: 16/24

60 An example of explaining away I saw an exhibition about the, uh... There are several causes of disfluency, including: An upcoming word is difficult to produce (e.g., low frequency, astrolabe) 16/24

61 An example of explaining away I saw an exhibition about the, uh... There are several causes of disfluency, including: An upcoming word is difficult to produce (e.g., low frequency, astrolabe) The speaker s attention was distracted by something in the non-linguistic environment 16/24

62 An example of explaining away I saw an exhibition about the, uh... There are several causes of disfluency, including: An upcoming word is difficult to produce (e.g., low frequency, astrolabe) The speaker s attention was distracted by something in the non-linguistic environment 16/24

63 An example of explaining away I saw an exhibition about the, uh... There are several causes of disfluency, including: An upcoming word is difficult to produce (e.g., low frequency, astrolabe) The speaker s attention was distracted by something in the non-linguistic environment A reasonable graphical model: W: hard word? A: attention distracted? D: disfluency? 16/24

64 An example of explaining away W: hard word? A: attention distracted? D: disfluency? Without knowledge of D, there s no reason to expect that W and A are correlated 17/24

65 An example of explaining away W: hard word? A: attention distracted? D: disfluency? Without knowledge of D, there s no reason to expect that W and A are correlated But hearing a disfluency demands a cause 17/24

66 An example of explaining away W: hard word? A: attention distracted? D: disfluency? Without knowledge of D, there s no reason to expect that W and A are correlated But hearing a disfluency demands a cause Knowing that there was a distraction explains away the disfluency, reducing the probability that the speaker was planning to utter a hard word 17/24

67 An example of the disfluency model W: hard word? A: attention distracted? Let s suppose that both hard words and distractions are unusual, the latter more so P(W = hard) = 0.25 P(A = distracted) = 0.15 D: disfluency? 18/24

68 An example of the disfluency model W: hard word? A: attention distracted? Let s suppose that both hard words and distractions are unusual, the latter more so P(W = hard) = 0.25 P(A = distracted) = 0.15 D: disfluency? Hard words and distractions both induce disfluencies; having both makes a disfluency really likely W A D=no disfluency D=disfluency easy undistracted easy distracted hard undistracted hard distracted /24

69 An example of the disfluency model W: hard word? A: attention distracted? P(W = hard) = 0.25 P(A = distracted) = 0.15 D: disfluency? W A D=no disfluency D=disfluency easy undistracted easy distracted hard undistracted hard distracted Suppose that we observe the speaker uttering a disfluency. What is P(W = hard D = disfluent)? 19/24

70 An example of the disfluency model W: hard word? A: attention distracted? P(W = hard) = 0.25 P(A = distracted) = 0.15 D: disfluency? W A D=no disfluency D=disfluency easy undistracted easy distracted hard undistracted hard distracted Suppose that we observe the speaker uttering a disfluency. What is P(W = hard D = disfluent)? Now suppose we also learn that her attention is distracted. What does that do to our beliefs about W 19/24

71 An example of the disfluency model W: hard word? A: attention distracted? P(W = hard) = 0.25 P(A = distracted) = 0.15 D: disfluency? W A D=no disfluency D=disfluency easy undistracted easy distracted hard undistracted hard distracted Suppose that we observe the speaker uttering a disfluency. What is P(W = hard D = disfluent)? Now suppose we also learn that her attention is distracted. What does that do to our beliefs about W That is, what is P(W = hard D = disfluent,a = distracted)? 19/24

72 An example of the disfluency model Fortunately, there is automated machinery to turn the Bayesian crank : P(W = hard) = /24

73 An example of the disfluency model Fortunately, there is automated machinery to turn the Bayesian crank : P(W = hard) = 0.25 P(W = hard D = disfluent) = /24

74 An example of the disfluency model Fortunately, there is automated machinery to turn the Bayesian crank : P(W = hard) = 0.25 P(W = hard D = disfluent) = 0.57 P(W = hard D = disfluent,a = distracted) = /24

75 An example of the disfluency model Fortunately, there is automated machinery to turn the Bayesian crank : P(W = hard) = 0.25 P(W = hard D = disfluent) = 0.57 P(W = hard D = disfluent,a = distracted) = 0.40 Knowing that the speaker was distracted (A) decreased the probability that the speaker was about to utter a hard word (W) A explained D away. 20/24

76 An example of the disfluency model Fortunately, there is automated machinery to turn the Bayesian crank : P(W = hard) = 0.25 P(W = hard D = disfluent) = 0.57 P(W = hard D = disfluent,a = distracted) = 0.40 Knowing that the speaker was distracted (A) decreased the probability that the speaker was about to utter a hard word (W) A explained D away. A caveat: the type of relationship among A, W, and D will depend on the values one finds in the probability table! P(W) P(A) P(D W,A) 20/24

77 Summary thus far Key points: Bayes Rule is a compelling framework for modeling inference under uncertainty DAGs/Bayes Nets are a broad class of models for specifying joint probability distributions with conditional independencies Classic Bayes Net references:??;?;?, Chapter 14;?, Chapter 8. 21/24

78 References I 22/24

79 An example of the disfluency model P(W = hard D = disfluent,a = distracted) hard W=hard easy W=easy disfl D=disfluent distr A=distracted undistr A=undistracted P(hard disfl,distr) = P(disfl hard,distr)p(hard distr) P(disfl distr) = P(disfl hard,distr)p(hard) P(disfl distr) P(disfl distr) = w P(disfl W = w )P(W = w ) (Bayes Rule) (Independence from the DAG) (Marginalization) = P(disfl hard)p(hard) + P(disfl easy)p(easy) = = P(hard disfl,distr) = = /24

80 An example of the disfluency model P(W = hard D = disfluent) P(hard disfl) = P(disfl hard)p(hard) P(disfl) (Bayes Rule) P(disfl hard) = a P(disfl A = a,hard)p(a = a hard) = P(disfl A = distr, hard)p(a = distr hard) + P(disfl undistr, hard)p(undistr hard) = = P(disfl) = P(disfl W = w )P(W = w ) w = P(disfl hard)p(hard) + P(disfl easy)p(easy) P(disfl easy) = a P(disfl A = a,easy)p(a = a easy) = P(disfl A = distr, easy)p(a = distr easy) + P(disfl undistr, easy)p(undistr easy) = = P(disfl) = = P(hard disfl) = = /24

81 Bayesian parameter estimation The scenario: you are a native English speaker in whose experience passivizable constructions are passivized with frequency q. 1. The ball hit the window. (Active) 2. The window was hit by the ball. (Passive) You encounter a new dialect of English and hear data y consisting of n passivizable utterances, m of which were passivized: Goal: X Bern(π) Estimate the success parameter π associated with passivization in the new English dialect; Or place a probability distribution on the number of passives in the next N passivizable utterances.

82 Anatomy of Bayesian inference Simplest possible scenario: I θ Y

83 Anatomy of Bayesian inference Simplest possible scenario: I θ Y The corresponding Bayesian inference: P(θ y,i) = P(y θ,i)p(θ I) P(y I)

84 Anatomy of Bayesian inference Simplest possible scenario: I θ Y The corresponding Bayesian inference: P(θ y,i) = P(y θ,i)p(θ I) P(y I) Likelihood for θ Prior over θ {}}{{}}{ P(y θ) P(θ I) = P(y I) }{{} Likelihood marginalized over θ (because y I θ)

85 Anatomy of Bayesian inference Simplest possible scenario: I θ Y The corresponding Bayesian inference: P(θ y,i) = P(y θ,i)p(θ I) P(y I) Likelihood for θ Prior over θ {}}{{}}{ P(y θ) P(θ I) = P(y I) }{{} Likelihood marginalized over θ (because y I θ) At the bottom of the graph, our model is the binomial distribution: P(y θ) Binom(n, θ) But to get things going we have to set the prior P(θ I).

86 Priors for the binomial distribution For a model with parameters θ, a prior distribution is just some joint probability distribution P(θ) Because the prior is often supposed to account for knowledge we bring to the table, we often write P(θ I) to be explciit

87 Priors for the binomial distribution For a model with parameters θ, a prior distribution is just some joint probability distribution P(θ) Because the prior is often supposed to account for knowledge we bring to the table, we often write P(θ I) to be explciit Model parameters are nearly always real-valued, so P(θ) is generally a multivariate continuous distribution

88 Priors for the binomial distribution For a model with parameters θ, a prior distribution is just some joint probability distribution P(θ) Because the prior is often supposed to account for knowledge we bring to the table, we often write P(θ I) to be explciit Model parameters are nearly always real-valued, so P(θ) is generally a multivariate continuous distribution In general, the sky is the limit as to what you choose for P(θ)

89 Priors for the binomial distribution For a model with parameters θ, a prior distribution is just some joint probability distribution P(θ) Because the prior is often supposed to account for knowledge we bring to the table, we often write P(θ I) to be explciit Model parameters are nearly always real-valued, so P(θ) is generally a multivariate continuous distribution In general, the sky is the limit as to what you choose for P(θ) But in many cases there are useful priors that will make your life easier

90 The beta distribution The beta distribution has two parameters α 1,α 2 > 0 and is defined as: P(π α 1,α 2 ) = 1 B(α 1,α 2 ) πα 1 1 (1 π) α 2 1 (0 π 1,α 1 > 0,α 2 > 0) where the beta function B(α 1,α 2 ) serves as a normalizing constant: asdf B(α 1,α 2 ) = 1 0 π α 1 1 (1 π) α 2 1 dπ

91 Some beta distributions If X B(α 1,α 2 ): p(π) Beta(1,1) Beta(0.5,0.5) Beta(3,3) Beta(3,0.5) E[X] = α 1 α 1 +α 2 If α 1,α 2 > 1, then X has a mode at α 1 1 α 1 +α π

92 Using the beta distribution as a prior 1. The ball hit the window. (Active) 2. The window was hit by the ball. (Passive) Let us use a beta distribution as a prior for our problem hence I = α 1,α 2.

93 Using the beta distribution as a prior 1. The ball hit the window. (Active) 2. The window was hit by the ball. (Passive) Let us use a beta distribution as a prior for our problem hence I = α 1,α 2. P(π y,α 1,α 2 ) = P(y π)p(π α 1,α 2 ) P(y α 1,α 2 ) (1)

94 Using the beta distribution as a prior 1. The ball hit the window. (Active) 2. The window was hit by the ball. (Passive) Let us use a beta distribution as a prior for our problem hence I = α 1,α 2. P(π y,α 1,α 2 ) = P(y π)p(π α 1,α 2 ) P(y α 1,α 2 ) Since the denominator is not a function of π, it is a normalizing constant. Ignore it and work in terms of proportionality: P(π y,α 1,α 2 ) P(y π)p(π α 1,α 2 ) (1)

95 Using the beta distribution as a prior 1. The ball hit the window. (Active) 2. The window was hit by the ball. (Passive) Let us use a beta distribution as a prior for our problem hence I = α 1,α 2. P(π y,α 1,α 2 ) = P(y π)p(π α 1,α 2 ) P(y α 1,α 2 ) Since the denominator is not a function of π, it is a normalizing constant. Ignore it and work in terms of proportionality: P(π y,α 1,α 2 ) P(y π)p(π α 1,α 2 ) Likelihood for the binomial distribution is ( ) n P(y π) = π m (1 π) n m m (1)

96 Using the beta distribution as a prior 1. The ball hit the window. (Active) 2. The window was hit by the ball. (Passive) Let us use a beta distribution as a prior for our problem hence I = α 1,α 2. P(π y,α 1,α 2 ) = P(y π)p(π α 1,α 2 ) P(y α 1,α 2 ) Since the denominator is not a function of π, it is a normalizing constant. Ignore it and work in terms of proportionality: P(π y,α 1,α 2 ) P(y π)p(π α 1,α 2 ) Likelihood for the binomial distribution is ( ) n P(y π) = π m (1 π) n m m Beta prior is P(π α 1,α 2 ) = 1 B(α 1,α 2 ) πα 1 1 (1 π) α 2 1 (1)

97 Using the beta distribution as a prior Ignore ( n m) and B(α1,α 2 ) (both constant in π): Likelihood Prior {}}{{}}{ P(π y,α 1,α 2 ) π m (1 π) n m π α1 1 (1 π) α 2 1

98 Using the beta distribution as a prior Ignore ( n m) and B(α1,α 2 ) (both constant in π): Likelihood Prior {}}{{}}{ P(π y,α 1,α 2 ) π m (1 π) n m π α1 1 (1 π) α 2 1 π m+α 1 1 (1 π) n m+α 2 1

99 Using the beta distribution as a prior Ignore ( n m) and B(α1,α 2 ) (both constant in π): Likelihood Prior {}}{{}}{ P(π y,α 1,α 2 ) π m (1 π) n m π α1 1 (1 π) α 2 1 π m+α 1 1 (1 π) n m+α 2 1 Crucial trick: this is itself a beta distribution! Recall that if θ Beta(α 1,α 2 ) then P(θ) = 1 B(α 1,α 2 ) πα 1 1 (1 π) α 2 1

100 Using the beta distribution as a prior Ignore ( n m) and B(α1,α 2 ) (both constant in π): Likelihood Prior {}}{{}}{ P(π y,α 1,α 2 ) π m (1 π) n m π α1 1 (1 π) α 2 1 π m+α 1 1 (1 π) n m+α 2 1 Crucial trick: this is itself a beta distribution! Recall that if θ Beta(α 1,α 2 ) then P(θ) = 1 B(α 1,α 2 ) πα 1 1 (1 π) α 2 1 Hence P(θ y,α 1,α 2 ) is distributed as Beta(α 1 +m,α 2 +n m).

101 Using the beta distribution as a prior Ignore ( n m) and B(α1,α 2 ) (both constant in π): Likelihood Prior {}}{{}}{ P(π y,α 1,α 2 ) π m (1 π) n m π α1 1 (1 π) α 2 1 π m+α 1 1 (1 π) n m+α 2 1 Crucial trick: this is itself a beta distribution! Recall that if θ Beta(α 1,α 2 ) then P(θ) = 1 B(α 1,α 2 ) πα 1 1 (1 π) α 2 1 Hence P(θ y,α 1,α 2 ) is distributed as Beta(α 1 +m,α 2 +n m). With a beta prior and a binomial likelihood, the posterior is still beta-distributed. This is called conjugacy.

102 Using our beta-binomial model Goal: Estimate the success parameter π associated with passivization in the new English dialect; Or place a probability distribution on the number of passives in the next N passivizable utterances.

103 Using our beta-binomial model Goal: Estimate the success parameter π associated with passivization in the new English dialect; Or place a probability distribution on the number of passives in the next N passivizable utterances. To estimate π it is common to use Maximum a-posteriori (MAP) estimation: choose the value of π with highest posterior probability

104 Using our beta-binomial model Goal: Estimate the success parameter π associated with passivization in the new English dialect; Or place a probability distribution on the number of passives in the next N passivizable utterances. To estimate π it is common to use Maximum a-posteriori (MAP) estimation: choose the value of π with highest posterior probability P(passive passivizable clause) 0.08 (Roland et al., 2007)

105 Using our beta-binomial model Goal: Estimate the success parameter π associated with passivization in the new English dialect; Or place a probability distribution on the number of passives in the next N passivizable utterances. To estimate π it is common to use Maximum a-posteriori (MAP) estimation: choose the value of π with highest posterior probability P(passive passivizable clause) 0.08 (Roland et al., 2007) The mode of a beta distribution is α 1 1 α 1 +α 2 2

106 Using our beta-binomial model Goal: Estimate the success parameter π associated with passivization in the new English dialect; Or place a probability distribution on the number of passives in the next N passivizable utterances. To estimate π it is common to use Maximum a-posteriori (MAP) estimation: choose the value of π with highest posterior probability P(passive passivizable clause) 0.08 (Roland et al., 2007) The mode of a beta distribution is α 1 1 α 1 +α 2 2 Hence we might use α 1 = 3,α 2 = 24 (note that 2 25 = 0.08)

107 Using our beta-binomial model Goal: Estimate the success parameter π associated with passivization in the new English dialect; Or place a probability distribution on the number of passives in the next N passivizable utterances. To estimate π it is common to use Maximum a-posteriori (MAP) estimation: choose the value of π with highest posterior probability P(passive passivizable clause) 0.08 (Roland et al., 2007) The mode of a beta distribution is α 1 1 α 1 +α 2 2 Hence we might use α 1 = 3,α 2 = 24 (note that 2 25 = 0.08) Suppose that n = 7,m = 2: our posterior will be Beta(5,29), hence ˆπ = 4 32 = 0.125

108 Beta-binomial posterior distributions P(p) Prior Likelihood (n=7) Posterior (n=7) Likelihood (n=21) Posterior (n=21) Likelihood(p) p

109 Fully Bayesian density estimation Goal: Estimate the success parameter π associated with passivization in the new English dialect; Or place a probability distribution on the number of passives in the next N passivizable utterances.

110 Fully Bayesian density estimation Goal: Estimate the success parameter π associated with passivization in the new English dialect; Or place a probability distribution on the number of passives in the next N passivizable utterances. In the fully Bayesian view, we don t summarize our posterior beliefs into a point estimate; rather, we marginalize over them in predicting the future:

111 Fully Bayesian density estimation Goal: Estimate the success parameter π associated with passivization in the new English dialect; Or place a probability distribution on the number of passives in the next N passivizable utterances. In the fully Bayesian view, we don t summarize our posterior beliefs into a point estimate; rather, we marginalize over them in predicting the future: P(y new y,i) = P(y new θ)p(θ y,i)dθ θ

112 Fully Bayesian density estimation Goal: Estimate the success parameter π associated with passivization in the new English dialect; Or place a probability distribution on the number of passives in the next N passivizable utterances. In the fully Bayesian view, we don t summarize our posterior beliefs into a point estimate; rather, we marginalize over them in predicting the future: P(y new y,i) = P(y new θ)p(θ y,i)dθ θ This leads to the beta-binomial predictive model: ( ) k B(α1 +m+r,α 2 +n m+k r) P(r k,i,y) = r B(α 1 +m,α 2 +n m)

113 Fully Bayesian density estimation P(k passives out of 50 trials y, I) Binomial Beta Binomial k

114 Fully Bayesian density estimation In this case (as in many others), marginalizing over the model parameters allows for greater dispersion in the model s predictions

115 Fully Bayesian density estimation In this case (as in many others), marginalizing over the model parameters allows for greater dispersion in the model s predictions This is because the new observations are only conditionally independent given θ with uncertainty about θ, they are linked!

116 Fully Bayesian density estimation In this case (as in many others), marginalizing over the model parameters allows for greater dispersion in the model s predictions This is because the new observations are only conditionally independent given θ with uncertainty about θ, they are linked! I θ y (1) new y (2) new... y (N) new

117 References I Roland, D., Dick, F., and Elman, J. L. (2007). Frequency of basic English grammatical structures: A corpus analysis. Journal of Memory and Language, 57: