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PGM: Tirgul 10 Parameter Learning and Priors

PGM: Tirgul 10 Parameter Learning and Priors. Why learning?. Knowledge acquisition bottleneck Knowledge acquisition is an expensive process Often we don’t have an expert Data is cheap Vast amounts of data becomes available to us Learning allows us to build systems based on the data. E. B.

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PGM: Tirgul 10 Parameter Learning and Priors

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  1. PGM: Tirgul 10Parameter Learningand Priors .

  2. Why learning? Knowledge acquisition bottleneck • Knowledge acquisition is an expensive process • Often we don’t have an expert Data is cheap • Vast amounts of data becomes available to us • Learning allows us to build systems based on the data

  3. E B P(A | E,B) B E .9 .1 e b e b .7 .3 .8 .2 e b R A .99 .01 e b C Data + Prior information Learning Bayesian networks Inducer

  4. E B P(A | E,B) .9 .1 e b e b .7 .3 .8 .2 e b .99 .01 e b E B P(A | E,B) B B E E ? ? e b A A e b ? ? ? ? e b ? ? e b Known Structure -- Complete Data E, B, A <Y,N,N> <Y,Y,Y> <N,N,Y> <N,Y,Y> . . <N,Y,Y> • Network structure is specified • Inducer needs to estimate parameters • Data does not contain missing values Inducer

  5. E B P(A | E,B) .9 .1 e b e b .7 .3 .8 .2 e b B E .99 .01 e b A E B P(A | E,B) B E ? ? e b A e b ? ? ? ? e b ? ? e b Unknown Structure -- Complete Data E, B, A <Y,N,N> <Y,Y,Y> <N,N,Y> <N,Y,Y> . . <N,Y,Y> • Network structure is not specified • Inducer needs to select arcs & estimate parameters • Data does not contain missing values Inducer

  6. E B P(A | E,B) .9 .1 e b e b .7 .3 .8 .2 e b .99 .01 e b E B P(A | E,B) B B E E ? ? e b A A e b ? ? ? ? e b ? ? e b Known Structure -- Incomplete Data E, B, A <Y,N,N> <Y,?,Y> <N,N,Y> <N,Y,?> . . <?,Y,Y> • Network structure is specified • Data contains missing values • We consider assignments to missing values Inducer

  7. Known Structure / Complete Data • Given a network structure G • And choice of parametric family for P(Xi|Pai) • Learn parameters for network Goal • Construct a network that is “closest” to probability that generated the data

  8. Example: Binomial Experiment(Statistics 101) • When tossed, it can land in one of two positions: Head or Tail • We denote by  the (unknown) probability P(H). Estimation task: • Given a sequence of toss samples x[1], x[2], …, x[M] we want to estimate the probabilities P(H)=  and P(T) = 1 -  Head Tail

  9. i.i.d. samples Statistical Parameter Fitting • Consider instances x[1], x[2], …, x[M] such that • The set of values that x can take is known • Each is sampled from the same distribution • Each sampled independently of the rest • The task is to find a parameter  so that the data can be summarized by a probability P(x[j]|  ). • Depends on the given family of probability distributions: multinomial, Gaussian, Poisson, etc. • For now, focus on multinomial distributions

  10. L( :D)  0 0.2 0.4 0.6 0.8 1 The Likelihood Function • How good is a particular ?It depends on how likely it is to generate the observed data • The likelihood for the sequence H,T, T, H, H is

  11. Sufficient Statistics • To compute the likelihood in the thumbtack example we only require NH and NT (the number of heads and the number of tails) • NH and NT are sufficient statistics for the binomial distribution

  12. Sufficient Statistics • A sufficient statistic is a function of the data that summarizes the relevant information for the likelihood • Formally, s(D) is a sufficient statistics if for any two datasets D and D’ • s(D) = s(D’ ) L( |D) = L( |D’) Datasets Statistics

  13. Maximum Likelihood Estimation MLE Principle: Choose parameters that maximize the likelihood function • This is one of the most commonly used estimators in statistics • Intuitively appealing

  14. L(:D) 0 0.2 0.4 0.6 0.8 1 Example: MLE in Binomial Data • Applying the MLE principle we get (Which coincides with what one would expect) Example: (NH,NT ) = (3,2) MLE estimate is 3/5 = 0.6

  15. B E A C Learning Parameters for a Bayesian Network • Training data has the form:

  16. B E A C Learning Parameters for a Bayesian Network • Since we assume i.i.d. samples,likelihood function is

  17. B E A C Learning Parameters for a Bayesian Network • By definition of network, we get

  18. B E A C Learning Parameters for a Bayesian Network • Rewriting terms, we get

  19. General Bayesian Networks Generalizing for any Bayesian network: • The likelihood decomposes according to the structure of the network. i.i.d. samples Network factorization

  20. General Bayesian Networks (Cont.) Decomposition  Independent Estimation Problems If the parameters for each family are not related, then they can be estimated independently of each other.

  21. From Binomial to Multinomial • For example, suppose X can have the values 1,2,…,K • We want to learn the parameters 1, 2. …, K Sufficient statistics: • N1, N2, …, NK - the number of times each outcome is observed Likelihood function: MLE:

  22. Likelihood for Multinomial Networks • When we assume that P(Xi | Pai) is multinomial, we get further decomposition:

  23. Likelihood for Multinomial Networks • When we assume that P(Xi | Pai) is multinomial, we get further decomposition: • For each value paiof the parents of Xi we get an independent multinomial problem • The MLE is

  24. Maximum Likelihood Estimation Consistency • Estimate converges to best possible value as the number of examples grow • To make this formal, we need to introduce some definitions

  25. KL-Divergence • Let P and Q be two distributions over X • A measure of distance between P and Q is the Kullback-Leibler Divergence • KL(P||Q) = 1 (when logs are in base 2) = • The probability P assigns to an instance is, on average, half the probability Q assigns to it • KL(P||Q)  0 • KL(P||Q) = 0 iff are P and Q equal

  26. Consistency • Let P(X| ) be a parametric family • We need to make various regularity condition we won’t go into now • Let P*(X) be the distribution that generates the data • Let be the MLE estimate given a dataset D Thm • As N , where with probability 1

  27. Consistency -- Geometric Interpretation P* Distributions that canrepresented by P(X| ) P(X| * ) Space of probability distribution

  28. Is MLE all we need? • Suppose that after 10 observations, • ML estimates P(H) = 0.7 for the thumbtack • Would you bet on heads for the next toss? • Suppose now that after 10 observations, • ML estimates P(H) = 0.7 for a coin • Would you place the same bet?

  29. Bayesian Inference Frequentist Approach: • Assumes there is an unknown but fixed parameter  • Estimates  with some confidence • Prediction by using the estimated parameter value Bayesian Approach: • Represents uncertainty about the unknown parameter • Uses probability to quantify this uncertainty: • Unknown parameters as random variables • Prediction follows from the rules of probability: • Expectation over the unknown parameters

  30. X[1] X[2] X[m] X[m+1] Observed data Query Bayesian Inference (cont.) • We can represent our uncertainty about the sampling process using a Bayesian network • The values of X are independent given  • The conditional probabilities, P(x[m] | ), are the parameters in the model • Prediction is now inference in this network

  31. X[1] X[2] X[m] X[m+1] Bayesian Inference (cont.) Prediction as inference in this network where Likelihood Prior Posterior Probability of data

  32. 0 0.2 0.4 0.6 0.8 1 Example: Binomial Data Revisited • Prior: uniform for in [0,1] • P() = 1 • Then P(|D) is proportional to the likelihood L(:D) (NH,NT) = (4,1) • MLE for P(X = H ) is 4/5 = 0.8 • Bayesian prediction is

  33. Bayesian Inference and MLE • In our example, MLE and Bayesian prediction differ But… If prior is well-behaved • Does not assign 0 density to any “feasible” parameter value Then: both MLE and Bayesian prediction converge to the same value • Both are consistent

  34. Dirichlet Priors • Recall that the likelihood function is • A Dirichlet prior with hyperparameters 1,…,K is defined as for legal 1,…, K Then the posterior has the same form, with hyperparameters 1+N 1,…,K +N K

  35. Dirichlet Priors (cont.) • We can compute the prediction on a new event in closed form: • If P() is Dirichlet with hyperparameters 1,…,K then • Since the posterior is also Dirichlet, we get

  36. Dirichlet Priors -- Example 5 Dirichlet(1,1) Dirichlet(2,2) 4.5 Dirichlet(0.5,0.5) Dirichlet(5,5) 4 3.5 3 2.5 2 1.5 1 0.5 0 0 0.2 0.4 0.6 0.8 1

  37. Prior Knowledge • The hyperparameters 1,…,K can be thought of as “imaginary” counts from our prior experience • Equivalent sample size = 1+…+K • The larger the equivalent sample size the more confident we are in our prior

  38. 0.55 0.6 Different strength H + T Fixed ratio H / T 0.5 Fixed strength H + T Different ratio H / T 0.5 0.45 0.4 0.4 0.35 0.3 0.3 0.2 0.25 0.1 0.2 0.15 0 0 20 40 60 80 100 0 20 40 60 80 100 Effect of Priors Prediction of P(X=H ) after seeing data with NH = 0.25•NT for different sample sizes

  39. MLE Dirichlet(.5,.5) Dirichlet(1,1) Dirichlet(5,5) Dirichlet(10,10) Effect of Priors (cont.) • In real data, Bayesian estimates are less sensitive to noise in the data 0.7 0.6 0.5 P(X = 1|D) 0.4 0.3 0.2 N 0.1 5 10 15 20 25 30 35 40 45 50 1 Toss Result 0 N

  40. Conjugate Families • The property that the posterior distribution follows the same parametric form as the prior distribution is called conjugacy • Dirichlet prior is a conjugate family for the multinomial likelihood • Conjugate families are useful since: • For many distributions we can represent them with hyperparameters • They allow for sequential update within the same representation • In many cases we have closed-form solution for prediction

  41. Y|X X X m Y|X X[m] X[M] X[M+1] X[1] X[2] Y[m] Y[M] Y[M+1] Y[1] Y[2] Plate notation Query Observed data Bayesian Networks and Bayesian Prediction • Priors for each parameter group are independent • Data instances are independent given the unknown parameters

  42. Y|X X X m Y|X X[m] X[M] X[M+1] X[1] X[2] Y[m] Y[M] Y[M+1] Y[1] Y[2] Plate notation Query Observed data Bayesian Networks and Bayesian Prediction (Cont.) • We can also “read” from the network: Complete data  posteriors on parameters are independent

  43. X X m Y|X m Refined model Y|X=0 X[m] X[m] Y|X=1 Y[m] Y[m] Bayesian Prediction(cont.) • Since posteriors on parameters for each family are independent, we can compute them separately • Posteriors for parameters within families are also independent: • Complete data independent posteriors on Y|X=0 and Y|X=1

  44. Bayesian Prediction(cont.) • Given these observations, we can compute the posterior for each multinomial Xi | pai independently • The posterior is Dirichlet with parameters (Xi=1|pai)+N (Xi=1|pai),…, (Xi=k|pai)+N (Xi=k|pai) • The predictive distribution is then represented by the parameters

  45. Assessing Priors for Bayesian Networks We need the(xi,pai) for each node xj • We can use initial parameters 0 as prior information • Need also an equivalent sample size parameter M0 • Then, we let (xi,pai) = M0P(xi,pai|0) • This allows to update a network using new data

  46. Learning Parameters: Case Study (cont.) Experiment: • Sample a stream of instances from the alarm network • Learn parameters using • MLE estimator • Bayesian estimator with uniform prior with different strengths

  47. MLE Bayes w/ Uniform Prior, M'=5 Bayes w/ Uniform Prior, M'=10 Bayes w/ Uniform Prior, M'=20 Bayes w/ Uniform Prior, M'=50 Learning Parameters: Case Study (cont.) 1.4 1.2 1 0.8 KL Divergence 0.6 0.4 0.2 0 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 M

  48. Bayesian (Dirichlet) MLE Learning Parameters: Summary • Estimation relies on sufficient statistics • For multinomial these are of the form N (xi,pai) • Parameter estimation • Bayesian methods also require choice of priors • Both MLE and Bayesian are asymptotically equivalent and consistent • Both can be implemented in an on-line manner by accumulating sufficient statistics

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