an introduction to bayesian inference

Transcription

1 with an application to network analysis january 13, 2010

2 motivation would like models that: provide predictive and explanatory power are complex enough to describe observed phenomena are simple enough to generalize to future observations

3 motivation would like models that: provide predictive and explanatory power are complex enough to describe observed phenomena are simple enough to generalize to future observations claim: provides a systematic framework to infer such models from observed data

4 motivation principles behind bayesian interpretation of probability and are well established (bayes, laplace, etc., 18th century) + recent advances in mathematical techniques and computational resources have enabled successful applications of these principles to real-world problems

5 motivation: a bayesian approach to network modularity

6 outline principles 1 principles (what we d like to do) : joint, marginal, and conditional probabilities : inverting conditional probabilities : unknowns as random variables : + 2 (what we re able to do) monte carlo methods: representative samples variational methods: bound optimization references 3

7 joint, marginal, and conditional probabilities joint distribution p XY (X = x, Y = y): probability X = x and Y = y conditional distribution p X Y (X = x Y = y): probability X = x given Y = y marginal distribution p X (X ): probability X = x (regardless of Y )

8 sum and product rules sum rule sum out settings of irrelevant variables: p (x) = y Ω Y p (x, y) (1) product rule the joint as the product of the conditional and marginal: p (x, y) = p (x y) p (y) (2) = p (y x) p (x) (3)

9 outline principles 1 principles (what we d like to do) : joint, marginal, and conditional probabilities : inverting conditional probabilities : unknowns as random variables : + 2 (what we re able to do) monte carlo methods: representative samples variational methods: bound optimization references 3

10 inverting conditional probabilities equate far right- and left-hand sides of product rule p (y x) p (x) = p (x, y) = p (x y) p (y) (4) and divide: (bayes and price 1763) the probability of Y given X from the probability of X given Y : p (y x) = p (x y) p (y) p (x) (5) where p (x) = y Ω Y p (x y) p (y) is the normalization constant

11 example: diagnoses a la bayes population 10,000 1% has (rare) disease 1 test is 99% (relatively) effective, i.e. given a patient is sick, 99% test positive given a patient is healthy, 99% test negative 1 subtlety: assuming this fraction is known

12 example: diagnoses a la bayes population 10,000 1% has (rare) disease test is 99% (relatively) effective, i.e. given a patient is sick, 99% test positive given a patient is healthy, 99% test negative given positive test, what is probability the patient is sick? 1 1 follows wiggins (2006)

13 example: diagnoses a la bayes sick population (100 ppl) healthy population (9900 ppl) 1% (99 ppl test +) 1% (1 ppl test -) 99% (99 ppl test +) 99% (9801 ppl test -) 99 sick patients test positive, 99 healthy patients test positive

14 example: diagnoses a la bayes sick population (100 ppl) healthy population (9900 ppl) 1% (99 ppl test +) 1% (1 ppl test -) 99% (99 ppl test +) 99% (9801 ppl test -) 99 sick patients test positive, 99 healthy patients test positive given positive test, 50% probability that patient is sick

15 example: diagnoses a la bayes know probability of testing positive/negative given sick/healthy use to invert to probability of sick/healthy given positive/negative test 99/100 1/100 {}}{{}}{ p (test + sick) p (sick) p (sick test +) = p (test +) }{{} 99/ /100 2 =198/100 2 = = 1 2 (6)

16 example: diagnoses a la bayes know probability of testing positive/negative given sick/healthy use to invert to probability of sick/healthy given positive/negative test 99/100 1/100 {}}{{}}{ p (test + sick) p (sick) p (sick test +) = p (test +) }{{} 99/ /100 2 =198/100 2 = = 1 2 (6) most work in calculating denominator (normalization)

17 outline principles 1 principles (what we d like to do) : joint, marginal, and conditional probabilities : inverting conditional probabilities : unknowns as random variables : + 2 (what we re able to do) monte carlo methods: representative samples variational methods: bound optimization references 3

18 interpretations of probabilities (just enough philosophy) frequentists: limit of relative frequency of events for large number of trials bayesians: measure of a state of knowledge, quantifying degrees of belief (jaynes 2003)

19 interpretations of probabilities (just enough philosophy) frequentists: limit of relative frequency of events for large number of trials bayesians: measure of a state of knowledge, quantifying degrees of belief (jaynes 2003) key difference: bayesians permit assignment of probabilities to unknown/unobservable hypotheses (frequentists do not)

20 interpretations of probabilities (just enough philosophy) e.g., inferring model parameters Θ from observed data D: frequentist approach: calculate parameter setting that maximizes likelihood of observed data (point estimate), Θ = argmax p(d Θ) (7) Θ bayesian approach: calculate distribution over parameter settings given data, p(θ D) =? (8)

21 interpretations of probabilities (just enough philosophy) using bayes rule being bayesian

22 interpretations of probabilities (just enough philosophy) using bayes rule being bayesian s/bayesian/subjective probabilist/g

23 outline principles 1 principles (what we d like to do) : joint, marginal, and conditional probabilities : inverting conditional probabilities : unknowns as random variables : + 2 (what we re able to do) monte carlo methods: representative samples variational methods: bound optimization references 3

24 + : treat unknown quantities as random variables use to systematically update prior knowledge in the presence of observed data posterior {}}{ p(θ D) = likelihood prior {}}{{}}{ p(d Θ) p(θ) p(d) }{{} evidence (9)

25 example: coin flipping observe independent coin flips (bernoulli trials) infer distribution over coin bias

26 example: coin flipping prior p(θ) over coin bias before observing flips

27 example: coin flipping observe flips: HTHHHTTHHHH

28 example: coin flipping update posterior p(θ D) using

29 example: coin flipping observe flips: HHHHHHHHHHHHTHHHHHHHHHH HHHHHHHHHHHHHHHHHHHHHHH HHHHHHTHHHHHHHHHHHHHHHH HHHHHHHHHHHHHHHHHHHHHHH HHHHHHHHT

30 example: coin flipping update posterior p(θ D) using

31 naive bayes for document classifiction model presence/absence of each word as independent coin flip p (word class) = Bernoulli(θ wc ) (10) p (words class) = p (word 1 class) p (word 2 class)...(11)

32 naive bayes for document classifiction model presence/absence of each word as independent coin flip p (word class) = Bernoulli(θ wc ) (10) p (words class) = p (word 1 class) p (word 2 class)...(11) maximum likelihood estimates of probabilities from word and class counts ˆθ wc = N wc N c (12)

33 naive bayes for document classifiction model presence/absence of each word as independent coin flip p (word class) = Bernoulli(θ wc ) (10) p (words class) = p (word 1 class) p (word 2 class)...(11) maximum likelihood estimates of probabilities from word and class counts ˆθ wc = N wc N c (12) use bayes rule to calculate distribution over classes given words p (class words, Θ) = p (words class, Θ) p (class, Θ) p (words, Θ) (13)

34 naive bayes for document classification example: spam filtering for enron using one word 2 2 code: lecture 03/enron naive bayes.sh

35 naive bayes for document classification example: spam filtering for enron using one word 2 2 code: lecture 03/enron naive bayes.sh

36 naive bayes for document classification example: spam filtering for enron using one word 2 2 code: lecture 03/enron naive bayes.sh

37 naive bayes for document classification example: spam filtering for enron using one word 2 2 code: lecture 03/enron naive bayes.sh

38 naive bayes for document classification example: spam filtering for enron using one word 2 guard against overfitting by smoothing counts, equivalent to maximum a posteriori (map) inference ˆθ wc = N wc + α N c + α + β (14) 2 code: lecture 03/enron naive bayes.sh

39 naive bayes for document classification naive bayes for document classification is neither naive nor bayesian! not so naive: works well in, not in theory not bayesian: point estimates ˆθ wc for parameters rather than distributions over parameters

40 quantities of interest maintains full posterior distributions over unknowns many quantities of interest require expectations under these posteriors, e.g. posterior mean and predictive distribution: Θ = E p(θ D) [Θ] = dθ Θ p(θ D) (15) p (x D) = E p(θ D) [p (x Θ, D)] = dθ p (x Θ, D) p(θ D) (16)

41 quantities of interest maintains full posterior distributions over unknowns many quantities of interest require expectations under these posteriors, e.g. posterior mean and predictive distribution: Θ = E p(θ D) [Θ] = dθ Θ p(θ D) (15) p (x D) = E p(θ D) [p (x Θ, D)] = dθ p (x Θ, D) p(θ D) often can t compute posterior (normalization), let alone expectations with respect to it approximation methods (16)

42 outline principles sampling methods variational methods references 1 principles (what we d like to do) : joint, marginal, and conditional probabilities : inverting conditional probabilities : unknowns as random variables : + 2 (what we re able to do) monte carlo methods: representative samples variational methods: bound optimization references 3

43 representative samples sampling methods variational methods references general approach: approximate intractable expectations via sum over representative samples 3 Φ = E p(x) [φ(x)] = dx φ(x) }{{} arbitrary function p(x) }{{} target density (17) 3 follows mackay (2003)

44 representative samples sampling methods variational methods references general approach: approximate intractable expectations via sum over representative samples 3 Φ = E p(x) [φ(x)] = dx φ(x) }{{} arbitrary function p(x) }{{} target density (17) Φ = 1 R R φ(x (r) ) (18) r=1 3 follows mackay (2003)

45 representative samples sampling methods variational methods references general approach: approximate intractable expectations via sum over representative samples 3 Φ = E p(x) [φ(x)] = dx φ(x) }{{} arbitrary function p(x) }{{} target density (17) Φ = 1 R R φ(x (r) ) (18) r=1 shifts problem to finding good samples 3 follows mackay (2003)

46 representative samples sampling methods variational methods references further complication: in general we can only evaluate the target density to within a multiplicative (normalization) constant, i.e. p(x) = p (x) Z and p (x (r) ) can be evaluated with Z unknown (19)

47 sampling methods sampling methods variational methods references monte carlo methods uniform sampling importance sampling rejection sampling... markov chain monte carlo (mcmc) methods metropolis-hastings gibbs sampling...

48 outline principles sampling methods variational methods references 1 principles (what we d like to do) : joint, marginal, and conditional probabilities : inverting conditional probabilities : unknowns as random variables : + 2 (what we re able to do) monte carlo methods: representative samples variational methods: bound optimization references 3

49 bound optimization sampling methods variational methods references general approach: replace integration with optimization construct auxiliary function upper-bounded by log-evidence, maximize auxiliary 9.4. The EM function Algorithm in General 453 he EM algorithm involves alterately computing a lower bound n the log likelihood for the curent parameter values and then aximizing this bound to obtain he new parameter values. See he text for a full discussion. ln p(x θ) L (q,θ) θ old θ new 4 omplete data) 4 image log likelihood from bishop function (2006) whose value we wish to maximize. We start ith some initial parameter value θ old, and injake the hofman first E step we an introduction evaluate thetoposte-

50 variational bayes sampling methods variational methods references bound log of expected value by expected value of log using jensen s inequality 5 : ln p(d) = ln dθ p(d Θ)p(Θ) = ln dθ p(d Θ)p(Θ) q(θ) q(θ) [ ] p(d Θ)p(Θ) dθ ln q(θ) q(θ) for sufficiently simple (i.e. factorized) approximating distribution q(θ), right-hand side can be easily evaluated and optimized 5 image from feynman (1972)

51 variational bayes sampling methods variational methods references iterative coordinate ascent algorithm provides controlled analytic approxmations to posterior and evidence approximate posterior q(θ) minimizes kullback-leibler distance to true posterior resulting deterministic algorithm is often fast and scalable

52 variational bayes sampling methods variational methods references iterative coordinate ascent algorithm provides controlled analytic approxmations to posterior and evidence approximate posterior q(θ) minimizes kullback-leibler distance to true posterior resulting deterministic algorithm is often fast and scalable complexity of approximation often limited (to, e.g., mean-field theory, assuming weak interaction between unknowns) iterative algorithm requires restarts, no guarantees on quality of approximation

53 outline principles sampling methods variational methods references 1 principles (what we d like to do) : joint, marginal, and conditional probabilities : inverting conditional probabilities : unknowns as random variables : + 2 (what we re able to do) monte carlo methods: representative samples variational methods: bound optimization references 3

54 sampling methods variational methods references information theory, inference, and learning algorithms, mackay (2003) pattern recognition and machine learning, bishop (2006) bayesian data analysis, gelman, et. al. (2003) probabilistic inference using markov chain monte carlo methods, neal (1993) graphical models, exponential families, and variational inference, wainwright & jordan (2006) probability theory: the logic of science, jaynes (2003) what is..., wiggins (2006) view on cran variational-bayes.org variational for network modularity

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56 example: a bayesian approach to network modularity

57 example: a bayesian approach to network modularity

58 example: a bayesian approach to network modularity inferred topological communities correspond to sub-disciplines

59 Thanks. Questions? 6 6