April 20th, Advanced Topics in Machine Learning California Institute of Technology. Markov Chain Monte Carlo for Machine Learning

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1 for for Advanced Topics in California Institute of Technology April 20th, / 50

2 Table of Contents for / 50

3 History of methods for Enrico Fermi used to calculate incredibly accurate predictions using statistical sampling methods when he had insomnia, in order to impress his friends. 3 / 50

4 History of methods for Enrico Fermi used to calculate incredibly accurate predictions using statistical sampling methods when he had insomnia, in order to impress his friends. Sam Ulam used sampling to approximate the probability that a solitaire layout would be successful after the combinatorics proved too difficult. 3 / 50

5 History of methods for Enrico Fermi used to calculate incredibly accurate predictions using statistical sampling methods when he had insomnia, in order to impress his friends. Sam Ulam used sampling to approximate the probability that a solitaire layout would be successful after the combinatorics proved too difficult. Both of these examples get at the heart of Monte Carlo methods: selecting a statistical sample to approximate a hard combinatorial problem by a much simpler problem. 3 / 50

6 History of methods for Enrico Fermi used to calculate incredibly accurate predictions using statistical sampling methods when he had insomnia, in order to impress his friends. Sam Ulam used sampling to approximate the probability that a solitaire layout would be successful after the combinatorics proved too difficult. Both of these examples get at the heart of Monte Carlo methods: selecting a statistical sample to approximate a hard combinatorial problem by a much simpler problem. These ideas have been extended to fields across science, spurred on by the development of computers 3 / 50

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8 Bayesian Inference and for We often need to solve the following intractable problems in Bayesian Statistics, given unknown variables x X and data y Y : Normalization: we need to compute the normalizing factor X p(y x )p(x )dx in Bayes theorem in order to compute the posterior p(x y) given p(x) and p(y x) 5 / 50

9 Bayesian Inference and for We often need to solve the following intractable problems in Bayesian Statistics, given unknown variables x X and data y Y : Normalization: we need to compute the normalizing factor X p(y x )p(x )dx in Bayes theorem in order to compute the posterior p(x y) given p(x) and p(y x) Marginalization: Computing the marginal posterior p(x y) = Z p(x, z y)dz given the joint posterior of (x, z) X Z 5 / 50

10 Bayesian Inference and for We often need to solve the following intractable problems in Bayesian Statistics, given unknown variables x X and data y Y : Normalization: we need to compute the normalizing factor X p(y x )p(x )dx in Bayes theorem in order to compute the posterior p(x y) given p(x) and p(y x) Marginalization: Computing the marginal posterior p(x y) = Z p(x, z y)dz given the joint posterior of (x, z) X Z Expectation: Obtain the expectation E p(x y) (f (x)) = Z f (x)p(x y)dx for some function f : X R n f, such as the conditional mean or the conditional covariance 5 / 50

11 Other problems that can be approximated with for Statistical mechanics: Computing the partition function of a system can involve a large sum over possible configurations Optimization: Finding the optimal solution over a large feasible set Penalized likelihood model selection: Avoiding wasting computing resources on computing maximum likelihood estimates over a large initial set of models 6 / 50

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13 Introduction to for We first draw N i.i.d. samples x (i)n i=1 from a target density p(x) on a high-dimensional space X. We can use these N samples to estimate the target density p(x) using a point-mass function: p N (x) = 1 N N δ x (i)(x) (1) i=1 where δ x (i)(x) is the Dirac-delta mass located at x(i) 8 / 50

14 Introduction to for This allows us to approximate integrals or large sums I (f ) with tractable, smaller sums I N (f ). These converge by the strong law of large numbers: I N (f ) = 1 N N i=1 f (x (i) ) a.s. I (f ) = f (x)p(x)dx (2) N X 9 / 50

15 Introduction to for Note that I N (f ) is unbiased. If the variance of f (x) satisfies σf 2(x) = E p(x)(f 2 (x)) I 2 (f ) <, then var(i N (f )) = σ2 f (x) N Bounded variance of f also guarantees convergence in distribution of the error via the central limit theorem: N(IN (f ) I (f )) = N (0, σf 2 (x)) (3) N 10 / 50

16 Inverse Transform for How do we sample from easy-to-sample distributions? Suppose we want to sample from a distribution f with CDF F. Sample u Uniform[0, 1] Compute x = F 1 (u) (This gives us the largest x : P( < f < x) <= u) Then x will be sampled according to f 11 / 50

17 Inverse Transform for Example: N (0, 1) But for more complicated distributions, we may not have an explicit formula for F 1 (u). In these cases we can consider other sampling methods. 12 / 50

18 Rejection for Main idea: It can be difficult to sample from complicated distributions Instead, we will sample from a simple distribution that contains our complicated distribution (an upper bound), and reject any sample that is not within the complicated distribution If we take enough samples, this will give us a good approximation under the principle 13 / 50

19 Rejection : from a density for This is how we sample if we can compute the inverse CDF 14 / 50

20 Rejection : from a density for And this is a good way to approximate if you can t 15 / 50

21 Rejection : from a density for 16 / 50

22 Rejection : Drawbacks for Consider approximating the area of the unit sphere by sampling uniformly from the simplex and rejecting any sample not inside the sphere. In low dimensions, this is fine, but as the number of dimensions gets large, the volume of the unit sphere goes to zero whereas the volume of the simplex stays one, resulting in an intractable number of rejected samples. Rejection sampling is impractical in high dimensions! 17 / 50

23 Importance for Importance sampling is used to estimate properties of a particular distribution of interest. Essentially we are transforming a difficult integral into an expectation over a simpler proposal distribution. Relies on drawing samples from a proposal distribution in order to develop the approximation Convergence time depends heavily on an appropriate choice of proposal distribution. 18 / 50

24 Importance : visual example for 19 / 50

25 Importance for If we choose proposal distribution q(x) such that supp(p(x)) supp(q(x)) then we can write I (f ) = f (x)w(x)dx (4) where w(x) = p(x) q(x) is known as the importance weight 20 / 50

26 Importance for If we can simulate N i.i.d. samples x (i)n i+1 from q(x) and evaluate w(x (i) ) then we can approximate I (f ) as I N (f ) = N f (x (i) )w(x (i) ) (5) i=1 which converges to I (f ) by the strong law of large numbers 21 / 50

27 Importance for Choosing an appropriate q(x) is very important. In importance sampling, we can choose an optimal q by finding one that will minimize the variance of I N (f ) By sampling from areas of high importance we can achieve very high sampling efficiency It can be hard to find a candidate q in high dimensions. One strategy is to use adaptive importance sampling which parametrizes q. 22 / 50

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29 for Main Idea: Given a target distribution p, construct a Markov chain over the sample space χ with invariant distribution equal to p. Perform a random walk and a sample at each iteration. Use the sample histogram to approximate the target distribution p. This is called the Metropolis-Hastings algorithm. 24 / 50

30 for Main Idea: Given a target distribution p, construct a Markov chain over the sample space χ with invariant distribution equal to p. Perform a random walk and a sample at each iteration. Use the sample histogram to approximate the target distribution p. This is called the Metropolis-Hastings algorithm. Two important things to note for : We assume we can evaluate p(x) for any x χ. We only need to know p(x) up to a normalizing constant. 24 / 50

31 Review: Markov chains for Suppose a stochastic process x (i) can only take n values in χ = {x 1, x 2,..., x n }. Then x (i) is a Markov chain if Pr(x (i) x (i 1),..., x (1) ) = T (x (i) x (i 1) ). Transition matrix T is n n with row-sum 1 (aka stochastic matrix). T ij = T (x j x i ) transition probability from state x i to state x j A row vector π is a stationary distribution if πt = π. A row vector π is a (unique) invariant/limiting distribution if lim n µt n = π for all initial distributions µ. An invariant distribution is also a stationary distribution. 25 / 50

32 Review: Markov chains for Theorem: A Markov chain has an invariant distribution if it is: Irreducible: There is a positive probability to get from any state to all other states. Aperiodic: The gcd of the lengths of all path from a state to itself is 1, for all states x i : gcd{n > 0 : Pr(x (n) = x i x (0) = x i ) > 0} 26 / 50

33 Review: Markov chains for Theorem: A Markov chain has an invariant distribution if it is: Irreducible: There is a positive probability to get from any state to all other states. Aperiodic: The gcd of the lengths of all path from a state to itself is 1, for all states x i : gcd{n > 0 : Pr(x (n) = x i x (0) = x i ) > 0} If we construct an irreducible and aperiodic Markov chain T such that pt = p (i.e. p is a stationary distribution), then p is necessarily the invariant distribution. This is the key behind the Metropolis-Hastings algorithm. 26 / 50

34 Reversibility / Detailed Balance Condition for How can we ensure that p is a stationary distribution of T? Check if T satisfies the reversibility / detailed balance condition: p(x (i+1) )T (x (i) x (i+1) ) = p(x (i) )T (x (i+1) x (i) ) (6) 27 / 50

35 Reversibility / Detailed Balance Condition for How can we ensure that p is a stationary distribution of T? Check if T satisfies the reversibility / detailed balance condition: p(x (i+1) )T (x (i) x (i+1) ) = p(x (i) )T (x (i+1) x (i) ) (6) Summing over x (i) on both sides gives us: p(x (i+1) ) = x (i) p(x (i) )T (x (i+1) x (i) ) (7) This is equivalent to pt = p. 27 / 50

36 Discrete vs. Continuous Markov chains for Transition matrix T becomes an integral/transition kernel K. Detailed balance condition is the same: p(x (i+1) )K(x (i) x (i+1) ) = p(x (i) )K(x (i+1) x (i) ) (8) Integral instead of summation: p(x (i+1) ) = p(x (i) )K(x (i+1) x (i) )dx (i) (9) χ 28 / 50

37 Metropolis-Hastings Algorithm for Initialize x (0) For i from 0 to N 1: Sample u Uniform[0, 1] Sample x q(x x (i) ) If u < A(x (i), x ) = min x (i+1) = x Else: x (i+1) = x (i) { } 1, p(x )q(x (i) x ) : p(x (i) )q(x x (i) ) q is called the proposal distribution and should be easy-to-sample. A(x (i), x ) is called the acceptance probability of moving from x (i) to x. 29 / 50

38 Metropolis-Hastings Algorithm for 30 / 50

39 Example of MH algorithm for p(x) 0.1 exp( 0.2x 2 ) exp( 0.2(x 10) 2 ) q(x x (i) ) = N (x (i), 100) bimodal distribution 31 / 50

40 Verify: detailed balance condition for The transition kernel for the MH algorithm is: K MH (x (i+1) x (i) ) = q(x (i+1) x (i) ) A(x (i), x (i+1) ) + δ x (i)(x (i+1) ) r(x (i) ) (10) where r(x (i) ) = χ q(x x (i) )(1 A(x (i), x ))dx is the rejection term 32 / 50

41 Verify: detailed balance condition for The transition kernel for the MH algorithm is: K MH (x (i+1) x (i) ) = q(x (i+1) x (i) ) A(x (i), x (i+1) ) + δ x (i)(x (i+1) ) r(x (i) ) (10) where r(x (i) ) = χ q(x x (i) )(1 A(x (i), x ))dx is the rejection term We can check that K MH satisfies the detailed balance condition: p(x (i+1) )K MH (x (i) x (i+1) ) = p(x (i) )K MH (x (i+1) x (i) ) (11) 32 / 50

42 Verify: irreducibility and aperiodicity for Irreducibility holds as long as the support of q includes the support of p. Aperiodicity holds by construction. By allowing for rejection, we are creating self-loops in our Markov chain. 33 / 50

43 Disadvantages of MH algorithm for Samples are not independent Nearby samples are correlated. Can use thinning by sampling every d steps, but this is often unnecessary. Just take more samples. Theoretically, the sample histogram is an unbiased estimator for p. 34 / 50

44 Disadvantages of MH algorithm for Initial samples are biased Starting state not chosen according to p, meaning the random walk might start in a low density region of p. We can employ a burn-in period, where the first few samples are discarded. How many samples to discard depends on the mixing time of the Markov chain. 35 / 50

45 Convergence of Random Walk for How many time-steps do we need to converge to p? Theoretically, infinite. But being close is often a good enough approximation. 36 / 50

46 Convergence of Random Walk for How many time-steps do we need to converge to p? Theoretically, infinite. But being close is often a good enough approximation. Total Variation norm: Finite case: δ(q, p) = 1 2 x χ q(x) p(x) General case: δ(q, p) = sup A χ q(a) p(a) ɛ-mixing time: τ x (ɛ) = min{t : δ(k (t ) x, p) ɛ, t > t} 36 / 50

47 Convergence of Random Walk for How many time-steps do we need to converge to p? Theoretically, infinite. But being close is often a good enough approximation. Total Variation norm: Finite case: δ(q, p) = 1 2 x χ q(x) p(x) General case: δ(q, p) = sup A χ q(a) p(a) ɛ-mixing time: τ x (ɛ) = min{t : δ(k (t ) x, p) ɛ, t > t} Mixing times of a Markov chain can be bounded by its second eigenvalue and its conductance (measure of connectivity). CS139 Advanced covers this in great detail. 36 / 50

48 Other algorithms for Many algorithms are just special cases of the MH algorithm. Independent sampler Assume q(x x (i) ) = q(x ), independent of current state { } { } A(x (i), x ) = min 1, p(x )q(x (i) ) = min 1, w(x ) p(x (i) )q(x ) w(x (i) ) Metropolis algorithm Assume q(x x (i) ) = q(x (i) x ), symmetric random walk { } A(x (i), x ) = min 1, p(x ) p(x (i) ) 37 / 50

49 Other algorithms for Simulated Annealing EM Hybrid Slice Sampler Reversible jump Gibbs Sampler 38 / 50

50 Gibbs Sampler for Assume each x is an n-dimensional vector and we know the full conditional probabilities: p(x j x 1,..., x j 1, x j+1,..., x n ) = p(x j x j ). Furthermore, assume the full conditional probabilities are easy-to-sample. Then we can choose proposal distribution for each j: { q(x x (i) p(xj x (i) j ) = ) if x j = x (i) j, meaning all other coordinates match 0 otherwise With acceptance probability: A(x (i), x ) = min {1, p(x )p(x (i) j x (i) j ) } { = min 1, p(x j ) } = 1 p(x (i) )p(xj x(i) j ) p(x (i) j ) 39 / 50

51 Gibbs Sampler for At each iteration, we are changing one coordinate of our current state. We can choose which coordinate to change randomly, or we can go in order. 40 / 50

52 Gibbs Sampler for At each iteration, we are changing one coordinate of our current state. We can choose which coordinate to change randomly, or we can go in order. For directed graphical models (aka Bayesian networks), we can write the full conditional probability of x j in terms of its Markov blanket: p(x j x j ) = p(x j x parent(j) ) k children(j) p(x k x parent(k) ) (12) Recall: the Markov blanket MB(x j ) of x j is the minimal set of nodes such that x j is independent from the rest of the graph if MB(x j ) is observed. For directed graphical models, MB(x j ) = {parents(x j ), children(x j ), parents(children(x j ))}. 40 / 50

53 Table of Contents for / 50

54 Sequential MC and Particle Filters for Sequential MC methods provide online approximations of probability distributions for inherently sequential data 42 / 50

55 Sequential MC and Particle Filters for Sequential MC methods provide online approximations of probability distributions for inherently sequential data Assumptions: Initial distribution: p(x 0 ) Dynamic model: p(x t x 0:t 1, y 1:t 1 ) (for t 1) Measurement model: p(y t x 0:t, y 1:t 1 ) (for t 1) 42 / 50

56 Sequential MC and Particle Filters for Sequential MC methods provide online approximations of probability distributions for inherently sequential data Assumptions: Initial distribution: p(x 0 ) Dynamic model: p(x t x 0:t 1, y 1:t 1 ) (for t 1) Measurement model: p(y t x 0:t, y 1:t 1 ) (for t 1) Aim: estimate the posterior p(x 0:t y 1:t ), including the filtering distribution, p(x t y 1:t ), and sample N particles {x (i) 0:t }N i=1 from that distribution 42 / 50

57 Sequential MC and Particle Filters for Sequential MC methods provide online approximations of probability distributions for inherently sequential data Assumptions: Initial distribution: p(x 0 ) Dynamic model: p(x t x 0:t 1, y 1:t 1 ) (for t 1) Measurement model: p(y t x 0:t, y 1:t 1 ) (for t 1) Aim: estimate the posterior p(x 0:t y 1:t ), including the filtering distribution, p(x t y 1:t ), and sample N particles {x (i) 0:t }N i=1 from that distribution Method: Generate a weighted set of paths { x (i) 0:t }N i=1, using an importance sampling framework with appropriate proposal distribution q( x 0:t y 1:t ) 42 / 50

58 Sequential MC and Particle Filters for Here, q( x 0:t y 1:t ) = p(x 0:t 1 y 1:t 1 )q( x x 0:t 1, y 1:t ) 43 / 50

59 Vision Application for Tracking Multiple Interacting Targets with an -based Particle Filter Aim: Generate sequence of states X it, for the i th ant at frame t Idea: Use an independent particle filter for each ant 44 / 50

60 Vision Application for Tracking Multiple Interacting Targets with an -based Particle Filter Particle Filter Assumptions: Motion Model: Normally distributed, centered at location in previous frame X t 1 Measurement Model: Image-comparison function ({how ant-like} - {how background-like}) 45 / 50

61 Vision Application for Independent Particle Filters P(X it Z t 1 ) {X (r) i,t 1, π(r) i,t 1 }N r=1 46 / 50

62 Vision Application for Independent Particle Filters P(X it Z t 1 ) {X (r) i,t 1, π(r) i,t 1 }N r=1 Proposal distribution: q(x it ) = r π(r) i,t 1 P(X i,t X (r) i,t 1 ) where π (r) i,t 1 = P(Z it X (r) it ) 46 / 50

63 Vision Application for Independent Particle Filters P(X it Z t 1 ) {X (r) i,t 1, π(r) i,t 1 }N r=1 Proposal distribution: q(x it ) = r π(r) i,t 1 P(X i,t X (r) i,t 1 ) where π (r) i,t 1 = P(Z it X (r) it ) 46 / 50

64 Vision Application for Markov Random Field (MRF) Model & the Joint MRF Particle Filter Modify motion model to include pairwise interactions: P(X t X t 1 ) i P(X it X i,t 1 ) ij E ψ(x it, X jt ) Proposal distribution (same as before!): q(x t ) = r π(r) t 1 i P(X i,t X (r) i,t 1 ) but now π (r) t 1 = n i=1 P(Z it X (r) it ) (r) ij E ψ(x i,t, X j,t r ) 47 / 50

65 Vision Application for -based Particle Filter Because of pairwise interactions, Joint MRF Particle Filter scales badly with number of targets (ants) Intractable instance of importance sampling... smells like a good time to use an algorithm! 48 / 50

66 Vision Application for Tracking Multiple Interacting Targets with an -based Particle Filter Use Metropolis-Hastings instead of importance sampling Proposal Density: Simply the Motion Model 49 / 50

67 References for intromontecarlomachinelearning.pdf pdf?sequence=1&isallowed=y nips-2015-monte-carlo-tutorial.pdf / 50