Probabilistic Graphical Models
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1 Probabilistic Graphical Models Introduction. Basic Probability and Bayes Volkan Cevher, Matthias Seeger Ecole Polytechnique Fédérale de Lausanne 26/9/2011 (EPFL) Graphical Models 26/9/ / 28
2 Outline 1 Motivation 2 Probability. Decisions. Estimation 3 Bayesian Terminology (EPFL) Graphical Models 26/9/ / 28
3 Benefits of Doubt Motivation Not to be absolutely certain is, I think, one of the essential things in rationality B. Russell (1947) Real-world problems are uncertain Measurement errors Incomplete, ambiguous data Model? Features? (EPFL) Graphical Models 26/9/ / 28
4 Benefits of Doubt Motivation Not to be absolutely certain is, I think, one of the essential things in rationality B. Russell (1947) If uncertainty is part of your problem... Ignore/remove it Costly Complicated Not always possible (EPFL) Graphical Models 26/9/ / 28
5 Benefits of Doubt Motivation Not to be absolutely certain is, I think, one of the essential things in rationality B. Russell (1947) If uncertainty is part of your problem... Ignore/remove it Costly Complicated Not always possible Live with it Quantify it: probabilities Compute it: Bayesian inference (EPFL) Graphical Models 26/9/ / 28
6 Benefits of Doubt Motivation Not to be absolutely certain is, I think, one of the essential things in rationality B. Russell (1947) If uncertainty is part of your problem... Ignore/remove it Costly Complicated Not always possible Exploit it Experimental design Robust decision making Multimodal data integration (EPFL) Graphical Models 26/9/ / 28
7 Motivation Image Reconstruction (EPFL) Graphical Models 26/9/ / 28
8 Motivation Reconstruction is Ill Posed (EPFL) Graphical Models 26/9/ / 28
9 Image Statistics Motivation Whatever images are... they are not Gaussian! Image gradient super-gaussian ( sparse ) Use sparsity prior distribution P(u) (EPFL) Graphical Models 26/9/ / 28
10 Motivation Posterior Distribution Likelihood P(y u): Data fit P(y u) (EPFL) Graphical Models 26/9/ / 28
11 Motivation Posterior Distribution Likelihood P(y u): Data fit Prior P(u): Signal properties P(y u) P(u) (EPFL) Graphical Models 26/9/ / 28
12 Motivation Posterior Distribution Likelihood P(y u): Data fit Prior P(u): Signal properties Posterior distribution P(u y ): Consistent information summary P(u y )= P(y u) P(u) P(y ) (EPFL) Graphical Models 26/9/ / 28
13 Estimation Motivation Maximum a Posteriori (MAP) Estimation u = argmax u P(y u)p(u) (EPFL) Graphical Models 26/9/ / 28
14 Estimation Motivation Maximum a Posteriori (MAP) Estimation There are many solutions. Why settle for any single one? (EPFL) Graphical Models 26/9/ / 28
15 Bayesian Inference Motivation Use All Solutions Weight each solution by our uncertainty Average over them. Integrate, don t prune (EPFL) Graphical Models 26/9/ / 28
16 Motivation Bayesian Experimental Design Posterior: Uncertainty in reconstruction Experimental design: Find poorly determined directions Sequential search with interjacent partial measurements (EPFL) Graphical Models 26/9/ / 28
17 Structure of Course Motivation Graphical Models [ 6 weeks] Probabilistic database. Expert system Query for making optimal decisions Graph separation $ conditional independence ) (More) efficient computation (dynamic programming) Approximate Inference [ 6 weeks] Bayesian inference is never really tractable Variational relaxations (convex duality) Propagation algorithms Sparse Bayesian models (EPFL) Graphical Models 26/9/ / 28
18 Course Goals Motivation This course is not: Exhaustive (but we will give pointers) Playing with data until it works Purely theoretical analysis of methods This course is: Computer scientist s view on Bayesian machine learning: Layers above and below formulae Understand concepts (what to do and why) Understand approximations, relaxations, generic algorithmic schemata (how to do, above formulae) Safe implementation on a computer (how to do, below formulae) Red line through models, algorithms. Exposing roots in specialized computational mathematics (EPFL) Graphical Models 26/9/ / 28
19 Probability. Decisions. Estimation Why Probability? Remember sleeping through Statistics 101 (p-value, t-test,...)? Forget that impression! Probability leads to beautiful, useful insights and algorithms. Not much would work today without probabilistic algorithms, decisions from incomplete knowledge. Numbers, functions, moving bodies ) Calculus Predicates, true/false statements ) Predicate logic Uncertain knowledge about numbers, predicates,... ) Probability Machine learning? Have to speak probability! Crash course here. But dig further, it s worth it: Grimmett, Stirzaker: Probability and Random Processes Pearl: Probabilistic Reasoning in Intelligent Systems (EPFL) Graphical Models 26/9/ / 28
20 Probability. Decisions. Estimation Why Probability? Reasons to use probability (forget classical straightjacket) We really don t / cannot know (exactly) It would be too complicated/costly to find out It would take too long to compute Nondeterministic processes (given measurement resolution) Subjective beliefs, interpretations (EPFL) Graphical Models 26/9/ / 28
21 Probability. Decisions. Estimation Probability over Finite/Countable Sets You know databases? You know probability! Probability distribution P: Joint table/hypercube ( 0; P = 1) Random variable F: Index of table Event E: Part of table Probability P(E): Sum over cells in E Marginal distribution P(F): Projection of table (sum over others) Box Fruit red blue apple 1/10 9/20 orange 3/10 3/20 (EPFL) Graphical Models 26/9/ / 28
22 Probability. Decisions. Estimation Probability over Finite/Countable Sets You know databases? You know probability! Probability distribution P: Joint table/hypercube ( 0; P = 1) Random variable F: Index of table Event E: Part of table Probability P(E): Sum over cells in E Marginal distribution P(F): Projection of table (sum over others) Marginalization (Sum Rule) Not interested in variable(s) right now? ) Marginalize over them! Box Fruit red blue apple 1/10 9/20 orange 3/10 3/20 P(F) = X B=r,b P(F, B) (EPFL) Graphical Models 26/9/ / 28
23 Probability. Decisions. Estimation Probability over Finite/Countable Sets (II) Conditional probability: Factorization of table Chop out part you re sure about (don t marginalize: you know!) Renormalize to 1 (/ marginal) Box Fruit red blue apple 1/10 9/20 orange 3/10 3/20 (EPFL) Graphical Models 26/9/ / 28
24 Probability. Decisions. Estimation Probability over Finite/Countable Sets (II) Conditional probability: Factorization of table Chop out part you re sure about (don t marginalize: you know!) Renormalize to 1 (/ marginal) Conditioning (Product Rule) Observed some variable/event? ) Condition on it! Joint = Conditional Marginal Box Fruit red blue apple 1/10 9/20 orange 3/10 3/20 P(F, B) =P(F B)P(B) (EPFL) Graphical Models 26/9/ / 28
25 Probability. Decisions. Estimation Probability over Finite/Countable Sets (II) Conditional probability: Factorization of table Chop out part you re sure about (don t marginalize: you know!) Renormalize to 1 (/ marginal) Conditioning (Product Rule) Observed some variable/event? ) Condition on it! Joint = Conditional Marginal Information propagation (B!F) Predict Marginalize Box Fruit red blue apple 1/10 9/20 orange 3/10 3/20 P(F, B) =P(F B)P(B) P(F) = X B P(F B)P(B) (EPFL) Graphical Models 26/9/ / 28
26 Probability. Decisions. Estimation Probability over Finite/Countable Sets (III) Bayes Formula P(B F)P(F) =P(F, B) =P(F B)P(B) Box Fruit red blue apple 1/10 9/20 orange 3/10 3/20 (EPFL) Graphical Models 26/9/ / 28
27 Probability. Decisions. Estimation Probability over Finite/Countable Sets (III) Bayes Formula P(B F) = P(F B)P(B) P(F) Inversion of information flow Causal! diagnostic (diseases! symptoms) Inverse problem Box Fruit red blue apple 1/10 9/20 orange 3/10 3/20 (EPFL) Graphical Models 26/9/ / 28
28 Probability. Decisions. Estimation Probability over Finite/Countable Sets (III) Bayes Formula P(B F) = P(F B)P(B) P(F) Inversion of information flow Causal! diagnostic (diseases! symptoms) Inverse problem Chain rule of probability Box Fruit red blue apple 1/10 9/20 orange 3/10 3/20 P(X 1,...,X n )=P(X 1 )P(X 2 X 1 )P(X 3 X 1, X 2 )...P(X n X 1,...,X n 1 ) Holds in any ordering Starting point for Bayesian networks [next lecture] (EPFL) Graphical Models 26/9/ / 28
29 Probability. Decisions. Estimation Probability over Continuous Variables (EPFL) Graphical Models 26/9/ / 28
30 Probability. Decisions. Estimation Probability over Continuous Variables (II) Caveat: Null sets [P({x = 5}) =0; P({x 2 N}) =0] Every observed event is a null set! P(y x) =P(y, x)/p(x) cannot work for P(x) =0 Define conditional density as P(y x) s.t. Z P(y x 2A)= P(y x)p(x) dx Most cases in practice: Look at y 7! P(y, x) ( plug in x ) Recognize density / normalize A for all events A (EPFL) Graphical Models 26/9/ / 28
31 Probability. Decisions. Estimation Probability over Continuous Variables (II) Caveat: Null sets [P({x = 5}) =0; P({x 2 N}) =0] Every observed event is a null set! P(y x) =P(y, x)/p(x) cannot work for P(x) =0 Define conditional density as P(y x) s.t. Z P(y x 2A)= P(y x)p(x) dx Most cases in practice: Look at y 7! P(y, x) ( plug in x ) Recognize density / normalize A for all events A Another (technical) caveat: Not all subsets can be events. ) Events: Nice subsets (measurable) (EPFL) Graphical Models 26/9/ / 28
32 Probability. Decisions. Estimation Expectation. Moments of a Distribution Expectation Z E[f (x )] = f (x )P(x ) dx or X f (x )P(x ) x (EPFL) Graphical Models 26/9/ / 28
33 Probability. Decisions. Estimation Expectation. Moments of a Distribution Expectation Z E[f (x )] = f (x )P(x ) dx or X f (x )P(x ) x P(x ) complicated. How does x P(x ) behave? Moments: Essential behaviour of distribution Mean (1st order) Z E[x ]= x P(x ) dx Covariance (2nd order) Cov[x, y ]=E[xy T ] E[x ](E[y ]) T = E[v x v T y ], v x = x E[x ] (EPFL) Graphical Models 26/9/ / 28
34 Probability. Decisions. Estimation Expectation. Moments of a Distribution Expectation Z E[f (x )] = f (x )P(x ) dx or X f (x )P(x ) x P(x ) complicated. How does x P(x ) behave? Moments: Essential behaviour of distribution Mean (1st order) Z E[x ]= x P(x ) dx Covariance (2nd order) Cov[x ]=Cov[x, x ] (EPFL) Graphical Models 26/9/ / 28
35 Probability. Decisions. Estimation Decision Theory in 30 Seconds Recipe for optimal decisions 1 Choose a loss function L (depends on problem, your valuation, susceptibility) 2 Model actions (A), outcomes (O). Compute P(O A) 3 Compute risks (expected losses) R(A) = R L(O)P(O A) do 4 Go for A = argmin A R(A) Special case: Pricing of A (option, bet, car) Choose R(A)+Margin Next best measurement? Next best scientific experiment? Harder if timing plays a role (optimal control, etc) (EPFL) Graphical Models 26/9/ / 28
36 Probability. Decisions. Estimation Maximum Likelihood Estimation Bayesian inversion hard. In simple cases, with enough data: Maximum Likelihood Estimation Observed {x i }. Interested in cause Construct sampling model P(x ) Likelihood L( ) =P(D ) = Q i P(x i ): Should be high close to true 0 Maximum likelihood estimator: = argmax L( ) =argmax log L( ) =argmax P i log P(x i ) (EPFL) Graphical Models 26/9/ / 28
37 Probability. Decisions. Estimation Maximum Likelihood Estimation Bayesian inversion hard. In simple cases, with enough data: Maximum Likelihood Estimation Observed {x i }. Interested in cause Construct sampling model P(x ) Likelihood L( ) =P(D ) = Q i P(x i ): Should be high close to true 0 Maximum likelihood estimator: = argmax L( ) =argmax log L( ) =argmax P i log P(x i ) Method of choice for simple, lots of data. Well understood asymptotically Knowledge about besides D? Not used Breaks down if larger than D And another problem... (EPFL) Graphical Models 26/9/ / 28
38 Overfitting Probability. Decisions. Estimation Overfitting Problem (Estimation) For finite D, more and more complicated models fit D better and better. With huge brain, you just learn by heart. Generalization only comes with a limit on complexity! Marginalization solves this problem, but even Bayesian estimation ( half-way marginalization ) embodies complexity control (EPFL) Graphical Models 26/9/ / 28
39 Bayesian Terminology Probabilistic Model Model Concise description of joint distribution (generative process) of all variables of interest Encoding assumptions: What are the entities? How do they relate? Variables have different roles. Roles may change depending on what model is used for Model specifies variables and their (in)dependencies ) Graphical models [next lecture] (EPFL) Graphical Models 26/9/ / 28
40 Bayesian Terminology The Linear Model (Polynomial Fitting) Fit data with polynomial (degree k) Prior P(w ) Restrictions on w Prior knowledge ) Prefer smaller kw k Likelihood P(y w ) Posterior P(w y )= P(y w )P(w ) P(y ) Linear Model y = Xw + " y Responses (observed) X Design (controlled) w Weights (query) " Noise (nuisance) (EPFL) Graphical Models 26/9/ / 28
41 Bayesian Terminology The Linear Model (Polynomial Fitting) Fit data with polynomial (degree k) Prior P(w ) Restrictions on w Prior knowledge ) Prefer smaller kw k Likelihood P(y w ) Posterior P(w y )= P(y w )P(w ) P(y ) Prediction: y = x T E[w y ] Marginal likelihood Z P(y )=P(y k) = Linear Model y = Xw + " y Responses (observed) X Design (controlled) w Weights (query) " Noise (nuisance) P(y w )P(w ) dw (EPFL) Graphical Models 26/9/ / 28
42 Bayesian Terminology The Linear Model (Polynomial Fitting) (EPFL) Graphical Models 26/9/ / 28
43 Bayesian Terminology The Linear Model (Polynomial Fitting) (EPFL) Graphical Models 26/9/ / 28
44 Bayesian Terminology Model Selection 4 Posterior P(w y, k = 3) 8 x Marginal Likelihood P(y k) Simpler hypotheses considered as well ) Occam s razor (EPFL) Graphical Models 26/9/ / 28
45 Bayesian Terminology Model Averaging Don t know polynomial order k ) Marginalize out E[y y ]= X k 1 P(k y )x (k) T E[w y, k] x (EPFL) Graphical Models 26/9/ / 28
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