Cognitive Computer Vision Kingsley Sage and Hilary Buxton Prepared under ECVision Specific Action 8-3

Lecture 4 A family of graphical models We will see some example models: – Mixture models – Factor analysis – Hidden Markov Models – Dynamic Bayesian Networks – Coupled Hidden Markov Models Inference and Learning

So why are graphical models relevant to Cognitive CV? Precisely because they allows us to see different methods as instances of a broader probabilistic framework These methods are the basis for our model of perception guided by expectation We can put our model of expectation on a solid theoretical foundation We can develop well-founded methods of learning rather than just being stuck with hand-coded models

Reminder from previous lecture … A probabilistic graphical model is a type of probabilistic network that has roots in AI, statistics and neural networks Provides a clean mathematical formalism that makes it possible to understand the relationships between a wide variety of network based approaches to computation Allows to see different methods as instances of a broader probabilistic framework

A taxonomy of graphical models © Julie Vogel

Notation - reminder Squares denote discrete nodes Circles denote continuous valued nodes Clear denotes hidden node Shaded denotes observed node B A C

Mixture model as a graphical model Data point drawn from a fixed set of class in Y, but the class label for each data point is ‘missing’ X = class label x (hidden) Y = observed data point P(X=x,Y=y) = P(X=x).P(Y=y|X=x) Learning problem is to find P(Y|X) Inference problem is P(X=x|Y=y) Let’s try to put this in a vision context … Y X

Mixture model as a graphical model Let’s say our class label X has 4 possible values (wall,bush,sign, road) Each pixel in the image has a grey level y But we don’t know which class each pixel belongs to P(X=x,Y=y) = P(X=x).P(Y=y|X=x) Learning problem is to find P(Y|X) Inference problem is P(X=x|Y=y) Haven’t said how to learn P(Y|X) yet

Mixture model as a graphical model But we might have a spec for P(Y|X) y p(y|x) x=sign x=bush x=road x=wall

Mixture model as a graphical model How is this a generative model? – From lecture 2: – Estimate P(X) somehow – Calculate P(X,Y) using the training data (the set of pixels in our image) using P(X,Y) = P(Y|X). P(X) from the graphical model – P(Y|X) is the previous slide – we didn’t say how we learned it – Calculate P(X|Y) using Bayes Rule. This would assign each pixel to a class based on its grey level (classifier mode) To use generatively, use P(X) and sample P(X,Y) although this would produce a set of pixels without any spatial structure!!!

Factor analysis as graphical model In fact, this example of a mixture model where the underlying variable X is continuous (e.g. a Gaussian or normal distribution) is actually factor analysis

Factor analysis as graphical model y y p(y|x) w Factor analysis represents a high-dimensional vector Y as a linear combination of low dimensional features X

State space models State space models “roll out” their structure over time The graphical model shows which variable sets dictate how the model parameters will change over time – Hidden Markov Models – Dynamic Bayesian Networks – Kalman Filter Models – Coupled Hidden Markov Models

Hidden Markov Models Y1Y1 Y2Y2 Y3Y3 YTYT X1X1 X2X2 X3X3 XTXT

Dynamic Bayesian Networks DBNs have more than one (here up to N) underlying models X of behaviour and can switch from model to model with time Y1Y1 Y2Y2 Y3Y3 YTYT X11X11 X22X22 X33X33 XTNXTN

Switching Kalman Filter Models TRANSITION FUNCTION OBSERVATION FUNCTION Y3Y3 Y2Y2 X2X2 X3X3 X1X1 Y1Y1 U2U2 U3U3 U1U1

Coupled Hidden Markov Models Observation variable 1 Observation variable 2 Oliver, Rosario & Pentland, 1999)

Coupled Hidden Markov Models Observation variable 1 Observation variable 2 For a coupled model, the timescales associated with each underlying hidden model do not need to be the same Useful for coupling multi-modal signals such as audio and video

Inference and Learning Inference – Calculating a probability over a set of nodes given the values of other nodes – Estimating the value of hidden nodes given the values of the observed nodes – Computing statistical and information theoretic quantities like fit between model and observed data (likelihood), mutual information … Exact and approximate methods

Inference and Learning Learning – Learn parameters and/or structure from data – Maximise fit between model and observed data – Fixed model structure - discover best parameter set – Learn structure using information criteria (“scoring”) Structure Observability FullPartial KnownClosed formExpectation Maximisation (EM) UnknownLocal searchStructural EM

Summary Graphical models can be seen a family of models that allow different vision problems to be solved For example, a mixture model is a directed Bayes net which could be used to classify pixels State space models, for example the HMM, show how variable sets determine model parameters over time

Next time … Bayes Rule and Bayesian Networks A lot of excellent reference material on graphical models can be found at: