Bayesian Networks. Marcello Cirillo PhD Student 11 th of May, 2007

Transcription

1 Bayesian Networks Marcello Cirillo PhD Student 11 th of May, 2007

2 Outline General Overview Full Joint Distributions Bayes' Rule Bayesian Network An Example Network Everything has a cost Learning with Bayesian Networks Plan Recognition with Bayesian Networks the 11th of May, 2007 Bayesian Networks 2 / 29

3 General Overview propositions and probability distributions evidence: prior and posterior probability conditional probability P(hear bark) dog out P a b = P a, b P b P(hear bark dog out) product rule P a,b =P a b P b the 11th of May, 2007 Bayesian Networks 3 / 29

4 General Overview probabilistic inference the computation from observed evidence of posterior probabilities for query propositions full joint distribution P Y = a knowledge base for marginalization or conditioning z P Y,z P Y = P Y z P z z n boolean variables >> input table O(2 n ) dog out not(dog out) hear bark not(hear bark) P(hear bark) = 0.41 the 11th of May, 2007 Bayesian Networks 4 / 29

5 General Overview family out bowel problem light on dog out hear bark independence P X, Y =P X P Y the full joint distribution can be factored into separate joint distributions on independent subsets the 11th of May, 2007 Bayesian Networks 5 / 29

6 General Overview Bayes' Rule a simple equation that underlies AI systems for probabilistic inference P X Y P Y P Y X = P X apparently simple, yet powerful: diagnostic knowledge is often more fragile than causal knowledge P meningitis stiff neck = P s m P m P s the 11th of May, 2007 Bayesian Networks 6 / 29

7 General Overview Bayesian Networks (1) data structure that represents the dependencies among the variables and that gives a concise specification of any full joint distribution specification: a set of random variables makes up the nodes of the network. Variables may be discrete or continuous (or both, as it happens in hybrid Bayesian networks) a set of directed links connects pairs of nodes. If there's an arrow from X to Y, then X is said to be parent of Y each node has a conditional probability distribution P(X parents(x)) that quantifies the effects of the parents on the node the graph has no directed cycles the 11th of May, 2007 Bayesian Networks 7 / 29

8 General Overview Bayesian Networks (2) the combination of the topology and the conditional distributions suffices to specify (implicitly) the full joint distribution decomposed representation of the joint distribution P x 1, x 2,, x n = i=1 n P x i parents X i locally structured systems size of the conditional probability table: O(2 k ) vs O(2 n ) the 11th of May, 2007 Bayesian Networks 8 / 29

9 An Example Network family out bowel problem light on dog out hear bark we want to do some reasoning about whether or not a family is out of the house: the dog is usually out when the family is out... the 11th of May, 2007 Bayesian Networks 9 / 29

10 An Example Network family out P(fo)=.15 bowel problem P(bp)=.01 P(lo fo)=.6 P(lo -fo)=.05 light on dog out P(do fo, bp)=.99 P(do fo, -bp)=.9 P(do -fo, bp)=.97 P(do -fo, -bp)=.3 P(hb do)=.7 P(hb -do)=.01 hear bark we need only 10 numbers to specify the full probability distribution example 1: P(fo lo) = ( P(lo fo)*p(fo) ) / P(lo) P(lo) = P(lo fo)*p(fo) + P(lo -fo)*p(-fo) the 11th of May, 2007 Bayesian Networks 10 / 29

11 An Example Network family out P(fo)=.15 bowel problem P(bp)=.01 P(lo fo)=.6 P(lo -fo)=.05 light on dog out P(do fo, bp)=.99 P(do fo, -bp)=.9 P(do -fo, bp)=.97 P(do -fo, -bp)=.3 P(hb do)=.7 P(hb -do)=.01 hear bark example 2 - it's not always simple as it seems... P(fo lo, -hb) = P(fo, lo, -hb) / P(lo, -hb) = = J(fo, lo, BP, DO, -hb) / J(FO, lo, BP, DO, -hb) 1. J(fo, lo, bp, do, -hb) = P(fo)*P(lo fo)*p(bp)*p(do fo, bp)*(1-p(hb do)) the 11th of May, 2007 Bayesian Networks 11 / 29

12 Everything has a cost Bayesian Networks as the ultimate solution? Exact inference in an arbitrary Bayesian network for discrete variables is NP-hard Even approximate inference is NP-hard For many applications structures are simple enough so that inference is efficient the 11th of May, 2007 Bayesian Networks 12 / 29

13 Learning with BN why Bayesian Networks? (1) can handle incomplete data sets allow one to learn about causal relationships causal relationship allows us to make predictions in the presence of interventions facilitate the combination of domain knowledge (prior knowledge) and data facilitate to avoid over fitting of data the 11th of May, 2007 Bayesian Networks 13 / 29

14 Learning with BN why Bayesian Networks? (2) data: evidence hypotheses: probabilistic theories of how the domain works hypotheses as intermediaries between the raw data and the predictions prior to penalize complexity the 11th of May, 2007 Bayesian Networks 14 / 29

15 Learning with BN how do we learn? - The simplest case complete data given network structure parameter independence we start with an hypothesis prior over the possible values of the parameters and we update this distribution as data arrives the 11th of May, 2007 Bayesian Networks 15 / 29

16 Learning with BN hypothesis prior: domain knowledge uniform distribution binomial sampling: beta distribution (conjugate distribution for binomial sampling) with hyperparameters equals to 0 Beta[a, b] = a 1 1 b 1 multinomial sampling: Dirichelet ditributions with each data sample, we update the parameters using Bayes' rule the 11th of May, 2007 Bayesian Networks 16 / 29

17 Learning with BN we start with a joint probability distribution p x s, S h n = i =1 p x i pa i, i,s h S h : structure of the network Ө s : vector of parameters given a random sample D, we compute the posterior distribution p s D, S h the 11th of May, 2007 Bayesian Networks 17 / 29

18 Learning with BN considering multinomial distributions: (1) each variable x i is discrete, having r i possible values we use vectors of parameters Ө ij (one for each couple x i, Pa j ) we assume that there are no missing data in each random sample D we assume that the parameter vectors are mutually independent we compute the posterior probability for each vector the 11th of May, 2007 Bayesian Networks 18 / 29

19 Learning with BN considering multinomial distributions: (2) assuming each vector Ө ij has the prior distribution Dir ij ijr 1,..., ijr i we obtain with each sample the posterior distribution p ij D,S h =Dir ij ijr 1 N ijr 1,..., ijr i N ijr i where N ijk is the number of cases in which x i = r k the 11th of May, 2007 Bayesian Networks 19 / 29

20 Learning with BN this is the simple case advanced learning techniques consider also hidden variables structure of the network (model search)... the 11th of May, 2007 Bayesian Networks 20 / 29

21 Plan Recognition with BN Towards a Bayesian Model for Keyhole Plan Recognition in Large Domains the goal to recognise the plans of an agent that is unaware that his plans are being inferred (next action, next location, current goal) the information available to the plan recognizer is gleaned from non-interactive and often incomplete observations of a user the 11th of May, 2007 Bayesian Networks 21 / 29

22 Plan Recognition with BN the domain Shattered Worlds Multi-User Dungeon, a text based virtual reality game where players compete for limited resources in an attempt to achieve various goals ( over 4700 locations more than 7200 possible actions 21 different goals (quest), including the nul l one NPCs controlled by the system many items to be acquired or used the 11th of May, 2007 Bayesian Networks 22 / 29

23 Plan Recognition with BN the problems not all the players enter the game to complete the quests the actions required to complete a quest (goal) are not always known to the users the users engage in activities that are not related to a specific quest (chatting, fighting,...) there may be more than one way to obtain a goal some actions leading to a goal are not order-dependent the domain is not always clearly defined the outcome of the actions is uncertain... the 11th of May, 2007 Bayesian Networks 23 / 29

24 Plan Recognition with BN data collection actions performed by each user location where the action was executed quests completed network structure nodes corresponding to Actions, Locations and Quests changes in the action and location over time are represented, but it is assumed that the current quest does not change the 11th of May, 2007 Bayesian Networks 24 / 29

25 Plan Recognition with BN run sequence of actionlocation pairs, starting with a new quest and ending when the quest is over 4981 runs 80% used for training 20% for testing the 11th of May, 2007 Bayesian Networks 25 / 29

26 Plan Recognition with BN results quite good predictions on the attempted quest (most of the time) the 11th of May, 2007 Bayesian Networks 26 / 29

27 Plan Recognition with BN results less accurate predictions for actions and locations the 11th of May, 2007 Bayesian Networks 27 / 29

28 Plan Recognition with BN what's interesting (for me) in this model simple network structure (although training and evaluation are a computationally complex task) complex domain (almost as in real life) predictions based on actions that are normally performed by users rather than necessary actions performance not compromised by extraneous actions the 11th of May, 2007 Bayesian Networks 28 / 29

29 References the 11th of May, 2007 Bayesian Networks 29 / 29