Where are we? Informatics 2D Reasoning and Agents Semester 2, - PowerPoint PPT Presentation

Introduction Introduction Constructing DBNs Constructing DBNs Inference in DBNs Inference in DBNs Summary Summary Where are we? Informatics 2D – Reasoning and Agents Semester 2, 2019–2020 Last time . . . ◮ Inference in temporal models Alex Lascarides ◮ Discussed general model (forward-backward, Viterbi etc.) alex@inf.ed.ac.uk ◮ Specific instances: HMMs ◮ But what is the connection to Bayesian networks? Today . . . ◮ Dynamic Bayesian Networks Lecture 28 – Dynamic Bayesian Networks 24th March 2020 Informatics UoE Informatics 2D 1 Informatics UoE Informatics 2D 183 Introduction Introduction Constructing DBNs Constructing DBNs Transient failure Inference in DBNs Inference in DBNs Persistent failure Summary Summary Dynamic Bayesian Networks Constructing DBNs ◮ We have to specify prior distribution of state variables P ( X 0 ), ◮ We’ve already seen an example of a DBN—Umbrella World transition model P ( X t +1 | X t ), and sensor model P ( E t | X t ) ◮ A DBN is a BN describing a temporal probability model that can ◮ Also, we have to fix topology of nodes have any number of state variables X t and evidence variables E t ◮ Stationarity assumption ◮ HMMs are DBNs with a single state and a single evidence variable most convenient to specify topology for first slice ◮ But recall that one can combine a set of discrete (evidence or ◮ Umbrella world example: state) variables into a single variable (whose values are tuples). R 0 P ( R ) ◮ So every discrete-variable DBN can be described as a HMM. 1 P ( R ) 0 t 0.7 0.7 f 0.3 ◮ So why bother with DBNs? Rain 0 Rain 1 ◮ Because decomposing a complex system into constituent variables, R 1 P ( U ) 1 t 0.9 as a DBN does, ameliorates sparseness in the temporal probability f 0.2 model Umbrella 1 Informatics UoE Informatics 2D 184 Informatics UoE Informatics 2D 185

Introduction Introduction Constructing DBNs Transient failure Constructing DBNs Transient failure Inference in DBNs Persistent failure Inference in DBNs Persistent failure Summary Summary An example Modelling failure ◮ Consider a battery-driven robot moving in the X × Y plane X t = ( ˙ X t , ˙ ◮ Let X t = ( X t , Y t ) and ˙ Y t ) state variables for position ◮ Assume Battery t and BMeter t take on discrete values (e.g. integer and velocity, and Z t measurements of position (e.g. GPS) between 0 and 5) ◮ Add Battery t for battery charge level and BMeter t for the ◮ These variables should be identically distributed (CPT=identity measurement of it matrix) unless error creeps in ◮ We obtain the following basic model: ◮ One way to model error is through Gaussian error model , i.e. a BMeter 1 small Gaussian error is added to the meter reading ◮ We can approximate this also for the discrete case through an Battery 0 Battery 1 appropriate distribution X 0 X 1 ◮ But problem is usually much worse: sensor failure rather than inaccurate measurements X 0 t X 1 X Z 1 Informatics UoE Informatics 2D 186 Informatics UoE Informatics 2D 187 Introduction Introduction Constructing DBNs Transient failure Constructing DBNs Transient failure Inference in DBNs Persistent failure Inference in DBNs Persistent failure Summary Summary Transient failure Transient failure ◮ Curves for prediction depending on whether BMeter t is only 0 for ◮ Transient failure : sensor occasionally sends inaccurate data t = 22 / 23 or whether it stays 0 indefinitely ◮ Robot example: after 20 consecutive readings of 5 suddenly BMeter 21 = 0 E ( Battery t |...5555005555...) ◮ In Gaussian error model belief about Battery 21 depends on: 5 ◮ Sensor model: P ( BMeter 21 = 0 | Battery 21 ) and 4 ◮ Prediction model: P ( Battery 21 | BMeter 1:20 ) E ( Battery t ) 3 ◮ If probability of large sensor error is smaller than sudden transition 2 to 0, then with high probability battery is considered empty 1 ◮ A measurement of zero at t = 22 will make this (almost) certain 0 ◮ After a reading of 5 at t = 23 the probability of full battery will go E ( Battery t |...5555000000...) -1 back to high level 15 20 25 30 ◮ But robot made completely wrong judgement . . . Time step t Informatics UoE Informatics 2D 188 Informatics UoE Informatics 2D 189

Introduction Introduction Constructing DBNs Transient failure Constructing DBNs Transient failure Inference in DBNs Persistent failure Inference in DBNs Persistent failure Summary Summary Transient failure model Transient failure model ◮ Handling transient failure with explicit error models ◮ In case of permanent failure the robot will (wrongly) believe the ◮ To handle failure properly, sensor model must include possibility of battery is empty failure E ( Battery t |...5555005555...) ◮ Simplest failure model: assign small probability to incorrect values, 5 e.g. P ( BMeter t = 0 | Battery t = 5) = 0 . 03 4 ◮ When faced with 0 reading, provided that predicted probability of E ( Battery t ) 3 empty battery is much less than 0.03, best explanation is failure 2 ◮ This model is much less susceptible to failure, because an 1 explanation is available 0 ◮ However, it cannot cope with persistent failure either E ( Battery t |...5555000000...) -1 15 20 25 30 Time step Informatics UoE Informatics 2D 190 Informatics UoE Informatics 2D 191 Introduction Introduction Constructing DBNs Transient failure Constructing DBNs Transient failure Inference in DBNs Persistent failure Inference in DBNs Persistent failure Summary Summary Persistent failure Persistent failure ◮ In case of temporary blip probability of broken sensor rises quickly ◮ Persistent failure models describe how sensor behaves under but goes back if 5 is observed normal conditions and after failure ◮ In case of persistent failure, robot assumes discharge of battery at ◮ Add additional variable BMBroken , and CPT to next BMBroken “normal” rate state has a very small probability if not broken, but 1.0 if broken before ( persistence arc ) E ( Battery t |...5555005555...) 5 ◮ When BMBroken is true, BMeter will be 0 regardless of Battery : 4 B P ( B ) E ( Battery t |...5555000000...) 0 1 E ( Battery t ) t 1.000 3 f 0.001 2 BMBroken BMBroken 1 0 P ( BMBroken t |...5555000000...) 1 0 BMeter 1 P ( BMBroken t |...5555005555...) -1 15 20 25 30 Battery 0 Battery 1 Time step Informatics UoE Informatics 2D 192 Informatics UoE Informatics 2D 193

Introduction Introduction Constructing DBNs Constructing DBNs Exact inference in DBNs Exact inference in DBNs Inference in DBNs Inference in DBNs Summary Summary Exact inference in DBNs Exact inference in DBNs ◮ Since DBNs are BNs, we already have inference algorithms like variable elimination ◮ Essentially DBN equivalent to infinite “unfolded” BN, but slices beyond required inference period are irrelevant ◮ Exact inference in DBNs is intractable, and this is a major problem. ◮ Unrolling : reproducing basic time slice to accommodate observation sequence ◮ There are approximate inference methods that work well in practice. R P ( R ) R P ( R ) R P ( R ) R P ( R ) R P ( R ) 1 0 1 0 1 2 2 3 3 4 ◮ This issue is currently a hot topic in AI. . . t t t t t 0.7 0.7 0.7 0.7 0.7 P ( R 0 ) P ( R 0 ) f f f f f 0.3 0.3 0.3 0.3 0.3 0.7 0.7 Rain Rain Rain 0 Rain 1 Rain 2 Rain 3 Rain 4 0 1 Umbrella 4 Umbrella 3 Umbrella 2 Umbrella Umbrella 1 1 R R R 2 P ( U ) R 3 P ( U ) R 4 P ( U ) P ( U ) P ( U ) 2 3 4 1 1 1 1 t t t t t 0.9 0.9 0.9 0.9 0.9 f f f f f 0.2 0.2 0.2 0.2 0.2 Informatics UoE Informatics 2D 194 Informatics UoE Informatics 2D 195 Introduction Constructing DBNs Inference in DBNs Summary Summary ◮ Account of time and uncertainty complete ◮ Looked at general Markovian models ◮ HMMs ◮ DBNs as general case ◮ Quite intractable, but powerful ◮ Next time: Decision Making under Uncertainty Informatics UoE Informatics 2D 196