introduction of probabilistic reasoning and bayesian networks
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Introduction of Probabilistic Reasoning and Bayesian Networks. Hongtao Du Group Presentation. Outline. Uncertain Reasoning Probabilistic Reasoning Bayesian Network (BN) Dynamic Bayesian Network (DBN). Reasoning. - PowerPoint PPT PresentationTRANSCRIPT
Introduction of Probabilistic
Reasoning and Bayesian Networks
Hongtao Du
Group Presentation
Outline Uncertain Reasoning
Probabilistic Reasoning
Bayesian Network (BN)
Dynamic Bayesian Network (DBN)
Reasoning The activity of guessing the state of the
domain from prior knowledge and observations.
Causal reasoning Diagnostic reasoning Combinations of these two
Uncertain Reasoning (Guessing) Some aspects of the domain are often
unobservable and must be estimated indirectly through other observations.
The relationships among domain events are often uncertain, particularly the relationship between the observables and non-observables.
The observations themselves may be unreliable.
Even though observable, very often we do not have sufficient resource to observe all relevant events.
Even though events relations are certain, very often it is impractical to analyze all of them
Probabilistic Reasoning Methodology founded on the Bayesian
probability theory.
Events and objects in the real world are represented by random variables.
Probabilistic models: Bayesian reasoning Evidence theory Robust statistics Recursive operators
Graphical Model A tool that visually illustrate conditional
independence among variables in a given problem.
Consisting of nodes (Random variables or States) and edges (Connecting two nodes, directed or undirected).
The lack of edge represents conditional independence between variables.
Chain, Path, Cycle, Directed Acyclic Graph (DAG), Parents and Children
Bayesian Network (BN) Probabilistic network, belief network,
causal network.
A specific type of graphical model that is represented as a Directed Acyclic Graph.
BN consists of variables (nodes) V={1, 2, …, k} A set of dependencies (edges) D A set of probability distribution functions
(pdf) of each variable P
Assumptions P(X)=1 if and only if X is certain If X and Y are mutually exclusive, then P(X v Y) = P(X) + P(Y) Joint probability P(X, Y)= P(X|Y) P(Y)
1)(0 XP
X represents hypothesis Y represents evidence P(Y|X) is likelihood P(X|Y) is the posterior probability If X, Y are conditionally independent P(X|Z, Y) = P(X|Z)
)(
)()|()|(
YP
XPXYPYXP
Given some certain evidence, BN operates by propagating beliefs throughout the network.
P(Z, Y, U, V) = P(Z) * P(Y|Z) * P(U|Y) * P(V|U)
where is the parents of node
Explaining away If a node is observed, its parents become
dependent. Two causes (parents) compete to explain the
observed data (child).
n
iiin XPaXPXXP
11 )](|[),,(
)( iXPa iX
Tasks in Bayesian Network Inference Learning
Inference Inference is the task of computing the
probability of each state of a node in a BN when other variables are known.
Method: dividing set of BN nodes into non-overlapping subsets of conditional independent nodes.
Example
Given Y is the observed variable.Goal: find the conditional pdf over
Case 1:
MNL YXZ },,{ 10 NN xxX },,{ 10 MM yyY
LK ZU
MNL
KU)|( YUP K
YUK
K
iiiK yuYUP
1
)()|( otherwise
xx
0
0
1)(
Case 2: XUK
KU K
KKK YUP
YUP
YP
YUPYUP
),(
),(
)(
),()|(
KUXx
KK YUxPYUP\
),,(),(
Learning Goal: completing the missing beliefs in
the network.
Adjusting the parameters of the Bayesian network so that the pdfs defined by the network sufficiently describes statistical behavior of the observed data.
M: a BN model : Parameter of probability of distribution : Observed data
Goal: Estimating to maximize the posterior probability
LZ
)|( LZMP
dMPMZPZP
MPZMP L
LL )|(),|(
)(
)()|(
Assume is highly peaked around maximum likelihood estimates
)|(logmaxarg
LML ZP
)|( LZMP
)|(),|()(
)()|( MPMZP
ZP
MPZMP MLMLL
LL
ML
Dynamic Bayesian Network (DBN) Bayesian network with time-series to represent
temporal dependencies.
Dynamically changing or evolving over time.
Directed graphical model of stochastic processes.
Especially aiming at time series modeling.
Satisfying the Markovian condition: The state of a system at time t depends only on its immediate
past state at time t-1.
Representation Time slice
t1 t2 tk
The transition matrix that represent these time dependencies is called Conditional Probability Table (CPT).
Description T: time boundary we are investigating : observable variables : hidden-state variables
: state transition pdfs, specifying time dependencies between states.
: observation pdfs, specifying dependencies of observation nodes regarding to other nodes at time slice t.
: initial state distribution.
},,{ 10 TyyY },,{ 10 TxxX
)()|()|(),( 0
1
1
1
11 xPxyPxxPYXPT
ttt
T
ttt
)|( 1tt xxP
)|( tt xyP
)( 0xP
Tasks in DBN Inference
Decoding
Learning
Pruning
Inference Estimating the pdf of unknown states
through given observations and initial probability distributions.
Goal: finding
: a finite set of T consecutive observations
: the set of corresponding hidden variables
0x
0y
0x
0y
0x
0y
)|( 10
10
TT YXP
},,{ 111
0 TT xxX
},,{ 111
0 TT yyY
Decoding Finding the best-fitting probability
values for the hidden states that have generated the known observations.
Goal: determine the sequence of hidden states with highest probabilities.
)|(maxargˆ 10
10
10
10
TT
X
T YXPXT
Learning Given a number of observations,
estimating parameters of DBN that best fit the observed data.
Goal: Maximizing the joint probability distribution.
: the model parameter vector
)(log)|(log)|(log)|,(log 0
1
1
1
11
10
10 xPxyPxxPYXP
T
itt
T
itt
TT
0)]|,([log 1
01
0
TT YXP
Pruning An important but difficult task in DBN.
Distinguishing which nodes are important for inference, and removing the unimportant nodes.
Actions: Deleting states from a particular node Removing the connection between nodes Removing a node from the network
Time slice t
: designated world nodes, a subset of the nodes, representing the part we want to inspect.
, If state of is known, , then are no longer relevant to the overall
goal of the inference.Thus, (1) delete all nodes (2) incorporate knowledge that
)(,),(1 tWtW q
)( ktV ttk )(tWi 1))(( ii stWP
)( ktV
ii stW )()( ktV
Future work Probabilistic reasoning in multiagent
systems.
Different DBNs and applications.
Discussion of DBN problems.