Bayesian Decision Theory

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Bayesian Decision Theory
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Learning Objectives

• Review probabilistic framework
• Understand what bayesian networks are
• Understand how bayesian networks work
• Understand inferences in bayesian networks

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Other Names

• Belief networks
• Probabilistic networks
• Graphical models
• Causal networks

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Probabilistic Framework

• Bayesian inference is a method based on
  probabilities to draw conclusions in the presence
  of uncertainty.
• It is an inductive method (pair
  deduction/induction)
• A ? B, IF A (is true) THEN B (is true)
  (deduction)IF B (is true) THEN A is plausible
  (induction)

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Probabilistic Framework

• Statistical framework
• State hypotheses and models (sets of hypotheses)
• Assign prior probabilities to the hypotheses
• Draw inferences using probability calculus
  evaluate posterior probabilities (or degrees of
  belief) for the hypotheses given the data
  available, derive unique answers.

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Probabilistic Framework

• Bayes theorem
• Complementary events
• Union of two events

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Probabilistic Framework

• Conditional probabilities (reads probability of X
  knowing Y)
• Events X and Y are said to be independent if

or
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Probabilistic Framework

• More generally, with mutually independent events
  X1, , Xk

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Probabilistic Framework

• Axioms of probabilities
• gt is an ordering relationship on degrees of
  confidence (belief) P(XI) gt P(YI) AND P(YI) gt
  P(ZI) ? P(XI) gt P(ZI)
• Complementary eventsP(AI) P(AI) 1
• There is a function such that P(X,Y I) G(
  P(X I), P(Y X,I) )
• Bayes theorem

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Bayesian Inference

• Bayesian inference and induction consists in
  deriving a model from a set of data
• M is a model (hypotheses), D are the data
• P(MD) is the posterior - updated belief that M
  is correct
• P(M) is our estimate that M is correct prior to
  any data
• P(DM) is the likelihood
• Logarithms are helpful to represent small numbers
• Permits to combine prior knowledge with the
  observed data.

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Bayesian Inference

• To be able to infer a model, we need to evaluate
• The prior P(M)
• The likelihood P(D/M)
• P(D) is calculated as the sum of all the
  numerators P(D/M)P(M) over all the hypotheses H,
  and thus we do not need to calculate it

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Bayesian Inference

• Example (from MacKay, Bayesian Methods for Neural
  Networks Theory and Applications, 1995)Given
  the sequence of numbers -1, 3, 7, 11predict
  what the next two numbers are likely to be, and
  infer what the underlying process probably was,
  that gave rise to this sequence.

H1

• 15, 19 (add four to the preceding number)
• -19.9, 1043.8 (evaluate x3/119/11x223/11
  where x is the preceding number)

H2
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Bayesian Inference

• H1 arithmetic progression add n
• H2 cubic function x ? cx3 dx2 e
• PRIOR
• H2 is less plausible than H1 because it is less
  simple, and less frequent, so we could give a
  higher odd to H1.
• But we decide to give the same prior probability
  to both H1 and H2.

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Bayesian Inference

• Likelihoodwe suppose that our numbers can vary
  between 50 and 50 (101 range).
• H1 depends on a first number, and n.
• H2 depends on c,d,e, and the first number. Each
  parameter c,d,e can be represented by a fraction
  where numerator is in -50, 50 integer range and
  denominator in 1,50 range.-1/11 has four
  representations (-1/11, -2/22, -3/33, -4/44),
  9/11 has four, and 23/11 had two)

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Bayesian Inference

• Thus the ratio between the two likelihood
  isis 40 million to one in favor of H1.
• Bayesian inference favors simple models because
  they have less parameters, each parameter
  introducing sources of variation and error.
• Occams razor

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Probabilistic Belief

• There are several possible worlds that
  areindistinguishable to a system given some
  priorevidence.
• The system believes that a logic sentence B is
  True with probability p and False with
  probability 1-p. B is called a belief
• In the frequency interpretation of probabilities,
  this means that the system believes that the
  fraction of possible worlds that satisfy B is p
• The distribution (p,1-p) is the strength of B

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Problem

• At a certain time t, the knowledge of a system is
  some collection of beliefs
• At time t systems gets an observation that
  changes the strength of one of its beliefs
• How should it update the strength of its other
  beliefs?

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Toothache Example

• A certain dentist is only interested in two
  things about any patient, whether he has a
  toothache and whether he has a cavity
• Over years of practice, she has constructed the
  following joint distribution

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Toothache Example

• Using the joint distribution, the dentist can
  compute the strength of any logic sentence built
  with the proposition Toothache and Cavity

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New Evidence

• She now makes an observation E that indicates
  that a specific patient x has high probability
  (0.8) of having a toothache, but is not directly
  related to whether he has a cavity

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Adjusting Joint Distribution

• She now makes an observation E that indicates
  that a specific patient x has high probability
  (0.8) of having a toothache, but is not directly
  related to whether he has a cavity
• She can use this additional information to create
  a joint distribution (specific for x) conditional
  to E, by keeping the same probability ratios
  between Cavity and ?Cavity

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Corresponding Calculus

• P(CT) P(C?T)/P(T) 0.04/0.05

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Corresponding Calculus

• P(CT) P(C?T)/P(T) 0.04/0.05
• P(C?TE) P(CT,E) P(TE)
  P(CT) P(TE)

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Corresponding Calculus

• P(CT) P(C?T)/P(T) 0.04/0.05
• P(C?TE) P(CT,E) P(TE)
  P(CT) P(TE) (0.04/0.05)0.8
  0.64

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Generalization

• n beliefs X1,,Xn
• The joint distribution can be used to update
  probabilities when new evidence arrives
• But
• The joint distribution contains 2n probabilities
• Useful independence is not made explicit

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Purpose of Belief Networks

• Facilitate the description of a collection of
  beliefs by making explicit causality relations
  and conditional independence among beliefs
• Provide a more efficient way (than by using joint
  distribution tables) to update belief strengths
  when new evidence is observed

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Alarm Example

• Five beliefs
• A Alarm
• B Burglary
• E Earthquake
• J JohnCalls
• M MaryCalls

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A Simple Belief Network
Intuitive meaning of arrow from x to y x has
direct influence on y
Directed acyclicgraph (DAG)
Nodes are random variables
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Assigning Probabilities to Roots
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Conditional Probability Tables
Size of the CPT for a node with k parents 2k
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Conditional Probability Tables
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What the BN Means
P(x1,x2,,xn) Pi1,,nP(xiParents(Xi))
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Calculation of Joint Probability
P(J?M?A??B??E) P(JA)P(MA)P(A?B,?E)P(?B)P(?E)
0.9 x 0.7 x 0.001 x 0.999 x 0.998 0.00062
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What The BN Encodes

• Each of the beliefs JohnCalls and MaryCalls is
  independent of Burglary and Earthquake given
  Alarm or ?Alarm

• The beliefs JohnCalls and MaryCalls are
  independent given Alarm or ?Alarm

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What The BN Encodes

• Each of the beliefs JohnCalls and MaryCalls is
  independent of Burglary and Earthquake given
  Alarm or ?Alarm

• The beliefs JohnCalls and MaryCalls are
  independent given Alarm or ?Alarm

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Structure of BN

• The relation P(x1,x2,,xn)
  Pi1,,nP(xiParents(Xi))means that each belief
  is independent of its predecessors in the BN
  given its parents
• Said otherwise, the parents of a belief Xi are
  all the beliefs that directly influence Xi
• Usually (but not always) the parents of Xi are
  its causes and Xi is the effect of these causes

E.g., JohnCalls is influenced by Burglary, but
not directly. JohnCalls is directly influenced
by Alarm
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Construction of BN

• Choose the relevant sentences (random variables)
  that describe the domain
• Select an ordering X1,,Xn, so that all the
  beliefs that directly influence Xi are before Xi
• For j1,,n do
• Add a node in the network labeled by Xj
• Connect the node of its parents to Xj
• Define the CPT of Xj

• The ordering guarantees that the BN will have
  no cycles
• The CPT guarantees that exactly the correct
  number of probabilities will be defined no
  missing, no extra

Use canonical distribution, e.g., noisy-OR, to
fill CPTs
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Locally Structured Domain

• Size of CPT 2k, where k is the number of parents
• In a locally structured domain, each belief is
  directly influenced by relatively few other
  beliefs and k is small
• BN are better suited for locally structured
  domains

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Inference In BN
P(Xobs) Se P(Xe) P(eobs) where e is an
assignment of values to the evidence variables

• Set E of evidence variables that are observed
  with new probability distribution, e.g.,
  JohnCalls,MaryCalls
• Query variable X, e.g., Burglary, for which we
  would like to know the posterior probability
  distribution P(XE)

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Inference Patterns

• Basic use of a BN Given new
• observations, compute the newstrengths of some
  (or all) beliefs

• Other use Given the strength of
• a belief, which observation should
• we gather to make the greatest
• change in this beliefs strength

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Singly Connected BN

• A BN is singly connected if there is at most one
  undirected path between any two nodes

is singly connected
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Types Of Nodes On A Path
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Independence Relations In BN
Given a set E of evidence nodes, two beliefs
connected by an undirected path are independent
if one of the following three conditions
holds 1. A node on the path is linear and in
E 2. A node on the path is diverging and in E 3.
A node on the path is converging and neither
this node, nor any descendant is in E
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Independence Relations In BN
Given a set E of evidence nodes, two beliefs
connected by an undirected path are independent
if one of the following three conditions
holds 1. A node on the path is linear and in
E 2. A node on the path is diverging and in E 3.
A node on the path is converging and neither
this node, nor any descendant is in E
Gas and Radio are independent given evidence on
SparkPlugs
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Independence Relations In BN
Given a set E of evidence nodes, two beliefs
connected by an undirected path are independent
if one of the following three conditions
holds 1. A node on the path is linear and in
E 2. A node on the path is diverging and in E 3.
A node on the path is converging and neither
this node, nor any descendant is in E
Gas and Radio are independent given evidence on
Battery
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Independence Relations In BN
Given a set E of evidence nodes, two beliefs
connected by an undirected path are independent
if one of the following three conditions
holds 1. A node on the path is linear and in
E 2. A node on the path is diverging and in E 3.
A node on the path is converging and neither
this node, nor any descendant is in E
Gas and Radio are independent given no evidence,
but they aredependent given evidence on Starts
or Moves
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BN Inference

• Simplest Case

P(B) ???
B
A
P(B) P(a)P(Ba) P(a)P(Ba)
P(C) ???
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BN Inference

• Chain

X2
X1
Xn
What is time complexity to compute P(Xn)?
What is time complexity if we computed the full
joint?
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Inference Ex. 2
Algorithm is computing not individual probabilitie
s, but entire tables

• Two ideas crucial to avoiding exponential blowup
• because of the structure of the BN,
  somesubexpression in the joint depend only on a
  small numberof variable
• By computing them once and caching the result,
  wecan avoid generating them exponentially many
  times

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Variable Elimination

• General idea
• Write query in the form
• Iteratively
• Move all irrelevant terms outside of innermost
  sum
• Perform innermost sum, getting a new term
• Insert the new term into the product

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A More Complex Example

• Asia network

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• We want to compute P(d)
• Need to eliminate v,s,x,t,l,a,b
• Initial factors

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• We want to compute P(d)
• Need to eliminate v,s,x,t,l,a,b
• Initial factors

Eliminate v
Note fv(t) P(t) In general, result of
elimination is not necessarily a probability term
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• We want to compute P(d)
• Need to eliminate s,x,t,l,a,b
• Initial factors

Eliminate s
Summing on s results in a factor with two
arguments fs(b,l) In general, result of
elimination may be a function of several variables
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• We want to compute P(d)
• Need to eliminate x,t,l,a,b
• Initial factors

Eliminate x
Note fx(a) 1 for all values of a !!
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• We want to compute P(d)
• Need to eliminate t,l,a,b
• Initial factors

Eliminate t
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• We want to compute P(d)
• Need to eliminate l,a,b
• Initial factors

Eliminate l
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• We want to compute P(d)
• Need to eliminate b
• Initial factors

Eliminate a,b
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Variable Elimination

• We now understand variable elimination as a
  sequence of rewriting operations
• Actual computation is done in elimination step
• Computation depends on order of elimination

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Dealing with evidence

• How do we deal with evidence?
• Suppose get evidence V t, S f, D t
• We want to compute P(L, V t, S f, D t)

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Dealing with Evidence

• We start by writing the factors
• Since we know that V t, we dont need to
  eliminate V
• Instead, we can replace the factors P(V) and
  P(TV) with
• These select the appropriate parts of the
  original factors given the evidence
• Note that fp(V) is a constant, and thus does not
  appear in elimination of other variables

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Dealing with Evidence

• Given evidence V t, S f, D t
• Compute P(L, V t, S f, D t )
• Initial factors, after setting evidence

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Variable Elimination Algorithm

• Let X1,, Xm be an ordering on the non-query
  variables
• For I m, , 1
• Leave in the summation for Xi only factors
  mentioning Xi
• Multiply the factors, getting a factor that
  contains a number for each value of the variables
  mentioned, including Xi
• Sum out Xi, getting a factor f that contains a
  number for each value of the variables mentioned,
  not including Xi
• Replace the multiplied factor in the summation

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Understanding Variable Elimination

• We want to select good elimination orderings
  that reduce complexity
• This can be done be examining a graph theoretic
  property of the induced graph we will not
  cover this in class.
• This reduces the problem of finding good ordering
  to graph-theoretic operation that is
  well-understoodunfortunately computing it is
  NP-hard!

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Bayesian Networks Classification
Bayes rule inverts the arc
diagnostic P (C x )
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Naive Bayes Classifier
Given C, xj are independent p(xC) p(x1C)
p(x2C) ... p(xdC)
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Summary

• Probabilistic framework
• Role of conditional independence
• Belief networks
• Causality ordering
• Inference in BN
• Naïve Bayes classifier