Probability, Conditional Independence


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1 Probability, Conditional Independence June 19, 2012 Probability, Conditional Independence
2 Probability Sample space Ω of events Each event ω Ω has an associated measure Probability of the event P(ω) Axioms of Probability: ω, P(ω) [0, 1] P(Ω) = 1 P(ω 1 ω 2 ) = P(ω 1 ) + P(ω 2 ) P(ω 1 ω 2 ) Note: We are being informal Probability, Conditional Independence
3 Random Variables Random variables are mappings of events (to real numbers) Mapping X : Ω R Any event ω maps to X (ω) Example: Tossing a coin has two possible outcomes Denoted by {H, T } or {0, 1} Fair coin has uniform probabilities P(X = 0) = 1 2 P(X = 1) = 1 2 Random variables (r.v.s) can be Discrete, e.g., Bernoulli Continuous, e.g., Gaussian Probability, Conditional Independence
4 Distribution, Density For a continuous r.v. Distribution function F (x) = P(X x) Corresponding density function f (x)dx = df (x) Note that F (x) = x t= f (t)dt For a discrete r.v. Probability mass function f (x) = P(X = x) = P(x) We will call this the probability of a discrete event Distribution function F (x) = P(X x) Probability, Conditional Independence
5 Joint Distributions, Marginals For two continuous r.v.s X 1, X 2 Joint distribution F (x 1, x 2 ) = P(X 1 x 1, X 2 x 2 ) Joint density function f (x 1, x 2 ) can be defined as before The marginal probability density f (x 1 ) = x 2= f (x 1, x 2 )dx 2 For two discrete r.v.s X 1, X 2 Joint probability f (x 1, x 2 ) = P(X 1 = x 1, X 2 = x 2 ) = P(x 1, x 2 ) The marginal probability P(X 1 = x 1 ) = x 2 P(X 1 = x 1, X 2 = x 2 ) Can be extended to joint distribution over several r.v.s Many hard problems involve computing marginals Probability, Conditional Independence
6 Expectation The expected value of a r.v. X For continuous r.v.s E[X ] = x xp(x)dx For discrete r.v. E[X ] = i x ip i Expectation is a linear operator E[aX + by + c] = ae[x ] + be[y ] + c Probability, Conditional Independence
7 Independence Joint probability P(X 1 = x 1, X 2 = x 2 ) Probability, Conditional Independence
8 Independence Joint probability P(X 1 = x 1, X 2 = x 2 ) X 1, X 2 are different dice Probability, Conditional Independence
9 Independence Joint probability P(X 1 = x 1, X 2 = x 2 ) X 1, X 2 are different dice X 1 denotes if grass is wet, X 2 denotes if sprinkler was on Probability, Conditional Independence
10 Independence Joint probability P(X 1 = x 1, X 2 = x 2 ) X 1, X 2 are different dice X 1 denotes if grass is wet, X 2 denotes if sprinkler was on Two r.v.s are independent if P(X 1 = x 1, X 2 = x 2 ) = P(X 1 = x 1 )P(X 2 = x 2 ) Probability, Conditional Independence
11 Independence Joint probability P(X 1 = x 1, X 2 = x 2 ) X 1, X 2 are different dice X 1 denotes if grass is wet, X 2 denotes if sprinkler was on Two r.v.s are independent if P(X 1 = x 1, X 2 = x 2 ) = P(X 1 = x 1 )P(X 2 = x 2 ) Two different dice are independent Probability, Conditional Independence
12 Independence Joint probability P(X 1 = x 1, X 2 = x 2 ) X 1, X 2 are different dice X 1 denotes if grass is wet, X 2 denotes if sprinkler was on Two r.v.s are independent if P(X 1 = x 1, X 2 = x 2 ) = P(X 1 = x 1 )P(X 2 = x 2 ) Two different dice are independent If sprinkler was on, then grass will be wet dependent Probability, Conditional Independence
13 Conditional Probability, Bayes Rule Grass Wet Grass Dry Sprinkler On Sprinkler Off Inference problems: Probability, Conditional Independence
14 Conditional Probability, Bayes Rule Grass Wet Grass Dry Sprinkler On Sprinkler Off Inference problems: Given grass wet what is P( sprinkler on grass wet ) Probability, Conditional Independence
15 Conditional Probability, Bayes Rule Grass Wet Grass Dry Sprinkler On Sprinkler Off Inference problems: Given grass wet what is P( sprinkler on grass wet ) Given symptom what is P( disease symptom ) Probability, Conditional Independence
16 Conditional Probability, Bayes Rule Grass Wet Grass Dry Sprinkler On Sprinkler Off Inference problems: Given grass wet what is P( sprinkler on grass wet ) Given symptom what is P( disease symptom ) For any r.v.s X, Y, the conditional probability P(x y) = P(x, y) P(y) Probability, Conditional Independence
17 Conditional Probability, Bayes Rule Grass Wet Grass Dry Sprinkler On Sprinkler Off Inference problems: Given grass wet what is P( sprinkler on grass wet ) Given symptom what is P( disease symptom ) For any r.v.s X, Y, the conditional probability P(x y) = P(x, y) P(y) Since P(x, y) = P(y x)p(x), we have P(y x) = P(x y)p(y) P(x) Probability, Conditional Independence
18 Conditional Probability, Bayes Rule Grass Wet Grass Dry Sprinkler On Sprinkler Off Inference problems: Given grass wet what is P( sprinkler on grass wet ) Given symptom what is P( disease symptom ) For any r.v.s X, Y, the conditional probability P(x y) = P(x, y) P(y) Since P(x, y) = P(y x)p(x), we have P(y x) = P(x y)p(y) P(x) Expressing posterior in terms of conditional : Bayes Rule Probability, Conditional Independence
19 Product Rule & Independence Product Rule: For X 1, X 2, P(X 1, X 2 ) = P(X 1 )P(X 2 X 1 ) For X 1, X 2, X 3, P(X 1, X 2, X 3 ) = P(X 1 )P(X 2 X 1 )P(X 3 X 1, X 2 ) In general, the chain rule n P(X 1,, X n ) = P(X i X 1,..., X i 1 ) i=1 Joint distribution of n Boolean variables Specification requires 2 n 1 parameters Recall Independence: For X 1, X 2, P(X 1, X 2 ) = P(X 1 )P(X 2 ) In general n P(X 1,, X n ) = P(X i ) i=1 Independence reduces specification to n parameters Probability, Conditional Independence
20 Independence Cavity Toothache Catch Weather decomposes into Cavity Toothache Catch Weather Consider 4 variables: Toothache,Catch,Cavity,Weather Probability, Conditional Independence
21 Independence Cavity Toothache Catch Weather decomposes into Cavity Toothache Catch Weather Consider 4 variables: Toothache,Catch,Cavity,Weather Independence implies P(Toothache,Catch, Cavity, Weather) = P(Toothache, Catch, Cavity)P(Weather) Probability, Conditional Independence
22 Independence Cavity Toothache Catch Weather decomposes into Cavity Toothache Catch Weather Consider 4 variables: Toothache,Catch,Cavity,Weather Independence implies P(Toothache,Catch, Cavity, Weather) = P(Toothache, Catch, Cavity)P(Weather) In terms of the joint distribution: Probability, Conditional Independence
23 Independence Cavity Toothache Catch Weather decomposes into Cavity Toothache Catch Weather Consider 4 variables: Toothache,Catch,Cavity,Weather Independence implies P(Toothache,Catch, Cavity, Weather) In terms of the joint distribution: 32 parameters reduced to 12 = P(Toothache, Catch, Cavity)P(Weather) Probability, Conditional Independence
24 Independence Cavity Toothache Catch Weather decomposes into Cavity Toothache Catch Weather Consider 4 variables: Toothache,Catch,Cavity,Weather Independence implies P(Toothache,Catch, Cavity, Weather) = P(Toothache, Catch, Cavity)P(Weather) In terms of the joint distribution: 32 parameters reduced to 12 For boolean variables 2 n 1 reduces to n Probability, Conditional Independence
25 Independence Cavity Toothache Catch Weather decomposes into Cavity Toothache Catch Weather Consider 4 variables: Toothache,Catch,Cavity,Weather Independence implies P(Toothache,Catch, Cavity, Weather) = P(Toothache, Catch, Cavity)P(Weather) In terms of the joint distribution: 32 parameters reduced to 12 For boolean variables 2 n 1 reduces to n Absolute independence powerful but rare Probability, Conditional Independence
26 Conditional Independence X and Y are conditionally independent given Z P(X, Y Z) = P(X Z)P(Y Z) Probability, Conditional Independence
27 Conditional Independence X and Y are conditionally independent given Z P(X, Y Z) = P(X Z)P(Y Z) Example: P(Toothache,Catch Cavity) = P(Toothache Cavity)P(Catch Cavity) Probability, Conditional Independence
28 Conditional Independence X and Y are conditionally independent given Z P(X, Y Z) = P(X Z)P(Y Z) Example: P(Toothache,Catch Cavity) = P(Toothache Cavity)P(Catch Cavity) Conditional Independence simplifies joint distributions Probability, Conditional Independence
29 Conditional Independence X and Y are conditionally independent given Z P(X, Y Z) = P(X Z)P(Y Z) Example: P(Toothache,Catch Cavity) = P(Toothache Cavity)P(Catch Cavity) Conditional Independence simplifies joint distributions Often reduces from exponential to linear in n P(X, Y, Z) = P(Z)P(X Z)P(Y Z) Probability, Conditional Independence
30 Naive Bayes Model If X 1,..., X n are independent given Y n P(Y, X 1,..., X n ) = P(Y ) P(X i Y ) i=1 Probability, Conditional Independence
31 Naive Bayes Model If X 1,..., X n are independent given Y P(Y, X 1,..., X n ) = P(Y ) Example: P(Cavity,Toothache, Catch) n P(X i Y ) i=1 = P(Cavity)P(Toothache Cavity)P(Catch Cavity) Probability, Conditional Independence
32 Naive Bayes Model If X 1,..., X n are independent given Y P(Y, X 1,..., X n ) = P(Y ) Example: P(Cavity,Toothache, Catch) More generally n P(X i Y ) i=1 = P(Cavity)P(Toothache Cavity)P(Catch Cavity) P(Cause, Effect 1,..., Effect n ) = P(Cause) n P(Effect i Cause) i=1 Probability, Conditional Independence
33 Naive Bayes Model If X 1,..., X n are independent given Y P(Y, X 1,..., X n ) = P(Y ) Example: P(Cavity,Toothache, Catch) More generally n P(X i Y ) i=1 = P(Cavity)P(Toothache Cavity)P(Catch Cavity) P(Cause, Effect 1,..., Effect n ) = P(Cause) n P(Effect i Cause) i=1 Cavity Cause Toothache Catch Effect 1 Effect n Probability, Conditional Independence
34 Bayesian networks A simple, graphical notation for conditional independence assertions and hence for compact specification of full joint distributions
35 Bayesian networks A simple, graphical notation for conditional independence assertions and hence for compact specification of full joint distributions Syntax A set of nodes, one per variable A directed, acyclic graph (link implies direct influence) A conditional distribution for each node given its parents
36 Bayesian networks A simple, graphical notation for conditional independence assertions and hence for compact specification of full joint distributions Syntax A set of nodes, one per variable A directed, acyclic graph (link implies direct influence) A conditional distribution for each node given its parents Conditional distributions For each X i, P(X i Parents(X i )) In the form of a conditional probability table (CPT) Distribution of X i for each combination of parent values
37 Example Topology of network encodes conditional independence assertions Weather Cavity Toothache Catch Weather is independent of the other variables Toothache, Catch are conditionally independent given Cavity
38 Example I m at work, neighbor John calls to say my alarm is ringing, but neighbor Mary doesn t call. Sometimes it s set off by minor earthquakes. Is there a burglar? Variables: Burglar, Earthquake, Alarm, JohnCalls, MaryCalls
39 Example I m at work, neighbor John calls to say my alarm is ringing, but neighbor Mary doesn t call. Sometimes it s set off by minor earthquakes. Is there a burglar? Variables: Burglar, Earthquake, Alarm, JohnCalls, MaryCalls Network topology reflects causal knowledge
40 Example I m at work, neighbor John calls to say my alarm is ringing, but neighbor Mary doesn t call. Sometimes it s set off by minor earthquakes. Is there a burglar? Variables: Burglar, Earthquake, Alarm, JohnCalls, MaryCalls Network topology reflects causal knowledge A burglar can set the alarm off
41 Example I m at work, neighbor John calls to say my alarm is ringing, but neighbor Mary doesn t call. Sometimes it s set off by minor earthquakes. Is there a burglar? Variables: Burglar, Earthquake, Alarm, JohnCalls, MaryCalls Network topology reflects causal knowledge A burglar can set the alarm off An earthquake can set the alarm off
42 Example I m at work, neighbor John calls to say my alarm is ringing, but neighbor Mary doesn t call. Sometimes it s set off by minor earthquakes. Is there a burglar? Variables: Burglar, Earthquake, Alarm, JohnCalls, MaryCalls Network topology reflects causal knowledge A burglar can set the alarm off An earthquake can set the alarm off The alarm can cause Mary to call
43 Example I m at work, neighbor John calls to say my alarm is ringing, but neighbor Mary doesn t call. Sometimes it s set off by minor earthquakes. Is there a burglar? Variables: Burglar, Earthquake, Alarm, JohnCalls, MaryCalls Network topology reflects causal knowledge A burglar can set the alarm off An earthquake can set the alarm off The alarm can cause Mary to call The alarm can cause John to call
44 Example (Contd.) Burglary P(B).001 Earthquake P(E).002 B T T F F E T F T F P(A B,E) Alarm JohnCalls A T F P(J A) MaryCalls A T F P(M A).70.01
45 Compactness B E A J M A CPT for Boolean X i with k Boolean parents
46 Compactness B E A J M A CPT for Boolean X i with k Boolean parents 2 k rows for the combinations of parent values
47 Compactness B E A J M A CPT for Boolean X i with k Boolean parents 2 k rows for the combinations of parent values Each row requires one number
48 Compactness B E A J M A CPT for Boolean X i with k Boolean parents 2 k rows for the combinations of parent values Each row requires one number Each variable has no more than k parents
49 Compactness B E A J M A CPT for Boolean X i with k Boolean parents 2 k rows for the combinations of parent values Each row requires one number Each variable has no more than k parents The complete network requires O(n 2 k ) numbers
50 Compactness B E A J M A CPT for Boolean X i with k Boolean parents 2 k rows for the combinations of parent values Each row requires one number Each variable has no more than k parents The complete network requires O(n 2 k ) numbers Grows linearly with n
51 Compactness B E A J M A CPT for Boolean X i with k Boolean parents 2 k rows for the combinations of parent values Each row requires one number Each variable has no more than k parents The complete network requires O(n 2 k ) numbers Grows linearly with n Full joint distribution requires O(2 n )
52 Compactness B E A J M A CPT for Boolean X i with k Boolean parents 2 k rows for the combinations of parent values Each row requires one number Each variable has no more than k parents The complete network requires O(n 2 k ) numbers Grows linearly with n Full joint distribution requires O(2 n ) Example: Burglary network
53 Compactness B E A J M A CPT for Boolean X i with k Boolean parents 2 k rows for the combinations of parent values Each row requires one number Each variable has no more than k parents The complete network requires O(n 2 k ) numbers Grows linearly with n Full joint distribution requires O(2 n ) Example: Burglary network Full joint distribution requires = 31 numbers
54 Compactness B E A J M A CPT for Boolean X i with k Boolean parents 2 k rows for the combinations of parent values Each row requires one number Each variable has no more than k parents The complete network requires O(n 2 k ) numbers Grows linearly with n Full joint distribution requires O(2 n ) Example: Burglary network Full joint distribution requires = 31 numbers Bayes net requires 10 numbers
55 Global semantics Full joint distribution
56 Global semantics Full joint distribution Can be written as product of local conditionals
57 Global semantics Full joint distribution Can be written as product of local conditionals Example: P(j, m, a, b, e) = P( b)p( e)p(a b, e)p(j a)p(m a)
58 Global semantics Full joint distribution Can be written as product of local conditionals Example: P(j, m, a, b, e) = P( b)p( e)p(a b, e)p(j a)p(m a) Example: P(j, m, a, b, e) = P(b)P( e)p(a b, e)p(j a)p( m a)
59 Global semantics Full joint distribution Can be written as product of local conditionals Example: P(j, m, a, b, e) = P( b)p( e)p(a b, e)p(j a)p(m a) Example: P(j, m, a, b, e) = P(b)P( e)p(a b, e)p(j a)p( m a) Can we compute P(b j, m)?
60 Local semantics Each node is conditionally independent of its nondescendants given its parents U 1... U m Z 1j X Z nj Y 1... Y n
61 Markov blanket Each node is conditionally independent of all others given its Markov blanket, i.e., parents + children + children s parents U 1... U m Z 1j X Z nj Y 1... Y n
62 Conditional Independence in BNs Which BNs support x 1 x 2 x 3 For (a), x 1, x 2 are dependent, x 3 is a collider
63 Conditional Independence in BNs Which BNs support x 1 x 2 x 3 For (a), x 1, x 2 are dependent, x 3 is a collider For (b)(d), x 1 x 2 x 3
64 Conditional Independence (Contd.) Which BNs support x y z For (a)(b), z is not a collider, so x y z
65 Conditional Independence (Contd.) Which BNs support x y z For (a)(b), z is not a collider, so x y z For (c), z is a collider, so x and y are conditionally dependent
66 Conditional Independence (Contd.) Which BNs support x y z For (a)(b), z is not a collider, so x y z For (c), z is a collider, so x and y are conditionally dependent For (d), w is a collider, and z is a descendent of w, so x and y are conditionally dependent
67 dconnection, dseparation Definition (dconnection): X, Y, Z be disjoint sets of vertices in a directed graph G. X, Y is dconnected by Z iff an undirected path U between some x X, y Y such that for every collider C on U
68 dconnection, dseparation Definition (dconnection): X, Y, Z be disjoint sets of vertices in a directed graph G. X, Y is dconnected by Z iff an undirected path U between some x X, y Y such that for every collider C on U Either C or a descendent of C is in Z
69 dconnection, dseparation Definition (dconnection): X, Y, Z be disjoint sets of vertices in a directed graph G. X, Y is dconnected by Z iff an undirected path U between some x X, y Y such that for every collider C on U Either C or a descendent of C is in Z No noncollider on U is in Z
70 dconnection, dseparation Definition (dconnection): X, Y, Z be disjoint sets of vertices in a directed graph G. X, Y is dconnected by Z iff an undirected path U between some x X, y Y such that for every collider C on U Either C or a descendent of C is in Z No noncollider on U is in Z Otherwise X and Y are dseparated by Z
71 dconnection, dseparation Definition (dconnection): X, Y, Z be disjoint sets of vertices in a directed graph G. X, Y is dconnected by Z iff an undirected path U between some x X, y Y such that for every collider C on U Either C or a descendent of C is in Z No noncollider on U is in Z Otherwise X and Y are dseparated by Z If Z dseparates X and Y, then X Y Z for all distributions represented by the graph
72 Conditional Independence (Contd.) Examples For (a), a e b; but a, e are dependent given {b, d}
73 Conditional Independence (Contd.) Examples For (a), a e b; but a, e are dependent given {b, d} For (b) a and e are dependent given b; c and e are unconditionally dependent
74 Conditional Independence: More Examples (a) (b) For (a), Is a c e? Is a e b? Is a e c? For (b), Is a e d? Is a e c? Is a c b
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78 Inference Burglary P(B).001 Earthquake P(E).002 B T T F F E T F T F P(A B,E) Alarm JohnCalls A T F P(J A) MaryCalls A T F P(M A) How can we compute P(b j, m)?
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