Tutorial Examples Uncertainty
November 27, 2020
Inference in Bayes Nets
EB
S
WG
P(E,S,B,W,G) = P(E)P(B)P(S|E,B)P(W|S)P(G|S)
Inference in Bayes Nets
P(E)
P(S|E,B)
e
-e
9/10
P(B)
b
-b
1/10
1/10
9/10
s
-s
P(W|S)
w
-w
e∧b
9/10
1/10
s
8/10
2/10
e ∧ -b
2/10
8/10
-s
2/10
8/10
-e ∧ b
8/10
2/10
-e ∧ -b
0
1
P(G|S)
g
-g
s
1/2
1/2
-s
0
1
Inference in Bayes Nets
Given the alarm went off (s) what is the probability that Mrs. Gibbons phones you (g)?
Inference in Bayes Nets
Given the alarm went off (s) what is the probability that Mrs. Gibbons phones you (g)? probability that the alarm went off (s)?
P(g|s) = 1/2
Inference in Bayes Nets
Given that Mrs. Gibbons phones you (g) what is the probability the alarm went off (s)?
Inference in Bayes Nets
Given that Mrs. Gibbons phones you (g) what is the probability the alarm went off (s)?
1. Bayes Rule says: P(S|g) = P(g|S) ∗ P(S)/P(g)
2. P(−s|g) = P(g| − s) ∗ P(−s)/P(g) = 0 ∗ P(−s)/P(g) = 0. 3. Therefore P(s|g) = 1 (P(s|g) + P(−s|g) must sum to 1.
P(s|g) = 1 P(−s|g) = 0 Alternatively: −s → −g, so g → s, so P(s|g) = 1.
Inference in Bayes Nets
Say that there was a burglary (b) and but no earthquake (-e), what is the expression specifying the posterior probability of Dr. Watson phoning you (w) given the evidence. (You do not need to calculate a numeric answer, just give the probability expression).
Inference in Bayes Nets
Say that there was a burglary (b) and but no earthquake (-e), what is the expression specifying the posterior probability of Dr. Watson phoning you (w) given the evidence. (You do not need to calculate a numeric answer, just give the probability expression).
P(w|b,−e)
Inference in Bayes Nets
What is P(G|S)? (i.e., the four probability values) P(g|s), P(−g|s), P(g| − s), P(−g| − s).
Inference in Bayes Nets
What is P(G|S)? (i.e., the four probability values P(g|s), P(−g|s), P(g| − s), P(−g| − s).
P(g|s) = 1/2 P(−g|s) = 1/2 P(−g|−s)=0 P(−g|−s)=1
Inference in Bayes Nets
What is P (G |S ∧ W )? (i.e., the 8 probability values P(g|s ∧w), P(g|s ∧−w), …, P(−g|−s ∧−w)).
Inference in Bayes Nets
What is P (G |S ∧ W )? (i.e., the 8 probability values P(g|s ∧w), P(g|s ∧−w), …, P(−g|−s ∧−w)).
P(g|s, −w) = P(g|s, w) = P(g|s) = 1/2 P(−g|s, −w) = P(−g|s, w) = P(−g|s) = 1/2 P(g|−s,−w) = P(g|−s,w) = P(g|−s) = 0 P(−g|−s,−w) = P(−g|−s,w) = P(g|−s) = 1
Inference in Bayes Nets
What do these values tell us about the relationship between G, W and S?
G is conditionally independent of W given S
Inference in Bayes Nets
What is P(G|W)? (i.e., the four probability values P(g|w), P(−g|w), P(g| − w), and P(−g| − w)).
Inference in Bayes Nets
What is P(G|W)? (i.e., the four probability values P(g|w), P(−g|w), P(g| − w), and P(−g| − w)).
Must do variable elimination.
Inference in Bayes Nets
What is P(G|W)? (i.e., the four probability values P(g|w), P(−g|w), P(g| − w), and P(−g| − w)).
Query variable is G.
First run of VE, evidence is W = w.
Second run of VE, evidence is W = −w.
Use same ordering for both runs of VE: E, B, S, G.
With same ordering some factors can be reused between the two runs of VE.
Inference in Bayes Nets
What is P(G|W)? (i.e., the four probability values P(g|w), P(−g|w), P(g| − w), and P(−g| − w)).
1. E: P(E), P(S|E,B) 2. B: P(B),
3. S: P(w|S), P(S|G) 4. G:
Inference in Bayes Nets
What is P(G|W)? (i.e., the four probability values P(g|w), P(−g|w), P(g| − w), and P(−g| − w)).
1. E: P(E), P(S|E,B) 2. B: P(B),
3. S: P(w|S), P(S|G)
4. G:
F1(S,B) = E P(E)×P(S|E,B)
= P(e)×P(S|e,B)+P(−e)×P(S|−e,B)
F1(−s,−b) =
= 0.1×0.8+0.9×1 = 0.98
P(e)P(−s, e, −b) + P(−e)P(−s, −e, −b)
F1(−s,b) =
= 0.1×0.1+0.9×0.2 = 0.19
P(e)P(−s, e, b) + P(−e)P(−s, −e, b)
F1(s,−b) =
= 0.1×0.2+0.9×0 = 0.02
P(e)P(s, e, −b) + P(−e)P(s, −e, −b)
P(e)P(s, e, b) + P(−e)P(s, −e, b)
F1(s,b) =
= 0.1×0.9+0.9×0.8 = 0.81
Inference in Bayes Nets
1. E: P(E), P(S|E,B) 2. B: P(B), F1(S,B)
3. S: P(w|S), P(S|G) 4. G:
F2(S) = =
F2(−s) = = F2(s) = =
B P(B)×F1(S,B)
P(b)F1(S, b) + P(−b)F1(S, −b)
P(b)F1(−s, b) + P(−b)F1(−s, −b) 0.1×0.19+0.9×0.98 = 0.901 P(b)F1(s, b) + P(−b)F1(s, −b) 0.1×0.81+0.9×0.02 = 0.099
Inference in Bayes Nets
1. E: P(E), P(S|E,B) 2. B: P(B), F1(S,B)
3. S: P(w|S), P(S|G), F2(S) 4. G:
S P(w|S) × P(S|G) × F2(S)
F3(G) =
= P(w|s)P(s|G)F2(s) + P(w| − s)P(−s|G)F2(−s)
F3(−g) =
= 0.8×0.5×0.099+0.2×1×0.901 = 0.2198
P(w|s)P(s| − g)F2(s) + P(w| − s)P(−s| − g)F2(−s)
P(w|s)P(s|g)F2(s) + P(w| − s)P(−s|g)F2(−s)
F3(g) =
= 0.8×0.5×0.099+0.2×0×0.901 = 0.0396
Inference in Bayes Nets
1. E: P(E), P(S|E,B) 2. B: P(B), F1(S,B)
3. S: P(w|S), P(S|G), F2(S) 4. G: F3(G)
Normalize F3(G):
P(−g|w) =
P(g|w) =
0.2198 = 0.8473 0.2198+0.0396
0.0396 = 0.1527 0.2198+0.0396
Inference in Bayes Nets
NowP(G|−w)?
1. E: P(E), P(S|E,B) 2. B: P(B),
3. S: P(−w|S), P(S|G) 4. G:
Already computed as F1(S,B)
Inference in Bayes Nets
1. E: P(E), P(S|E,B) 2. B: P(B), F1(S,B)
3. S: P(−w|S), P(S|G) 4. G:
Already computed as F2(S)
Inference in Bayes Nets
1. E: 2. B: 3. S: 4. G:
F3(G) = =
F3(−g) = = F3(g) = =
P(E), P(S|E,B)
P(B), F1(S,B) P(−w|S), P(S|G), F2(S)
S P(−w|S) × P(S|G) × F2(S) P(−w|s)P(s|G)F2(s) + P(−w| − s)P(−s|G)F2(−s)
P(−w|s)P(s| − g)F2(s) + P(−w| − s)P(−s| − g)F2(−s) 0.2×0.5×0.099+0.8×1×0.901 = 0.7307 P(−w|s)P(s|g)F2(s) + P(−w| − s)P(−s|g)F2(−s) 0.2×0.5×0.099+0.8×0×0.901 = 0.0099
Inference in Bayes Nets
1. E: P(E), P(S|E,B) 2. B: P(B), F1(S,B)
3. S: P(−w|S), P(S|G), F2(S) 4. G: F3(G)
Normalize F3(G):
P(−g| − w) = 0.7307
P(g| − w) = 0.0099 0.2198+0.00099
0.7307+0.0099
= 0.9866
= 0.0134
Inference in Bayes Nets
What do these values tell us about the relationship between G and W , and why does this relationship differ when we know S?
Inference in Bayes Nets
What do these values tell us about the relationship between G and W , and why does this relationship differ when we know S?
G and W are not independent of each other. But when S is known they become independent.