calculus 1 Pr h 0 If e deductively implies h then Prhe 1 disjunction rule If h and g are mutually exclusive then Pr h or g Pr h Pr g disjunction rule If h and g are ID: 656896
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Slide1
Advanced ProbabilitySlide2
Probability
calculus
1 ≥
Pr
(h) ≥ 0
If e deductively implies h, then Pr(h|e) = 1.
(disjunction rule) If h and g are mutually exclusive, then Pr(h or g) = Pr(h) + Pr(g)
(disjunction rule) If h and g are mutually exclusive, then Pr(h or g) = Pr(h) + Pr(g).
h and g can’t both be true at the same time.
(conditional probability) Pr(h|g) = Pr(h&g)/Pr(g)
5. (conjunction rule)
Pr
(
h&g
) =
Pr
(h) x
Pr
(
g|h
)Slide3
Probability
calculus
6
. (conjunction rule)
Pr(h&g
) = Pr(h) x Pr(g|h)
Independent: the occurrence of one doesn’t affect the probability of the other.Pr(g) = Pr(g|h)
7. (conjunction rule) If h and g are independent, then Pr(h&g) = Pr(h) x Pr(g).Slide4
Probability
calculus
1 ≥
Pr
(h) ≥ 0
If e deductively implies h, then Pr(h|e) = 1.
(disjunction rule) If h and g are mutually exclusive, then Pr(h or g) = Pr(h) + Pr(g).
(conditional probability) Pr(h|g) = Pr(h&g)/Pr(g)
5. (conjunction rule) Pr(
h&g) = Pr(h) x Pr(g|h)Slide5
Bayes’s
theorem
Pr(e|h) x Pr(h)
Pr(h|e) =
Pr
(e)Slide6
Bayes’s
theorem
Pr
(
e|h
) x Pr(h)
Pr(h|e) =
Pr(e)
“likelihood”
probability of the hypothesis given the evidence
“expectedness” of the evidence
“prior probability” of the hypothesis
probability of evidence given hypothesis
base rate
How plausible was the hypothesis
before
the new evidence?
How un
surprising
is the new evidence?
How strongly does the hypothesis lead us to expect this evidence?
[Pr(e|h) x Pr(h)] + [Pr(e|not-h) x Pr(not-h)]
What is the probability of the hypothesis, given this new
new
evidence
?
“posterior probability”Slide7
Suppose there
is
a 40% chance of rain today and a 90% chance that you get wet if it rains. What is the chance that
you get wet?Pr(r) = .4Pr(not-r) = .6Pr
(w|r) = .9so, Pr(w) = Pr(w|r) x Pr(r)
+[ ]
[Pr(w|not-r) x Pr(not-r)]
.9 x .4+?
0 x .6
.36+0
.36
.2 x .6
.36
+
.12
.48
Pr(w|not-r)Slide8
Bayes’s
theorem
Pr(e|h) x Pr(h)
Pr(h|e) =
[
Pr(e|h) x Pr
(h)] + [Pr(e|not-h) x Pr(not-h)]
Pr(e|h) x Pr(h)
Pr(h|e) =
Pr
(e)Slide9
Bayes and Frequency TreesSlide10
The probability that a woman in this group has breast cancer is .8%. If a woman has breast cancer, the probability is 90% that she will have a positive mammogram. If a woman does
not
have breast cancer, the probability is 7% that she will still have a positive mammogram. x has a positive mammogram. What is the probability that
x has breast cancer?Slide11
The
probability that a woman in this group has breast cancer is .8%
. If a woman has breast cancer, the probability is 90% that she will have a positive mammogram. If a woman does not
have breast cancer, the probability is 7% that she will still have a positive mammogram. x has a positive mammogram. What is the probability that x has breast cancer?
Base rate (prior probability)Slide12
The
probability that a woman in this group has breast cancer is .8%
. If a woman has breast cancer, the probability is
90% that she will have a positive mammogram. If a woman does not have breast cancer, the probability
is 7% that she will still have a positive mammogram. x has a positive mammogram. What is the probability that x has breast cancer?
Base rate (prior probability)Sensitivity inverse of false negatives likelihood Slide13
The
probability that a woman in this group has breast cancer is .8%
. If a woman has breast cancer, the probability is
90% that she will have a positive mammogram. If a woman does not have breast cancer, the probability is
7% that she will still have a positive mammogram. x has a positive mammogram. What is the probability that x has breast cancer?
Base rate (prior probability)Sensitivity
False positiveSlide14
1
,
000women
8
have cancer992 don’t
~7positive~1negative
~70positive~922negative
base rate = .8%
sensitivity = 90%
false positives = 7%
~7
positive
~70
positive
O
dds are 10:1 you
don’t
have cancer!
P
robability of having cancer given positive screen is 7/77, approximately 9%
.Slide15
W
omen in group
w
omen
with cancer
tested positive
base rate = .8%
sensitivity = 90%
false positives = 7%
base rate = .8%
sensitivity = 90%
false positives = 7%Slide16
Pr(e|h)
x
Pr(h)
Pr(h|e)
=
Pr
(e)posterior
likelihoodsensitivity
=
Don’t confuse sensitivity with posterior probability!