Bayesian Decision Theory Souce Alpaypin with modifications by Christoph F Eick Remark Belief Networks will be covered in April Utility theory will be covered as part of reinforcement learning ID: 385297
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Slide1
CHAPTER 3:Bayesian Decision Theory
Souce
:
Alpaypin
with modifications by
Christoph
F.
Eick
;
Remark: Belief Networks will be covered
in April.
Utility theory will be covered as
part
of reinforcement learning.Slide2
2
Probability and Inference
Result of tossing a coin is
Î
{Heads,Tails}Random var X Î{1,0} Bernoulli: P {X=1} = poSample: X = {xt }Nt =1 Estimation: po = # {Heads}/#{Tosses} = ∑t xt / NPrediction of next toss: Heads if po > ½, Tails otherwise
P(X=k)=
In the theory of
probability
and
statistics
, a
Bernoulli trial
is an experiment whose outcome is random and can be either of two possible outcomes, "success" and "failure".Slide3
Binomial Distribution
Lecture Notes for E Alpaydın 2004 Introduction to Machine Learning © The MIT Press (V1.1)
3Slide4
Lecture Notes for E Alpaydın 2004 Introduction to Machine Learning © The MIT Press (V1.1)
4
Classification
Credit scoring: Inputs are income and savings.
Output is low-risk vs high-riskInput: x = [x1,x2]T ,Output: C Î {0,1}Prediction: Slide5
5
Bayes’ Rule
posterior
likelihood
prior
evidence
see:
http
://en.wikipedia.org/wiki/Bayes'_
theoremSlide6
6
Bayes’ Rule: K>2 Classes
Remember: The disease/symptom exampleSlide7
7
Losses and Risks
Actions:
α
i Loss of αi when the state is Ck : λik Expected risk (Duda and Hart, 1973)
Remark:λ
ik is the cost of choosing i when k is correct!If we use accuracy/error, then λik := If i=k then 0 else 1! Slide8
8
Losses and Risks: 0/1 Loss
For minimum risk, choose the most probable class
Remark
: This strategy is not optimal in other cases Slide9
9
Losses and Risks: Reject
Risk for
rejectSlide10
Example and Homework!
C1=has cancer
C2=has not cancer12=9 21=72Homework:
a) Determine the optimal decision making strategy
Inputs: P(C1|x), P(C2|x)Decision Making Strategy:…b) Now assume we also have a reject option and the cost for making no decision are 3: reject,2=3 reject, 1=3 Inputs: P(C1|x), P(C2|x) Decision Making Strategy: …
10
Ungraded Homework: to be discussed Feb. 6! Slide11
Homework:
a) Determine the optimal decision making strategy
Input:
P(C1|x
),R(a1|x)=9xP(C2)R(a2|x)=72xP(C1)R(areject
|x)=3Setting those equal receive:
9xP(C2)=72xP(C1) (P(C2)/P(C1))=8; additionally using P(C1)+P(C2)=1 we receive: P(C1)=1/9 and P(C2)=8/9 and the risk-minimizing decision rule becomes: IF P(C1)>1/9 THEN choose C1 ELSE choose C2b) Now assume we also have a reject option and the cost for making no decision are 3: reject,2=3 reject, 1=3
Input:
P(C1|x
)
First we find equating R(
a
reject
|x
)
with
R(
a
1
|x
) and
R(
a2|x):
If P(C2)≥1/3 P(C1) ≤2/3 reject should be preferred over class1 and P(C1)≥1/24 reject should be preferred over class2. Combining this knowledge with the previous decision rule we receive:IF P(C1)[0,1/24] THEN choose class2
ELSE IF P(C1)[2/3,1] THEN choose class1
ELSE choose reject
je
11Slide12
Lecture Notes for E Alpaydın 2004 Introduction to Machine Learning © The MIT Press (V1.1)
12
Discriminant Functions
K
decision regions
R
1
,...,
R
K