Download presentation

Presentation is loading. Please wait.

Published byAntonio Latimer Modified over 3 years ago

1
Lecture 4A: Probability Theory Review Advanced Artificial Intelligence

2
Outline Axioms of Probability Product and chain rules Bayes Theorem Random variables PDFs and CDFs Expected value and variance

3
Introduction Sample space - set of all possible outcomes of a random experiment – Dice roll: {1, 2, 3, 4, 5, 6} – Coin toss: {Tails, Heads} Event space - subsets of elements in a sample space – Dice roll: {1, 2, 3} or {2, 4, 6} – Coin toss: {Tails}

5
examples Coin flip – P(H) – P(T) – P(H,H,H) – P(x1=x2=x3=x4) – P({x1,x2,x3,x4} contains more than 3 heads)

6
Set operations

7
Conditional Probability

9
examples Coin flip – P(x1=H)=1/2 – P(x2=H|x1=H)=0.9 – P(x2=T|x1=T)=0.8 – P(x2=H)=?

10
Conditional Probability

11
P(A, B)0.005 P(B)0.02 P(A|B)0.25

12
Quiz P(D1=sunny)=0.9 P(D2=sunny|D1=sunny)=0.8 P(D2=rainy|D1=sunny)=? P(D2=sunny|D1=rainy)=0.6 P(D2=rainy|D1=rainy)=? P(D2=sunny)=? P(D3=sunny)=?

13
Joint Probability Multiple events: cancer, test result 13 Has cancer?Test positive?P(C,TP) yes 0.018 yesno0.002 noyes0.196 no 0.784

14
Joint Probability The problem with joint distributions It takes 2 D -1 numbers to specify them! 14

15
Conditional Probability Describes the cancer test: Put this together with: Prior probability 15

16
Has cancer?Test positive?P(TP, C) yes no yes no Has cancer?Test positive?P(TP, C) yes 0.018 yesno0.002 noyes0.196 no 0.784 Conditional Probability We have: We can now calculate joint probabilities 16

17
Conditional Probability “Diagnostic” question: How likely do is cancer given a positive test? 17 Has cancer?Test positive?P(TP, C) yes 0.018 yesno0.002 noyes0.196 no 0.784

18
Bayes Theorem

19
Posterior Probability Likelihood Normalizing Constant Prior Probability

20
Bayes Theorem

21
Random Variables

22
Cumulative Distribution Functions

23
Probability Density Functions

26
f(X) X

27
Probability Density Functions f(X) X

28
Probability Density Functions f(x) x F(x) 1 x

29
Probability Density Functions f(x) x F(x) 1 x

30
Expectation

32
Variance

33
Gaussian Distributions

Similar presentations

Presentation is loading. Please wait....

OK

Information Theory and Security

Information Theory and Security

© 2018 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google