ProbIntro1 - This Thisintroductionisadaptedfrom...

Info iconThis preview shows pages 1–15. Sign up to view the full content.

View Full Document Right Arrow Icon
his introduction is adapted from This introduction is adapted from Bishop’s slides
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
robability Theory Probability Theory Apples and Oranges
Background image of page 2
robability Theory Probability Theory ( ) ,1 ij pxy = ∑∑ Marginal Probability Conditional Probability Joint Probability
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
robability Theory Probability Theory Sum Rule Product Rule
Background image of page 4
e Rules of Probability The Rules of Probability Sum Rule Product Rule Independence ()( ) ( ) , pXY pXpY = () ( ) | pY X pY =
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
ayes’ heorem Bayes Theorem posterior likelihood×prior
Background image of page 6
robability Densities Probability Densities
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
ontinuous variables Continuous variables Y f XYdX = ( ) ( ) () , , | fY fX Y y fXY y f Y y = == = ( )
Background image of page 8
Transformed Densities and Expectation X EX μ == ⎣⎦ ( ) ( ) pxx d xp x x dx Conditional Expectation (discrete) Approximate Expectation (discrete and continuous) ( ) ( ) 2 22 2 var XX X XE X E X σμ μ ⎡⎤ = JFMS5
Background image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
| 2 2 | || ar | | o oY X x o YX EY X x x X x E Y X x μ μ = ⎡⎤ == = = ⎣⎦ = = ( ) var oo o x xo σμ ⎢⎥ var[ ] var | var | XX YE Y X E Y X =+ () | o pY X x = ( ) cov , XY ρ σ = 11 σσ ρ −≤ Observed correlation does NOT imply causation
Background image of page 10
rove the follwing () Prove the follwing var var var 2 cov , v 0 XY X Y X
Background image of page 11

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 12
Background image of page 13

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 14
Background image of page 15
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 15

ProbIntro1 - This Thisintroductionisadaptedfrom...

This preview shows document pages 1 - 15. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online