This document discusses properties of univariate and multivariate normal distributions. It covers topics such as: - Univariate normal distributions are described by two parameters: the mean and variance. - Multivariate normal distributions describe multiple continuous variables using a mean vector and covariance matrix. - The covariance matrix can take different forms such as diagonal or full covariance. - Properties of normal distributions include how marginal, conditional, and transformed distributions remain normal. - The product of two normal distributions on the same variable is proportional to another normal distribution.

1. Computer vision: models, learning
and inference
Chapter 5
The Normal Distribution
EEE To The

2. Univariate Normal Distribution
For short we write:
Univariate normal distribution
describes single continuous
variable.
Takes 2 parameters m and s2>0
2Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
-
EEE EEE Top
If
#
Ako
'
it 0 EYE a
¥ Ea
N our
µ q2

3. Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
Multivariate Normal Distribution
For short we write:
Multivariate normal distribution describes multiple continuous
variables. Takes 2 parameters
• a vector containing mean position, mm
• a symmetric “positive definite” covariance matrix SS
Positive definite: is positive for any real
3
HEISLER'S Tip
XD
' Kean
"
'7taHjDEX2
PELEE
①
ftp.gfutta8#fETE4
Wu
ME
¥45474477187745 . # EGIL Ey

4. Types of covariance
Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 4
Symmetri
c
I 'RE
a
EEZ
B TG a
EdoEs
on
-
-
on -
HE -
EY tis
in
.
¥5437 Ea An
t
④
aft 45
TG
gknitter
few
! ftp.Eo
u

5. Diagonal Covariance =
Independence
Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 5
If FA FK To a 4 a ¥ To FEE ET 7 '
I Tt fifth # ¥582 a
-
Atk I
= 'RE E ERE if a
EINE
qq.az E ¥28 if a ¥8 FakET Ft to
"
Ft ¥79 49
KEELa Egg ⇐ .
⇒ 8k HI "
%Fh¥
.
⇒ REFER to The
pie
ke a Aa E ¥ To top
Effect Ri To e KLERK c E fit
I 42 Eats ¥254
to "
2M¥

6. Consider green frame of reference:
Relationship between pink and green frames of
reference:
Substituting in:
Conclusion
:
Full covariance can be decomposed
into rotation matrix and diagonal
Decomposition of Covariance
Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 6
x.
'
ex
!af¥t¥¥uA
'
heffern
'd '
'
I
I
I
I
I
I
✓
I
y
I
-
I
.
-
-
'
,
-
,
-
-
I
'tFine.EE#fEaEuE22 diagonal -
I
,
I
I
I
I
Ffftke
→
- n
full covariance
ftp.yxa-3EKKJFY

7. Transformation of Variables
I
f
and we transform the variable as
The result is also a normal distribution:
Can be used to generate data from arbitrary Gaussians from standard one
Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 7
77 a FEE 2%4 At c 2 E 3 a EE ¥ a BE FED if if I # BEG .
qq.AE
EEK BIG a ETIEe Esh K Ealy HE Fax a
ME FEE Fi IT 2-
'
I C
fait a zEFF 4*27FEETAE
EEEg FIFEE c 2 E EFI'¥¥t*k
Ya in F it IEEE the
-
-
FEE M due¥ FEE e KE
¥5 9*24574 E II FEE late

8. Marginal Distributions
Marginal distributions of a
multivariate normal are
also normal
If
then
Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 8
I # ¥8 TG a # I 5 Fp
EEE 5 The a # & B Try € I FAK DF R I 3 .
✓
Eff n
"
9 t a
9
443Tak4TH

9. Conditional Distributions
If
then
Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 9
HIM tf EGG
§
→
)
" "" "
"
Xii#HE tf H F- Xin FAI B Ep

10. Conditional Distributions
Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 10
For spherical / diagonal case, x1 and x2 are independent so all of the
conditional distributions are the same.
own
M at
X i 42
spherical
e diagonal a IER B ka a Eh # HE BE HE
"
is a a it ARE z a a 3 .
-1

11. Product of two normals
(self-conjugacy w.r.t mean)
The product of any two normal distributions in the same
variable is proportional to a third normal distribution
Amazingly, the constant also has the form of a normal!
Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 11
Self -
conjugacy HE E In KE HE
San FEAR 5 The a FIE
se p
FER B TG x I E'¥459 -
It Btp
279 IF 'k Gq ( E5 's EE Tak H
a ftFa E Is
futz -432 .
Fk cu ZEKE Fp
9 ZIG oh E La 27 a £43899
4th a 4 FEI -
-
H 2 .
-
-
Etta I¥x FEE 's tu B ¥14734
T 7th 4741
~
KEKE no
¥fz c =L a ! (as
Ia HEEL E I ¥25 it a Eq Zulu 3

12. Change of Variables
where
If the mean of a normal in x is proportional to y then this can be re-expressed as a
normal in y that is proportional to x
Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 12
7-784-22
" E th 447 .
-
-
E¥* -
-

13. Conclusion
13Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
• Normal distribution is used ubiquitously in
computer vision
• Important properties:
• Marginal dist. of normal is normal
• Conditional dist. of normal is normal
• Product of normals prop. to normal
• Normal under linear change of variables
FLEET he if FB 2 I I ⇒ e
"
KEK K 2 .
FEET Ep a # I 8 E i II FEIG
HERB R a ¥14 HE G q to I The
EEE B HI A I a EEE IFBB Ep
IFBB top a 873EeThe¥4
z 8344¥HE HE ZEE ta ft