Conditional Joint DistributionsChris Piech

CS109, Stanford University

Tracking in 2D Space?

Joint Random Variables

Use a joint table, density function or CDF to solve probability question

Use and find independence of random variables

Think about conditional probabilities with joint variables (which might be continuous)

What happens when you add random variables?

Use and find expectation of random variables

Joint Random Variables

Use a joint table, density function or CDF to solve probability question

Use and find independence of random variables

Think about conditional probabilities with joint variables (which might be continuous)

What happens when you add random variables?

Use and find expectation of random variables

Joint Probability Table

Roommates 2RoomDblShared Partner Single

Frosh 0.30 0.07 0.00 0.00 0.37Soph 0.12 0.18 0.00 0.03 0.32Junior 0.04 0.01 0.00 0.10 0.15Senior 0.01 0.02 0.02 0.01 0.05

5+ 0.02 0.00 0.05 0.04 0.110.49 0.27 0.07 0.18 1.00

Prob

abili

ty

Continuous Joint Random Variables

0

y

x900

900

A joint probability density function gives the relative likelihood of more than one continuous random variable each taking on a specific value.

ò ò=£<£<2

1

2

1

),( ) ,(P ,2121

a

a

b

bYX dxdyyxfbYbaXa

Joint Probability Density Function

0

y

x 900

900

0 900

900

Four Prototypical Trajectories

End Review

• Cumulative Density Function (CDF):

ò ò¥- ¥-

=a b

YXYX dxdyyxfbaF ),( ),( ,,

),(),( ,

2

, baFbaf YXYX ba ¶¶¶=

Jointly Continuous

ò ò=£<£<2

1

2

1

),( ) ,(P ,2121

a

a

b

bYX dxdyyxfbYbaXa

!",$ %, & = ( ) ≤ %, + ≤ &

%

&

to 0 asx → -∞,y → -∞,

to 1 asx → +∞,y → +∞,

plot by Academo

Jointly CDF

! "# < % ≤ "',)# < * ≤ )' = ,-,. "',)'

a1

a2

b1

b2

Probabilities from Joint CDF

! "# < % ≤ "',)# < * ≤ )' = ,-,. "',)'

a1

a2

b1

b2

Probabilities from Joint CDF

! "# < % ≤ "',)# < * ≤ )' = ,-,. "',)'

a1

a2

b1

b2

Probabilities from Joint CDF

! "# < % ≤ "',)# < * ≤ )' = ,-,. "',)'

−,-,. "#,)'

a1

a2

b1

b2

Probabilities from Joint CDF

! "# < % ≤ "',)# < * ≤ )' = ,-,. "',)'

−,-,. "#,)'

a1

a2

b1

b2

Probabilities from Joint CDF

! "# < % ≤ "',)# < * ≤ )' = ,-,. "',)'

−,-,. "#,)'

−,-,. "',)#

a1

a2

b1

b2

Probabilities from Joint CDF

! "# < % ≤ "',)# < * ≤ )' = ,-,. "',)'

−,-,. "#,)'

−,-,. "',)#

a1

a2

b1

b2

Probabilities from Joint CDF

! "# < % ≤ "',)# < * ≤ )' = ,-,. "',)'

−,-,. "#,)'

−,-,. "',)#

+,-,. "#,)#

a1

a2

b1

b2

Probabilities from Joint CDF

! "# < % ≤ "',)# < * ≤ )' = ,-,. "',)'

−,-,. "#,)'

−,-,. "',)#

+,-,. "#,)#

a1

a2

b1

b2

Probabilities from Joint CDF

Probability for Instagram!

Gaussian BlurIn image processing, a Gaussian blur is the result of blurring an image by a Gaussian function. It is a widely used effect in graphics software, typically to reduce image noise.

0.0000 0.0000 0.0000 0.0001 0.0001 0.0001 0.0000 0.0000 0.00000.0000 0.0001 0.0005 0.0020 0.0032 0.0020 0.0005 0.0001 0.00000.0000 0.0005 0.0052 0.0206 0.0326 0.0206 0.0052 0.0005 0.00000.0001 0.0020 0.0206 0.0821 0.1300 0.0821 0.0206 0.0020 0.00010.0001 0.0032 0.0326 0.1300 0.2060 0.1300 0.0326 0.0032 0.00010.0001 0.0020 0.0206 0.0821 0.1300 0.0821 0.0206 0.0020 0.00010.0000 0.0005 0.0052 0.0206 0.0326 0.0206 0.0052 0.0005 0.00000.0000 0.0001 0.0005 0.0020 0.0032 0.0020 0.0005 0.0001 0.00000.0000 0.0000 0.0000 0.0001 0.0001 0.0001 0.0000 0.0000 0.0000

Gaussian BlurIn image processing, a Gaussian blur is the result of blurring an image by a Gaussian function. It is a widely used effect in graphics software, typically to reduce image noise.

Gaussian blurring with StDev = 3, is based on a joint probability distribution:

fX,Y (x, y) =1

2⇡ · 32 e� x2+y2

2·32

FX,Y (x, y) = �⇣x3

⌘· �

⇣y3

⌘

Joint PDF

Joint CDF

Used to generate this weight matrix

Gaussian Blur

fX,Y (x, y) =1

2⇡ · 32 e� x2+y2

2·32

FX,Y (x, y) = �⇣x3

⌘· �

⇣y3

⌘

Joint PDF

Joint CDF

Each pixel is given a weight equal to the probability that X and Y are both within the pixel bounds. The center pixel covers the area where

-0.5 ≤ x ≤ 0.5 and -0.5 ≤ y ≤ 0.5What is the weight of the center pixel?

P (�0.5 < X < 0.5,�0.5 < Y < 0.5)

=P (X < 0.5, Y < 0.5)� P (X < 0.5, Y < �0.5)

� P (X < �0.5, Y < 0.5) + P (X < �0.5, Y < �0.5)

=�

✓0.5

3

◆· �

✓0.5

3

◆� 2�

✓0.5

3

◆· �

✓�0.5

3

◆

+ �

✓�0.5

3

◆· �

✓�0.5

3

◆

=0.56622 � 2 · 0.5662 · 0.4338 + 0.43382 = 0.206

P (�0.5 < X < 0.5,�0.5 < Y < 0.5)

=P (X < 0.5, Y < 0.5)� P (X < 0.5, Y < �0.5)

� P (X < �0.5, Y < 0.5) + P (X < �0.5, Y < �0.5)

=�

✓0.5

3

◆· �

✓0.5

3

◆� 2�

✓0.5

3

◆· �

✓�0.5

3

◆

+ �

✓�0.5

3

◆· �

✓�0.5

3

◆

=0.56622 � 2 · 0.5662 · 0.4338 + 0.43382 = 0.206

P (�0.5 < X < 0.5,�0.5 < Y < 0.5)

=P (X < 0.5, Y < 0.5)� P (X < 0.5, Y < �0.5)

� P (X < �0.5, Y < 0.5) + P (X < �0.5, Y < �0.5)

=�

✓0.5

3

◆· �

✓0.5

3

◆� 2�

✓0.5

3

◆· �

✓�0.5

3

◆

+ �

✓�0.5

3

◆· �

✓�0.5

3

◆

=0.56622 � 2 · 0.5662 · 0.4338 + 0.43382 = 0.206

P (�0.5 < X < 0.5,�0.5 < Y < 0.5)

=P (X < 0.5, Y < 0.5)� P (X < 0.5, Y < �0.5)

� P (X < �0.5, Y < 0.5) + P (X < �0.5, Y < �0.5)

=�

✓0.5

3

◆· �

✓0.5

3

◆� 2�

✓0.5

3

◆· �

✓�0.5

3

◆

+ �

✓�0.5

3

◆· �

✓�0.5

3

◆

=0.56622 � 2 · 0.5662 · 0.4338 + 0.43382 = 0.206

P (�0.5 < X < 0.5,�0.5 < Y < 0.5)

=P (X < 0.5, Y < 0.5)� P (X < 0.5, Y < �0.5)

� P (X < �0.5, Y < 0.5) + P (X < �0.5, Y < �0.5)

=�

✓0.5

3

◆· �

✓0.5

3

◆� 2�

✓0.5

3

◆· �

✓�0.5

3

◆

+ �

✓�0.5

3

◆· �

✓�0.5

3

◆

=0.56622 � 2 · 0.5662 · 0.4338 + 0.43382 = 0.206

P (�0.5 < X < 0.5,�0.5 < Y < 0.5)

=P (X < 0.5, Y < 0.5)� P (X < 0.5, Y < �0.5)

� P (X < �0.5, Y < 0.5) + P (X < �0.5, Y < �0.5)

=�

✓0.5

3

◆· �

✓0.5

3

◆� 2�

✓0.5

3

◆· �

✓�0.5

3

◆

+ �

✓�0.5

3

◆· �

✓�0.5

3

◆

=0.56622 � 2 · 0.5662 · 0.4338 + 0.43382 = 0.206

Weight Matrix

Four Prototypical Trajectories

Pedagogic Pause

Four Prototypical Trajectories

Properties of Joint Distributions

Boolean Operation on Variable = Event

P (X 5)

P (Y = 6)

Recall: any boolean question about a random variable makes for an event. For example:

P (5 Z 10)

Four Prototypical Trajectories

Conditionals with multiple variables

• Recall that for events E and F:

0)( )()()|( >= FP

FPEFPFEP where

Discrete Conditional Distribution

EF F

FE

• Recall that for events E and F:

• Now, have X and Y as discrete random variables§ Conditional PMF of X given Y:

0)( )()()|( >= FP

FPEFPFEP where

)(),(

)(),()|()|( ,

| ypyxp

yYPyYxXP

yYxXPyxPY

YXYX =

===

====

Discrete Conditional Distributions

Different notations, same idea.

Joint Probability Table

Roommates 2RoomDblShared Partner Single

Frosh 0.30 0.07 0.00 0.00 0.37Soph 0.12 0.18 0.00 0.03 0.32Junior 0.04 0.01 0.00 0.10 0.15Senior 0.01 0.02 0.02 0.01 0.05

5+ 0.02 0.00 0.05 0.04 0.110.49 0.27 0.07 0.18 1.00

Prob

ability

Room | Year

Transport | Year

Lunch | Year

Relationship Status | Year

Number or Function?

Number

Function(or 1D table)

Number or Function?

Number or Function?

2D Function(or 2D table)

P(Buy Book Y | Bought Book X)

And It Applies to Books Too

P (X = x|Y = y) =P (X = x, Y = y)

P (Y = y)

fX|Y (x|y) · ✏x =

fX|Y (x|y) · ✏x · ✏y

fY (y) · ✏y

fX|Y (x|y) =

fX|Y (x|y)fY (y)

Continuous Conditional DistributionsLet X and Y be continuous random variables

P (X = x|Y = y) =P (X = x, Y = y)

P (Y = y)

fX|Y (x|y) · ✏x =

fX,Y (x, y) · ✏x · ✏yfY (y) · ✏y

fX|Y (x|y) =

fX,Y (x, y)

fY (y)

P (X = x|Y = y) =P (X = x, Y = y)

P (Y = y)

fX|Y (x|y) · ✏x =

fX,Y (x, y) · ✏x · ✏yfY (y) · ✏y

fX|Y (x|y) =

fX,Y (x, y)

fY (y)

Warmup: Bayes Revisited

P(B|E) = P(E|B) P(B)

P(E)

Posterior b

eliefPrior belief

Likelihood of evidence

Normalization constant

Mixing Discrete and ContinuousLet X be a continuous random variable

Let N be a discrete random variable

P (X = x|N = n) =P (N = n|X = x)P (X = x)

P (N = n)

PX|N (x|n) =

PN|X (n|x)PX (x)

PN (n)

fX|N (x|n) · ✏x =

PN|X (n|x)fX (x) · ✏x

PN (n)

fX|N (x|n) =

PN|X (n|x)fX (x)

PN (n)

P (X = x|N = n) =P (N = n|X = x)P (X = x)

P (N = n)

PX|N (x|n) =

PN|X (n|x)PX (x)

PN (n)

fX|N (x|n) · ✏x =

PN|X (n|x)fX (x) · ✏x

PN (n)

fX|N (x|n) =

PN|X (n|x)fX (x)

PN (n)

P (X = x|N = n) =P (N = n|X = x)P (X = x)

P (N = n)

PX|N (x|n) =

PN|X (n|x)PX (x)

PN (n)

fX|N (x|n) · ✏x =

PN|X (n|x)fX (x) · ✏x

PN (n)

fX|N (x|n) =

PN|X (n|x)fX (x)

PN (n)

P (X = x|N = n) =P (N = n|X = x)P (X = x)

P (N = n)

PX|N (x|n) =

PN|X (n|x)PX (x)

PN (n)

fX|N (x|n) · ✏x =

PN|X (n|x)fX (x) · ✏x

PN (n)

fX|N (x|n) =

PN|X (n|x)fX (x)

PN (n)

pN|X (n|x) =

fX|N (x|n)pN (n)

fX (x)

P (X = x|N = n) =P (N = n|X = x)P (X = x)

P (N = n)

PX|N (x|n) =

PN|X (n|x)PX (x)

PN (n)

fX|N (x|n) · ✏x =

PN|X (n|x)fX (x) · ✏x

PN (n)

fX|N (x|n) =

PN|X (n|x)fX (x)

PN (n)

All the Bayes Belong to Us

pM|N (m|n) =

PN|M (n|m)pM (m)

pN (n)

M,N are discrete. X, Y are continuous

OG Bayes

Mix Bayes

#1

Mix Bayes

#2

All Continu

ous fX|Y (x|y) =

fY |X (y|x)fX (x)

fY (y)

Warmup: Bayes Revisited

P(B|E) = P(E|B) P(B)

P(E)

Posterior b

eliefPrior belief

Likelihood of evidence

Normalization constant

• X, Y follow a symmetric bivariate normal distribution if they have joint PDF:

Warmup: Bivariate Normal

fX,Y (x, y) =1

2⇡�2· e�

[(x�µx)2+(y�µy)2]

2·�2

Here is an example where:

µx = 3

µy = 3

� = 2

x

y

fX,Y (x, y)

fX,Y (x, y)

-5 53-5

5

3

33xy(top view)

(side view)

Tracking in 2D Space?

You have a prior belief about the 2D location of an object.

What is your updated belief about the 2D location of the object after

observing a noisy distance measurement?

Tracking in 2D Space?

Tracking in 2D Space: Prior

fX,Y (x, y) = K · e�[(x�3)2+(y�3)2]

8

fX,Y (x, y) =1

2⇡�2· e�

[(x�µx)2+(y�µy)2]

2·�2

x

y

fX,Y (x, y)

-5 53-5

5

3

Prior belief:

Prior belief with K:

µx = 3

µy = 3

� = 2

Relative to Satellite at (0, 0)

You will observe a noisy distance reading. It will say that your object is distance D away

We can say how likely that reading is if we know the

actual location of the object…

P(D | X, Y) is knowable!

µ = actual distance

σ = 1

Tracking in 2D Space: Observation!

Observe a ping of the object that is distance D away from satellite!

Know that the distance of a ping is normal with respect to the true distance.

µ = actual distance

σ = 1

D|X,Y ⇠ N(µ =p

x2 + y2,�2 = 1)

Tracking in 2D Space: Observation!

Tracking in 2D Space: Observation!Observe a ping of the object that is distance D = 4 away!

Know that the distance of a ping is normal with respect to the true distance

px2 + y2 = 4 µ = actual distance

σ = 1

Tracking in 2D Space: Observation!Observe a ping of the object that is distance D = 4 away!

Know that the distance of a ping is normal with respect to the true distance

px2 + y2 = 4 µ = actual distance

σ = 1

Observe a ping of the object that is distance D = 4 away from satellite!

D|X,Y ⇠ N(µ =p

x2 + y2,�2 = 1)

Tracking in 2D Space: Observation!

=1p2⇡

e�(d�µ)2

2

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f(D = d|X = x, Y = y) =1

�p2⇡

e�(d�µ)2

2�2

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= K2 · e�(d�µ)2

2<latexit sha1_base64="GoBopu5AfJrX1rOJCi26svjyF2E=">AAACEHicbVDLSsNAFJ34rPUVdelmsIh10ZIEwW6EghvBTQX7gCYNk8mkHTp5MDMRSsgnuPFX3LhQxK1Ld/6N0zYLbT1w4XDOvdx7j5cwKqRhfGsrq2vrG5ulrfL2zu7evn5w2BFxyjFp45jFvOchQRiNSFtSyUgv4QSFHiNdb3w99bsPhAsaR/dykhAnRMOIBhQjqSRXP7uCt64FbezHEhI4yOyAI5zVqj6sQTtMzwdWnll57uoVo27MAJeJWZAKKNBy9S/bj3EakkhihoTom0YinQxxSTEjedlOBUkQHqMh6SsaoZAIJ5s9lMNTpfgwiLmqSMKZ+nsiQ6EQk9BTnSGSI7HoTcX/vH4qg4aT0ShJJYnwfFGQMihjOE0H+pQTLNlEEYQ5VbdCPEIqEakyLKsQzMWXl0nHqptG3by7qDQbRRwlcAxOQBWY4BI0wQ1ogTbA4BE8g1fwpj1pL9q79jFvXdGKmSPwB9rnDxiMmrg=</latexit><latexit sha1_base64="GoBopu5AfJrX1rOJCi26svjyF2E=">AAACEHicbVDLSsNAFJ34rPUVdelmsIh10ZIEwW6EghvBTQX7gCYNk8mkHTp5MDMRSsgnuPFX3LhQxK1Ld/6N0zYLbT1w4XDOvdx7j5cwKqRhfGsrq2vrG5ulrfL2zu7evn5w2BFxyjFp45jFvOchQRiNSFtSyUgv4QSFHiNdb3w99bsPhAsaR/dykhAnRMOIBhQjqSRXP7uCt64FbezHEhI4yOyAI5zVqj6sQTtMzwdWnll57uoVo27MAJeJWZAKKNBy9S/bj3EakkhihoTom0YinQxxSTEjedlOBUkQHqMh6SsaoZAIJ5s9lMNTpfgwiLmqSMKZ+nsiQ6EQk9BTnSGSI7HoTcX/vH4qg4aT0ShJJYnwfFGQMihjOE0H+pQTLNlEEYQ5VbdCPEIqEakyLKsQzMWXl0nHqptG3by7qDQbRRwlcAxOQBWY4BI0wQ1ogTbA4BE8g1fwpj1pL9q79jFvXdGKmSPwB9rnDxiMmrg=</latexit><latexit sha1_base64="GoBopu5AfJrX1rOJCi26svjyF2E=">AAACEHicbVDLSsNAFJ34rPUVdelmsIh10ZIEwW6EghvBTQX7gCYNk8mkHTp5MDMRSsgnuPFX3LhQxK1Ld/6N0zYLbT1w4XDOvdx7j5cwKqRhfGsrq2vrG5ulrfL2zu7evn5w2BFxyjFp45jFvOchQRiNSFtSyUgv4QSFHiNdb3w99bsPhAsaR/dykhAnRMOIBhQjqSRXP7uCt64FbezHEhI4yOyAI5zVqj6sQTtMzwdWnll57uoVo27MAJeJWZAKKNBy9S/bj3EakkhihoTom0YinQxxSTEjedlOBUkQHqMh6SsaoZAIJ5s9lMNTpfgwiLmqSMKZ+nsiQ6EQk9BTnSGSI7HoTcX/vH4qg4aT0ShJJYnwfFGQMihjOE0H+pQTLNlEEYQ5VbdCPEIqEakyLKsQzMWXl0nHqptG3by7qDQbRRwlcAxOQBWY4BI0wQ1ogTbA4BE8g1fwpj1pL9q79jFvXdGKmSPwB9rnDxiMmrg=</latexit><latexit sha1_base64="GoBopu5AfJrX1rOJCi26svjyF2E=">AAACEHicbVDLSsNAFJ34rPUVdelmsIh10ZIEwW6EghvBTQX7gCYNk8mkHTp5MDMRSsgnuPFX3LhQxK1Ld/6N0zYLbT1w4XDOvdx7j5cwKqRhfGsrq2vrG5ulrfL2zu7evn5w2BFxyjFp45jFvOchQRiNSFtSyUgv4QSFHiNdb3w99bsPhAsaR/dykhAnRMOIBhQjqSRXP7uCt64FbezHEhI4yOyAI5zVqj6sQTtMzwdWnll57uoVo27MAJeJWZAKKNBy9S/bj3EakkhihoTom0YinQxxSTEjedlOBUkQHqMh6SsaoZAIJ5s9lMNTpfgwiLmqSMKZ+nsiQ6EQk9BTnSGSI7HoTcX/vH4qg4aT0ShJJYnwfFGQMihjOE0H+pQTLNlEEYQ5VbdCPEIqEakyLKsQzMWXl0nHqptG3by7qDQbRRwlcAxOQBWY4BI0wQ1ogTbA4BE8g1fwpj1pL9q79jFvXdGKmSPwB9rnDxiMmrg=</latexit>

= K2 · e�(d�

px2+y2)2

2<latexit sha1_base64="Y9dwE6JscUnF9hp6E6AyGpr5Y68=">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</latexit><latexit sha1_base64="Y9dwE6JscUnF9hp6E6AyGpr5Y68=">AAACHXicbVBNS8NAEN3Ur1q/qh69LBahIi1JKOhFELwIXipYFZo0bDYbXbr5cHcilpA/4sW/4sWDIh68iP/Gbe1BWx8MPN6bYWaenwquwDS/jNLM7Nz8QnmxsrS8srpWXd+4UEkmKevQRCTyyieKCR6zDnAQ7CqVjES+YJd+/3joX94xqXgSn8MgZW5ErmMeckpAS161dYhPPRs7NEgAM9zLnVASmjfqAW5gR91KyO97Nt7Dg55d7OrK7aLwqjWzaY6Ap4k1JjU0RturfjhBQrOIxUAFUaprmSm4OZHAqWBFxckUSwntk2vW1TQmEVNuPvquwDtaCXCYSF0x4JH6eyInkVKDyNedEYEbNekNxf+8bgbhgZvzOM2AxfRnUZgJDAkeRoUDLhkFMdCEUMn1rZjeEB0P6EArOgRr8uVpcmE3LbNpnbVqRwfjOMpoC22jOrLQPjpCJ6iNOoiiB/SEXtCr8Wg8G2/G+09ryRjPbKI/MD6/AVden5Y=</latexit><latexit sha1_base64="Y9dwE6JscUnF9hp6E6AyGpr5Y68=">AAACHXicbVBNS8NAEN3Ur1q/qh69LBahIi1JKOhFELwIXipYFZo0bDYbXbr5cHcilpA/4sW/4sWDIh68iP/Gbe1BWx8MPN6bYWaenwquwDS/jNLM7Nz8QnmxsrS8srpWXd+4UEkmKevQRCTyyieKCR6zDnAQ7CqVjES+YJd+/3joX94xqXgSn8MgZW5ErmMeckpAS161dYhPPRs7NEgAM9zLnVASmjfqAW5gR91KyO97Nt7Dg55d7OrK7aLwqjWzaY6Ap4k1JjU0RturfjhBQrOIxUAFUaprmSm4OZHAqWBFxckUSwntk2vW1TQmEVNuPvquwDtaCXCYSF0x4JH6eyInkVKDyNedEYEbNekNxf+8bgbhgZvzOM2AxfRnUZgJDAkeRoUDLhkFMdCEUMn1rZjeEB0P6EArOgRr8uVpcmE3LbNpnbVqRwfjOMpoC22jOrLQPjpCJ6iNOoiiB/SEXtCr8Wg8G2/G+09ryRjPbKI/MD6/AVden5Y=</latexit><latexit sha1_base64="Y9dwE6JscUnF9hp6E6AyGpr5Y68=">AAACHXicbVBNS8NAEN3Ur1q/qh69LBahIi1JKOhFELwIXipYFZo0bDYbXbr5cHcilpA/4sW/4sWDIh68iP/Gbe1BWx8MPN6bYWaenwquwDS/jNLM7Nz8QnmxsrS8srpWXd+4UEkmKevQRCTyyieKCR6zDnAQ7CqVjES+YJd+/3joX94xqXgSn8MgZW5ErmMeckpAS161dYhPPRs7NEgAM9zLnVASmjfqAW5gR91KyO97Nt7Dg55d7OrK7aLwqjWzaY6Ap4k1JjU0RturfjhBQrOIxUAFUaprmSm4OZHAqWBFxckUSwntk2vW1TQmEVNuPvquwDtaCXCYSF0x4JH6eyInkVKDyNedEYEbNekNxf+8bgbhgZvzOM2AxfRnUZgJDAkeRoUDLhkFMdCEUMn1rZjeEB0P6EArOgRr8uVpcmE3LbNpnbVqRwfjOMpoC22jOrLQPjpCJ6iNOoiiB/SEXtCr8Wg8G2/G+09ryRjPbKI/MD6/AVden5Y=</latexit>

px2 + y2 = 4

Tracking in 2D Space: New Belief

What is your new belief for the location of the object being tracked? Your joint probability density function can be expressed with a constant

µ = actual distance

σ = 1

x

y

-5 53-5

5

3

(top view)

Prior

Observation

f(D = d|X = x, Y = y) = K · e�[d�p

x2+y2]2

f(X = x, Y = y) = K · e�[(x�3)2+(y�3)2]

81

2 2

Tracking in 2D Space: New Belief

For your notes…

f(X = x, Y = y|D = 4) =f(D = 4|X = x, Y = y) · f(X = x, Y = y)

f(D = 4)

=K1 · e�

[4�p

x2+y2)2]2 ·K2 · e�

[(x�3)2+(y�3)2]8

f(D = 4)

=K3 · e�

⇥[4�

px2+y2)2]2 + [(x�3)2+(y�3)2]

8

⇤

f(D = 4)

= K4 · e�⇥

(4�p

x2+y2)2

2 + [(x�3)2+(y�3)2]8

⇤

Tracking in 2D Space: Posterior

x

y

-5 5-5

5

(top view)

Prior Posterior

(top view) x

y

-5 5

-5

5fX,Y

fX,Y |D

Tracking in 2D Space: CS221