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Joint Probability Distribution and Conditional Independence in Weather and Fun Scenarios, Study notes of Probability and Statistics

The concepts of joint probability distribution and conditional independence through the example of weather conditions and the presence or absence of fun. the difference between probability distributions and probabilities, the concept of discrete random variables, and the idea of conditional probability. It also covers the concept of conditional independence and its implications, using the example of weather given the presence or absence of fun. The document also touches upon the topic of estimating parameters from data and the use of Bayesian networks for modeling complex systems.

What you will learn

  • What is the difference between probability distributions and probabilities?
  • What is the concept of conditional probability?
  • What is a discrete random variable?
  • What is conditional independence and what are its implications?
  • How can Bayesian networks be used to model complex systems?

Typology: Study notes

2021/2022

Uploaded on 09/27/2022

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Joint Probability Distribution
Sunny Cloudy Rainy Snowy
Yes 0.25 0.15 0.05 0.13
No 0.05 0.1 0.25 0.02
Weather
Have
Fun?
pf3
pf4
pf5
pf8
pf9
pfa
pfd
pfe
pff

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Joint Probability Distribution

Sunny Cloudy Rainy Snowy

Yes 0.25 0.15 0.05 0.

No 0.05 0.1 0.25 0.

Weather

Have

Fun?

Distribution vs. Probability

discrete random variable

P(A) denotes a distribution

Probabilities for each value

P(A | B) denotes a different distribution across the

same values

P(A = a

2

) denotes the probability that A takes on the

value a

2

P(A = a

2

, B = b

3

) denotes the probability that A takes on

the value a

2

and B takes on the value b

3

Distribution

P(Weather | Fun=Yes)

P(Weather = Sunny | Fun = Yes) = 0.431+

0

Sunny Cloudy Rainy Snowy

Weather given Fun=Yes

Distribution Statements

P(A) = P(A | C)

vs

P(A) = P(A | C=c

5

Joint Probability Distribution

Sunny Cloudy Rainy Snowy

Yes 0.25 0.15 0.05 0.

No 0.05 0.1 0.25 0.

Weather

Have

Fun?

Now add RV for Time Of Day in 6hr

blocks

Conditional Independence

• Model the disease influencing the symptoms

• But no symptom interactions given the

disease

• Conditional independence:

P(A|B,C) = P(A|C)

“A is conditionally independent of B given C”

More Conditional Independence

If A and B are conditionally independent given C

C can still depend on both A and B

P(C|A)  P(C) and P(C|B)  P(C)

A and B are not necessarily independent:

P(B|A)  P(B) and P(A|B)  P(A)

But, Given C,

A and B do not influence each other

Knowledge of C separates A and B

Suppose we do NOT know C,

then A and B MAY influence each other

(think about this one until it’s intuitive

  • what does “influence” mean here?)

Suppose that Symptoms are Conditionally

Independent given the Disease

  • We know John has a cold
  • Congestion, a sore throat, headache, rash

are more likely but not necessary

  • We discover he does in fact have a sore throat
  • Is he now more or less likely to also have a

headache?

Dentist Example

A

C

B

3 Boolean Random Variables:

C – Patient has a cavity

A – Patient reports a

toothache

B – Dentist’s probe catches

on tooth

C directly influences A and B

A and B are conditionally independent given C

CPT at each node specifies a probability distribution for each context

(configuration of parents)

How many numbers do we need for the full joint?

How many for the Bayesian net?

How Many Numbers

Joint:

3

  • 1 = 7 (why?)

Bayes Net:

Distribution(s) at C

P(cavity), P(cavity)

Need one number

Distribution(s) at A

P(ache|cavity), P(ache|cavity)

P(ache| cavity), P(ache| cavity)

Distributions at B, like A

1 + 2 + 2 = 5 (a savings of 2!)

A

C

B

How Many Numbers?

• Suppose A,B,C,D,W are trinary

• How many numbers for

the joint?

• How many numbers

for the BN?

• Suppose A is binary,

B is trinary, C takes on 4 values

D takes on 5, W takes on 6?

A

C

B

D

W

5

  • 1 = 242

W 5, D 4, C 3, A 4, B 8 = 24

Notation

  • W, C, D, A, B are random variables
    • C (binary presence of cavity) ranges over values
    • C’s values might be Yes or No; 1 or 0; c 1

or c 2

  • Probability distribution: P(C)
  • Probability value: P(C=c 1

) or P(c

1

) or P(c)

  • P(A,B,C,W,D) full joint probability distribution
  • P(a,b,c,w,d) some particular entry in the joint