Baye's Theorem w/ Multiple Events

In summary, Baye's Theorem with Multiple Events is a mathematical theorem used to calculate the probability of an event occurring based on prior knowledge or information. It is commonly used in various fields of science, such as genetics and epidemiology, to update predictions as new data becomes available. The assumptions of the theorem include independence of events, finite probabilities, and consideration of all possible outcomes. Baye's Theorem with Multiple Events differs from Baye's Theorem with a single event as it allows for multiple pieces of evidence and can update probabilities over time. However, some limitations of the theorem include the need for accurate data, potential bias, and complexity when dealing with multiple events.
  • #1
dnbwise
2
0
So, Bayes' Theorem states

P(H|D) = P(D|H) X P(H) / [P(D|H) X P(H) + P(D|~H) x P(~H)], where ~ = negation

If my hypothesis consists of two events - S and O - would this be

P(S,O|D) = P(D|S,O) X P(S,O) / [P(D|S,O) X P(S,O) + P(D|~S,~O) x P(~S,~O)]

or this

P(S,O|D) = P(D|S,O) X P(S,O) / [P(D|S,O) X P(S,O) + P(D|~S,O) x P(~S,O) + P(D|S,~O) x P(S,~O) + P(D|~S,~O) x P(~S,~O)] ?
 
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  • #2
dnbwise said:
So, Bayes' Theorem states

P(H|D) = P(D|H) X P(H) / [P(D|H) X P(H) + P(D|~H) x P(~H)], where ~ = negation

If my hypothesis consists of two events - S and O - would this be

P(S,O|D) = P(D|S,O) X P(S,O) / [P(D|S,O) X P(S,O) + P(D|~S,~O) x P(~S,~O)]

or this

P(S,O|D) = P(D|S,O) X P(S,O) / [P(D|S,O) X P(S,O) + P(D|~S,O) x P(~S,O) + P(D|S,~O) x P(S,~O) + P(D|~S,~O) x P(~S,~O)] ?

the second formulation looks right.
 
  • #3
dnbwise said:
So, Bayes' Theorem states

P(H|D) = P(D|H) X P(H) / [P(D|H) X P(H) + P(D|~H) x P(~H)], where ~ = negation

Show how P(D)=P(D|H)P(H)+P(D|1-H)P(1-H)
 
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  • #4
SW VandeCarr said:
Show how P(D)=P(D|H)P(H)+P(D|1-H)(P|1-H)

HnH' = 1

D = HnD+ H'nD

P(D)= P (HnD) + (H'nD)= P(D | H) P(H) + P(D | H') P(H')
 
  • #5
ych22 said:
HnH' = 1

D = HnD+ H'nD

P(D)= P (HnD) + (H'nD)= P(D | H) P(H) + P(D | H') P(H')

Can you derive this from Bayes' Theorem?:

P(D)=P(D|H)P(H)/P(H|D) is equivalent to P(D)=P(D|H)P(H)+P(D|1-H)P(1-H)?

EDIT: I think it's more clear to use probability notation P(1-H) rather than logical notation P(~H) in what is essentially a conditional probability problem.
 
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  • #6
I think you meant

HuH' = 1

Given that D = HnD + H'nD can we assume that P(D) = P(HnD) + P(H'nD) ?

HuH' = 1
or H' = 1 - H
HnD = P(D|H)P(H)
H'nD = P(D|H - 1)(1 - P(H))

P(D) = P(D|H)P(H) + P(D|H - 1)(1 - P(H))

but bayes theorem says P(D)=P(D|H)P(H)/P(H|D)

so i think it's Ok
 
  • #7
Note:

D=P(D|H)P(H)+P(D|1-H)P(1-H) is in terms of joint probabilities (terms cancel);

D=P(D|H)P(H)/P(H|D) is in terms of a conditional probability.

You can express D either way and you can write a conditional probability in terms of joint probabilities as the OP has done. However, you cannot write D in terms of Bayes Theorem without a conditional probability.
 
  • #8
Sorry, that's P(D) where D appears in the previous post.
 

What is Baye's Theorem with Multiple Events?

Baye's Theorem with Multiple Events is a mathematical theorem that helps calculate the probability of an event occurring based on prior knowledge or information. It allows for the incorporation of new information to update the probability of an event.

How is Baye's Theorem with Multiple Events used in science?

Baye's Theorem with Multiple Events is used in various fields of science, including genetics, epidemiology, and decision-making. It allows scientists to update their knowledge and predictions as new data becomes available.

What are the assumptions of Baye's Theorem with Multiple Events?

The assumptions of Baye's Theorem with Multiple Events include independence of events, each event having a finite probability, and that all possible outcomes are included in the calculation.

How is Baye's Theorem with Multiple Events different from Baye's Theorem with a single event?

Baye's Theorem with Multiple Events allows for the consideration of multiple pieces of evidence or information, whereas Baye's Theorem with a single event only considers one piece of evidence. Additionally, Baye's Theorem with Multiple Events can be used to update the probability of an event over time as new information is obtained.

What are some limitations of Baye's Theorem with Multiple Events?

Some of the limitations of Baye's Theorem with Multiple Events include the assumption of independence between events, the need for accurate and reliable data, and the potential for bias in the prior probabilities used. It also requires a thorough understanding of conditional probabilities and can become complex when dealing with multiple events.

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