Conditional Probability
Examples
Wherever Mary went, her lamb was sure to go.
Wherever Mary went, her brother was sure to avoid.
Mary and Tom pay no attention to each other
Bayes Theorem
Rev Thomas Bayes (1764 [posthumously])
Conditionalization
Condition
Breast Cancer
A screening test has a 90% chance of registering breast cancer if it exists, as well as a 20%
chance of falsely registering cancer when it does not exist. About one in one hundred
women requesting the screening test end up diagnosed with bre
DNA Example 2
A DNA match between the defendant and a crime scene blood sample has a
probability of 1/100000 if the defendant is innocent. There is no other signicant
evidence. Suppose we agree that the prior probability of guilt under the (unspecied)
cir
Conditional Probability
Conditional probability describes how to change your probabilities
over time
incorporating new evidence
-> idealization, but a very useful one
Since we are constantly trying to assimilate evidence learn about the world
conditionin
Betting and Odds
Given P (h), what is a fair bet on h?
A bet is fair (relative to P () if and only if the
expected value of the bet is zero.
I.e.,
This will be satised in case betting odds are set
Betting and Odds
Example:
A fair bet that a (fair) die wil
Risk
There are two aspects to risk:
1. Probability
2. Value
The expected value of an action is the weighted value of its possible outcomes.
What is the expected value of a bet on heads that returns $3 and costs $1?
Tthis is not a fair bet.
A bet is fair w
Suppose we are more pessimistic and think P (drug) is 50%.
Counting we get: P (drugjpos) = 95=100.
FF: Vaccine
There is a disease X whose prevalence amongst people like you is expected to
reach 1/1000. X kills 2% of the people it strikes. A vaccine protec
Odds-Likelihood Bayes
P (ejh)=P (ej:h) is the likelihood ratio (ejh)
(aka Bayes Factor)
So, Odds-Likelihood Bayes says:
Posterior betting odds are equal to prior odds times
Since, odds and probabilities are interchangeable, is conrmatory power of the
evid
FF: Breast Cancer
Let P (h) = 0:01 (one in 100 women tested have it)
P (ejh) = 0:8 and P (ej:h) = 0:1
(true and false positive rates). What is P (hje)? Look at
frequencies among 1000 women.
Counting we get: P (hje) = 9=(9 + 99).
Recall odds-likelihood com
Korb
1
Bayesian Reasoning Notation
X \ Y : Intersection of X and Y
X
Y : Union of X and Y
X : Complement of X (whats not in X)
8X : For all X
X
Y : X is a subset of Y
:h : Not h
P (X ) : Probability of X
P (X jY ) : Probability of X given Y
P (h) : Prior