For example, when flipping a coin twice, knowing whether the first flip is Heads or Tails gives us no information about whether the second flip is Heads. However, knowing whether the first flip is Heads gives us information about whether both flips are Tails.
Let's define the event "people with cancer" as $A$ and "people with no cancer" as $\neg A$.
What is the probability of A?
$$P(A) = \frac{|A|}{|U|}$$Say we are studying cancer, so we observe people and see whether they have cancer or not. If we take as our Universe all people participating in our study, then there are two possible outcomes for any particular individual, either he has cancer or not.
Questions:
What is the max probability of A?
Since $|A| <= |U|$ (number of elements of A <= number of elements of U), $P(A) <= 1$ (probability <= 100%).
Let's define the event "people who tested positive for cancer" as $B$ and "people who tested negative for cancer" as $\neg B$.
What is the probability of B?
$$P(B) = \frac{|B|}{|U|}$$Let’s say there is a new screening test that is supposed to test for cancer. That test will be “positive” for some people, and “negative” for some other people. If we take the event $B$ to mean “people for which the test is positive”.
What is the probability of AB?
$$P(AB) = \frac{|AB|}{|U|}$$Given that the test is positive for a randomly selected individual, what is the probability that said individual has cancer?
$$P(A|B) = \frac{|AB|}{|B|}$$Questions:
How would you describe the “cancer status” and “test status” of people in each portion of the diagram (by color)?
Conditional Probability Notes
The notation for this is P(A|B) and it is read “the probability of A given B”.
What we’ve effectively done is change the Universe from U (all people), to B (people for whom the test is positive).
This is known as transforming the sample space.
Probability that one event occurs given that another event has occurred.
Probability of A given B (prob of cancer given that the test is positive)
$$ P(A|B) = \frac{P(AB)}{P(B)} $$Probability of B given A (prob of testing positive given that you have cancer)
$$ P(B|A) = \frac{P(AB)}{P(A)} $$Note that when writing a joint probability the order does not matter $P(AB) == P(BA)$.
Researchers randomly assigned 72 chronic users of cocaine into three groups: desipramine (antidepressant), lithium (standard treatment for cocaine) and placebo. Results of the study are summarized below.
| relapse | no relapse | total | |
|---|---|---|---|
| desipramine | 10 | 14 | 24 |
| lithium | 18 | 6 | 24 |
| placebo | 20 | 4 | 24 |
| total | 48 | 24 | 72 |
WRITE THESE QUESTIONS ON THE BOARD AND HAVE PEOPLE SOLVE THEM
Marginal Probability
P(relapsed) = 48 / 72 ~ 0.67
Joint Probability
P(relapsed and desipramine) = 10 / 72 ~ 0.14
Conditional Probability
talk about how discriminative learning algorithms learn the DIFFERENCE between multiple classes. i.e. a logistic regression trying to find the best fit line between the classes
generative learning models looks at each class individually and tries to learn that class in of itself. then it looks at a new observation and sees which model (for each class) it more closely resembles