M.F No Yes
Female 0.2823685 0.1230667
Male 0.3298719 0.2646929
prop.table(table(UCBAdmit$M.F,UCBAdmit$Admit))
No Yes
Female 0.2823685 0.1230667
Male 0.3298719 0.2646929
Marginal Probability
The row/column sums to one. We collapse the table to a single margin. Here, two can be identified. The probability of Admit and the probability of M.F.
UCBAdmit %>%tabyl(M.F)
M.F n percent
Female 1835 0.4054353
Male 2691 0.5945647
UCBAdmit %>%tabyl(Admit)
Admit n percent
No 2771 0.6122404
Yes 1755 0.3877596
prop.table(table(UCBAdmit$M.F))
Female Male
0.4054353 0.5945647
prop.table(table(UCBAdmit$Admit))
No Yes
0.6122404 0.3877596
Conditional Probability
How does one margin of the table break down given values of another? Each row or column sums to one
Four can be identified, the probability of admission/rejection for Male, for Female; the probability of male or female for admits/rejects.
The University says no. Why? The most important factor in the probability of admission is likely to be the department. This has a huge impact on what we see.
To find the joint probability [the intersection] of x and y, we can use either of the aforementioned methods. To turn this into a conditional probability, we simply take it is a proportion of the relevant margin.
Pr(x | y) = \frac{Pr(y | x) Pr(x)}{Pr(y)}
Applications of these Topics
Churn
Prior customers and current customers; engagement metrics, etc. The rows are prior state: customer/user and not. The columns are current state: customer/user and not.
The churn rate is the rate at which prior customers become current non-customers: a conditional probability.
Three nodes: guilty and not at each, convict at the third.
The Magic of Bayes Rule
To find the joint probability [the intersection] of x and y, we can use either of the aforementioned methods. To turn this into a conditional probability, we simply take it is a proportion of the relevant margin.
Pr(x | y) = \frac{Pr(y | x) Pr(x)}{Pr(y)}
By itself, this is algebra. It is magic in an application.
This poses the question: what does a positive test mean?
Working an Example
Suppose a test is 99% accurate for Users and 95% accurate for non-Users. Moreover, suppose that Users make up 10% of the population. So given some population to which this applies, we have:
Sensitivity refers to the ability of a test to designate an individual with a disease as positive. Specificity refers to the ability of a test to designate an individual without a disease as negative.
False positives are then the complement/opposite of specificity and false negatives are the complement/opposite of sensitivity.
Truth
Positive Test
Negative Test
Positive
Sensitivity
False Negative
Negative
False Positive.
Specificity
Applied to Hypothesis Testing
When we get to hypothesis testing in inference, this comes up again with null and alternative hypotheses and the related decision.
Truth
Reject Null
Accept Null
Alternative
Correct
Type II error
Null
Type I error
Correct
A Core Idea: Independence
What does it mean to say something is independent of something else? The simplest way to think about it is, “do I learn something more about x by knowing y than not”. If two things are independent, I don’t need to care about y if x is my objective.
Random Variables
I do not love the book definition of this. Technically, it is a variable whose values are generated according to some random process; your book implies that these are limited to quantities. It is really a measurable function defined on a probability space that maps from the sample space [the set of possible outcomes] to the real numbers.
