Today, we continue probability.
Slides
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.
\[ Pr(User | +) = \frac{Pr(+ | User) Pr(User)}{Pr(+)} \]
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:
\(Pr(User, +) = Pr(+ | User)*Pr(User)\)
\(Pr(User, -) = Pr(- | User)*Pr(User)\)
\(Pr(\overline{User}, +) = Pr(+ | \overline{User})*Pr(\overline{User})\)
and
\(Pr(\overline{User}, -) = Pr(- | \overline{User})*Pr(\overline{User})\)
The Table
| Status | Positive | Negative | Total |
|---|---|---|---|
| User | 0.099 | 0.001 | 0.1 |
| non-User | 0.045 | 0.855 | 0.9 |
| ——— | ———– | ———- | —— |
| Total | 0.144 | 0.856 | 1.0 |
\(Pr(User | +) = \frac{Pr(+ | User) Pr(User)}{Pr(+)}\)
yields:
\(Pr(User | +) = \frac{0.099 [0.99*0.1]}{0.144[0.99*0.1 + 0.05*0.9]} = 0.6875\)
One key bit of data
UCB Admissions
Is there evidence of sex bias in admission practices? Data on admissions to the University of California at Berkeley graduate programs in six departments are presented for the population of applicants. Admit measures whether or not the applicant was admitted; M.F measures the gender of the applicant; Dept measures the department in a classification ranging from A to F.
The class plan:
- Defining probability.
- Pivots and tables.
- Visualizing tables.
Your assignment for this time:
- Do the reading.
- The assignment: use Gemini and/or Notebook LM to complete the three excerpts from Jaynes.
For Next Class:
- Reading: Chapter 3 if not already and read through Jaynes on Probability.
- Deliverable: A quiz on probability [Assignment 10].