Statistical Inference
Business Analytics
Why Learn About a population?
Example: When Learning is Costly
- Maybe you are a hedge fund that wants to identify who might be a good trader.
- Hire everyone that applies
- One Idea
- Allow everyone you hired to trade for one year
- Fire everyone who is not good at trading
- Evaluate everyone's trading performance
Why is this not a good idea?
- Letting people trade who are not good at trading is expensive
- Firing people is expensive
- The avg. severance pay at Meta for the 11,000 employees laid of in 2022 was $88,000
It's really expensive to find out through experience who is good at trading and who is not.
- Initially, hire a small random subset of people who apply and evaluate them in a simulated trading environment
- Allow everyone you hired to trade
- In the future, hire only those who have a high estimated likelihood of real world trading success
- Learn the relationship between individual characteristics (like their score in the simulated trading environment) and real-world trading success
What's a Better Idea?
Why is this a Better Idea?
Probability Spaces
Motivation
- Composition is a fundamental way in which we can build new ideas using our existing ideas
- By learning about probability spaces, ideas in statistics become composable, which means we can build with them
Probability Space
- A probability space is a mathematical structure for representing uncertainty
- One can think of it as a tule with three elements
\Big(\Omega, \mathcal{F}, \mathbb{P}\Big)
The set of all possible outcomes
The set of all events that we can assign probability to
Sample Space
Event Space
A Function that assigns the probabilities
Probability Measure
Probability Space Example
- Let's say that we are considering hiring some unknown individual to be a trader, how can we use a probability space to represent our uncertainty?
\Big(\Omega, \mathcal{F}, \mathbb{P}\Big)
The set of all people we could hire (finite set)
The set of all subsets of people that we could hire
Sample Space
Event Space
A Function that assigns 1/n
probability to each {i}
Probability Measure
Probability Space Example (Continued)
- Our hedge fund doesn't care about the exact person that we hire
- We don't care if the person is a Chicago Cubs fan
- We don't care if that person has a dog
- What we care about is the performance of the person as a trader. That is, how much $ they will generate for the firm
Probability Space Example (Continued)
- Our hedge fund doesn't care about the exact person that we hire
- We don't care if the person is a Chicago Cubs fan
- We don't care if that person has a dog
- What we care about is the performance of the person as a trader. That is, how much $ they will generate for the firm