CHAPTER 06 / 40
Probability and Distributions
7 lessons · Explore, experiment and understand.
Let’s figure it out.
Ask me about this course or where to start.
Distributions, expectation and Bayes' rule are how models score predictions, calibrate confidence, and combine evidence — the substrate of every probabilistic model.
- softmax
- Bayesian inference
- calibration
- Step 32~30m111⚡Challenge
Probability Spaces
A probability space is a mathematical stage for reasoning about uncertainty. It consists of three ingredients: a sample space (the set of al…
Open lesson - Step 33~20m25⚡Challenge
Discrete and Continuous Probabilities
Random variables come in two flavors. A discrete random variable takes values in a finite or countable set (like or ) and is described by a…
Open lesson - Step 34~15m15⚡Challenge
Sum Rule, Product Rule, Bayes' Theorem
Three rules underpin all of probability. The sum rule (or marginalization): (or in the continuous case). Given a joint distribution, you rec…
Open lesson - Step 35~15m15⚡Challenge
Summary Statistics and Independence
Distributions are high-dimensional objects, but we often compress them into a few summary statistics. The expected value (mean) (or ) gives…
Open lesson - Step 36~20m3D · 125★Most cinematic⚡Challenge
Gaussian Distribution
The Gaussian (or Normal) distribution is the most important distribution in statistics and ML. A univariate Gaussian has density , parameter…
Open lesson - Step 37~20m25⚡Challenge
Conjugacy and Exponential Family
In Bayesian inference, we update a prior with a likelihood to get a posterior . The posterior is generally messy — but for certain conjugate…
Open lesson - Step 38~15m15⚡Challenge
Change of Variables
Given a random variable with density , what is the density of ? The answer is the change of variables formula: for a smooth, invertible , .…
Open lesson