Basic Math Concepts – Bayesian Regression
1. Algebra
- Understanding of variables, equations, and linear expressions
- Ability to manipulate formulas (e.g., solve for x in y = mx + b)
- Basic use of sums, averages, and fractions
Why: Bayesian regression uses equations to model the relationship between inputs and outputs.
2. Probability
- Basic idea of probability (e.g., 0.8 means 80% chance)
- Conditional probability: P(A|B) = chance of A given B
- Understanding of prior, likelihood, posterior
Why: Bayesian thinking is all about updating beliefs based on probabilities.
3. Bayes’ Theorem
- Know the formula:
P(H∣D)=P(D∣H)⋅P(H) / P(D)
(Posterior = Likelihood × Prior / Evidence)- Intuition of updating a guess when new data arrives
Why: This is the foundation of all Bayesian learning.
4. Linear Functions and Regression
- Understanding of lines and slopes: y = mx + b
- How regression finds the best line that fits data
- Concept of multivariable linear regression: y = b0 + b1*x1 + b2*x2 + …
Why: Bayesian regression is just linear regression + uncertainty.
5. Variance and Standard Deviation
- Know that variance measures spread in the data
- Variance of data and prediction errors
- How confidence depends on variance
Why: In Bayesian models, uncertainty is explicitly modeled using variance.
6. Matrix Basics (for multivariate regression)
- Idea of a matrix as a table of numbers
- Simple operations like matrix multiplication
- Concept of dot product (for vector multiplication)
Why: Multivariate Bayesian regression often uses matrices to manage multiple variables efficiently.
MATH WARM-UP KIT for Bayesian Regression
1. Algebra Basics
Concept: Manipulating equations and expressions
| Topic | Exercise |
|---|---|
| Solve for x | If 3x + 5 = 20, what is x? |
| Plug into equations | If y = 2x + 7, what is y when x = 3? |
| Rearranging terms | Rewrite y = mx + b to solve for x |
Practice Tip: Think of data points like (x, y) — this helps with regression understanding.
2. Probability Fundamentals
Concept: Understanding chances and likelihoods
| Topic | Exercise |
|---|---|
| Basic probability | A dice shows 1–6. What’s the chance of getting a 4? |
| Conditional probability | If 70% people have a car, and 80% of them also have insurance, what’s P(insurance | car)? |
| Complement rule | If it rains 30% of the time, what’s the chance it doesn’t rain? (1 – P) |
Practice Tip: Think in terms of real-world examples — weather, emails, choices.
3. Bayes’ Theorem (Intuition)
Concept: Updating beliefs with new evidence
| Topic | Exercise |
|---|---|
| Formula fill | Given: P(Disease) = 0.01, P(Pos | Disease) = 0.9, P(Pos | No Disease) = 0.05. What is P(Disease | Pos)? |
| Conceptual | Why does the prior matter when data is limited? (Try using fake data) |
| Application | You think a product has 50% chance to sell. After 10 trials & 7 buys, how will you update belief? |
Practice Tip: Sketch a tree diagram of possibilities to help visualize updates.
4. Linear Regression Intuition
Concept: Predicting using a best-fit line
| Topic | Exercise |
|---|---|
| Equation plugging | If salary = 10000 + 50000 × years, what is salary at 3 years? |
| Slope meaning | If slope = 2000, what does it mean in real life? |
| Feature extension | Add education_score to salary formula: salary = b₀ + b₁×exp + b₂×edu and compute a value. |
Practice Tip: Try building a mini table with X and Y values and guess the line.
5. Variance & Uncertainty
Concept: Understanding spread or inconsistency in data
| Topic | Exercise |
|---|---|
| Variance by hand | Compute variance of [4, 6, 8, 10] |
| Mean vs variability | Why is “mean alone” not enough in predictions? |
| Use case thinking | How does higher variance affect our confidence in salary predictions? |
Practice Tip: Use small lists and work with pen & paper. Visualize.
Bayesian Regression – Summary