Little Bits of Artificial Intelligence

1. Define the Model Structure

• Decide your regression equation:
  

  y=b0+b1x1+b2x2+⋯+bnxn+ϵ

• Identify our features (inputs) and target variable (output).

2. Assign Priors to Parameters

• Define prior beliefs about each weight (e.g., slope, intercept).
  

  bi∼N(μi,σi2)

• Choose prior mean and variance based on experience or domain knowledge.

3. Collect Observed Data

• Get training data: (X,y)(X, y)(X,y), where X is the feature matrix and y is the output vector.
• Example: experience, education level, city → salary

4. Calculate the Likelihood

• Define how likely the data is given our current model.
  

  yi∼N(Xi⋅b,σ2)

• This expresses how much you trust the data.

5. Apply Bayes’ Theorem

Posterior=Likelihood×Prior / Evidence

• Update our parameter beliefs based on the data.
• This gives us the posterior distribution of each parameter.

6. Make Predictions

• Use the posterior means (or sample from posterior) to make predictions:
  

  y^=X⋅b^

• Include uncertainty intervals if needed (e.g., 95% confidence).

7. Iterate with More Data

• If more data arrives, update the posteriors again (posterior becomes new prior).
• Keeps the model adaptive and learning.

8.Bayesian Regression Flow Diagram

Here’s a visual flowchart that captures all the steps in a clean and linear way:

┌─────────────────────┐
│ Define the Model │
│ (Linear structure) │
└─────────┬───────────┘
↓
┌────────────────────────────┐
│ Assign Prior Distributions │
│ (For intercept & weights) │
└─────────┬──────────────────┘
↓
┌──────────────────────────────┐
│ Collect Observed Data (X,y)│
└────────────┬─────────────────┘
↓
┌──────────────────────────┐
│ Compute Likelihood │
│ P(Data | Parameters) │
└────────────┬─────────────┘
↓
┌──────────────────────┐
│ Apply Bayes’ Rule │
│ Posterior ∝ L × Prior│
└──────────┬───────────┘
↓
┌────────────────────────────┐
│ Make Predictions with │
│ Posterior Means/Distribution│
└────────────┬───────────────┘
↓
┌──────────────────────────────┐
│ Optional: Update with New Data│
│ (Posterior → New Prior) │
└──────────────────────────────┘