10. Bayes' Theorem & Bayesian Inference: Prior, Likelihood, Posterior & Evidence • Mathematics, Statistics & Machine Learning (Big Tech Prep) • Rishabh Ki Kaksha

Mathematics, Statistics & Machine Learning (Big Tech Prep) › 10. Bayes' Theorem & Bayesian Inference: Prior, Likelihood, Posterior & Evidence

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Course Syllabus

Module 1: Linear Algebra for Machine Learning & Deep Learning

01. Vectors, Vector Spaces, Dot Products & Norms (L1, L2, L-Infinity) in High Dimensions 02. Matrices, Matrix Multiplication & Linear Transformations (Rotation, Scaling, Shear) 03. Eigenvalues & Eigenvectors: Geometric Intuition, Diagonalization & Spectral Theorem 04. Principal Component Analysis (PCA) & Singular Value Decomposition (SVD) Math

Module 2: Multivariable Calculus, Gradients & Optimization Algorithms

05. Partial Derivatives, Directional Derivatives & The Gradient Vector (Del Operator) 06. The Jacobian Matrix & Hessian Matrix (Second Derivative Curvature Analysis) 07. Multivariate Chain Rule & The Mathematical Backbone of Neural Backpropagation 08. Optimization Algorithms: Gradient Descent, SGD with Momentum, RMSprop & Adam Optimizer Math

Module 3: Probability Theory & Random Variables

09. Probability Foundations: Axioms of Probability, Conditional Probability & Independence 10. Bayes' Theorem & Bayesian Inference: Prior, Likelihood, Posterior & Evidence 11. Probability Distributions: Gaussian (Normal), Bernoulli, Binomial, Poisson & Exponential Distributions

Module 4: Inferential Statistics & Hypothesis Testing

12. Expectation, Variance, Covariance & Pearson Correlation Coefficient 13. Law of Large Numbers & Central Limit Theorem (CLT) with Monte Carlo Simulations 14. Hypothesis Testing: Null Hypothesis, p-Values, Z-Test, Student's t-Test & Chi-Square Test 15. Maximum Likelihood Estimation (MLE) vs Maximum A Posteriori (MAP) Estimation

Module 5: Classical Machine Learning Algorithms from Scratch (NumPy)

16. Linear Regression from Scratch: Ordinary Least Squares (OLS) Closed Form vs Gradient Descent 17. Logistic Regression & Binary Cross-Entropy Loss from Scratch with NumPy 18. Regularization Mathematics: L1 Lasso (Sparsity) vs L2 Ridge (Weight Decay) Penalties

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# 10. Bayes' Theorem & Bayesian Inference: Prior, Likelihood, Posterior & Evidence

> Mathematics & ML Foundations Notes by Rishabh - Topic 10

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### 1. Bayes' Theorem: The Engine of Belief Updating 🧠

$$\text{Posterior} = \frac{\text{Likelihood} \times \text{Prior}}{\text{Evidence}}$$

$$P(\theta \mid D) = \frac{P(D \mid \theta) \cdot P(\theta)}{P(D)} = \frac{P(D \mid \theta) \cdot P(\theta)}{\int P(D \mid \theta') P(\theta') d\theta'}$$

*   **Prior $P(\theta)$:** Data dekhne se pehle hamara belief.
*   **Likelihood $P(D \mid \theta)$:** Diye gaye model parameters $\theta$ ke under observed data $D$ create hone ki probability.
*   **Posterior $P(\theta \mid D)$:** Naya evidence/data dekhne ke baad updated refined belief!

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### 2. Spam Detection (Naive Bayes Classifier)

$$P(\text{Spam} \mid \text{words}) \propto P(\text{Spam}) \times \prod_{i=1}^N P(\text{word}_i \mid \text{Spam})$$

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### 3. Master Comparison Matrix

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### 4. Interview Ke 3 Sawal Is Topic Se

1. **Conjugate Prior kya hota hai?** -> Jab prior distribution aur posterior distribution same probability family ke member hon (e.g. Beta prior + Binomial likelihood = Beta posterior).
2. **Evidence $P(D)$ high dimensions me intractable kyu ho jata hai?** -> Kyunki denominator continuous integral across millions of parameters solve nahi ho sakta (Requires MCMC sampling / Variational Inference).
3. **Laplace Smoothing (Additive Smoothing) Naive Bayes me kyu zaroori hoti hai?** -> Unseen words par zero probability ($P(\text{word} \mid \text{class}) = 0$) multiply hone par poora prediction score zero na ho jaye.

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**Next Topic:** 11. Probability Distributions: Gaussian (Normal), Bernoulli, Binomial, Poisson & Exponential Distributions

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