15. Maximum Likelihood Estimation (MLE) vs Maximum A Posteriori (MAP) Estimation • Mathematics, Statistics & Machine Learning (Big Tech Prep) • Rishabh Ki Kaksha

Mathematics, Statistics & Machine Learning (Big Tech Prep) › 15. Maximum Likelihood Estimation (MLE) vs Maximum A Posteriori (MAP) Estimation

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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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# 15. Maximum Likelihood Estimation (MLE) vs Maximum A Posteriori (MAP) Estimation

> Mathematics & ML Foundations Notes by Rishabh - Topic 15

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### 1. Maximum Likelihood Estimation (MLE) 🎯

Parameters $\theta$ aise dhoondho jo observed data $D$ create hone ki probability ko maximize karein:
$$\hat{\theta}_{\text{MLE}} = \arg\max_\theta P(D \mid \theta) = \arg\min_\theta -\log P(D \mid \theta) \quad (\text{Negative Log Likelihood - NLL})$$

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### 2. Maximum A Posteriori (MAP): Adding Prior Regularization 🧠

$$\hat{\theta}_{\text{MAP}} = \arg\max_\theta P(\theta \mid D) = \arg\max_\theta \left[ \log P(D \mid \theta) + \log P(\theta) \right]$$

> **THE GRAND UNIFICATION OF MACHINE LEARNING:**
> 1. **MLE with Gaussian Error** $\equiv$ **Ordinary Least Squares (OLS) Linear Regression (MSE Loss)**!
> 2. **MAP with Gaussian Prior $\mathcal{N}(0, \sigma^2)$** $\equiv$ **L2 Ridge Regularization (Weight Decay)**!
> 3. **MAP with Laplace Prior** $\equiv$ **L1 Lasso Regularization (Sparsity)**!

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### 3. Master Comparison Matrix

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

1. **Log-Likelihood product of probabilities se computational perspective se behtar kyu hai?** -> Chhote probabilities ($10^{-5}$) multiply karne par floating-point arithmetic underflow ho jata hai; Log lene se products simple additions ($\sum \log p_i$) me convert ho jaate hain.
2. **MAP estimate full Bayesian inference se kaise alag hai?** -> MAP sirf posterior distribution ka single peak maximum point (Mode) nikalta hai; Full Bayesian inference poore posterior distribution ko integrate karti hai.
3. **Bernoulli distribution ka Negative Log-Likelihood kiske barabar hota hai?** -> Exact **Binary Cross-Entropy (BCE) Loss** ke barabar!

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**Next Topic:** 16. Linear Regression from Scratch: Ordinary Least Squares (OLS) Closed Form vs Gradient Descent

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