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Foundations of Probability

  • What is Probability?
  • Theoretical vs Empirical Probability
  • Three Views of Probability
  • Sample Space and Events
  • Axioms of Probability
  • Independence and Expectation
  • Variance and Standard Deviation
  • Covariance and Correlation
  • Key Inequalities

Set Theory & Combinatorics

  • Set Operations in Probability
  • Counting Methods
  • Advanced Counting

Conditional & Bayesian Probability

  • Conditional Probability
  • Bayes' Theorem
  • Law of Total Probability

Random Variables & Distributions

  • What is a Random Variable?
  • Discrete vs Continuous
  • PDFs and CDFs
  • Expectation, Variance, and Moments

Discrete Distributions

  • Bernoulli and Binomial
  • Poisson and Geometric
  • Negative Binomial and Hypergeometric

Continuous Distributions

  • Uniform and Normal
  • Exponential, Gamma, Beta
  • Heavy-Tailed Distributions

Limit Theorems

  • Law of Large Numbers
  • Central Limit Theorem
  • Convergence in Probability vs Distribution

Frequentist Inference

  • Confidence Intervals
  • Hypothesis Testing
  • p-values and Statistical Decisions
  • Type I and Type II Errors
  • Power and Effect Size
  • Bootstrapping and Resampling

Advanced Probability Tools

  • Law of the Unconscious Statistician
  • Moment Generating Functions
  • Characteristic Functions
  • Markov Chains
  • Stationary Distributions

Bayesian Inference

  • Bayesian Philosophy
  • Prior, Likelihood, Posterior
  • Conjugate Priors
  • MCMC and Modern Computation

Regression Analysis

  • Ordinary Least Squares
  • Multiple Linear Regression
  • Regression Diagnostics
  • Regularization
  • Logistic and Generalized Linear Models

Multivariate Statistics

  • Joint, Marginal, and Conditional
  • Multivariate Normal
  • Covariance Matrices
  • Correlation vs Causation
  • Principal Component Analysis

Stochastic Processes

  • Random Walks
  • Poisson Processes
  • Brownian Motion
  • Itô's Lemma
  • Martingales
  • Geometric Brownian Motion

Simulation & Approximation

  • Monte Carlo Simulation
  • Variance Reduction
  • Bootstrapping for Finance
  • Quasi-Monte Carlo

Time Series

  • Stationarity and Autocorrelation
  • AR, MA, and ARIMA
  • GARCH and Volatility Clustering
  • Cointegration and Pairs Trading
  • Kalman Filters

Information Theory

  • Shannon Entropy
  • Kullback–Leibler Divergence
  • Mutual Information
  • Maximum Entropy

Linear Algebra

  • Vectors, Norms, and Inner Products
  • Matrix Operations
  • Eigenvalues and Eigenvectors
  • Singular Value Decomposition
  • Positive Definite Matrices
  • Numerical Stability

Calculus & Optimization

  • Multivariate Calculus
  • Lagrange Multipliers
  • Convex Optimization
  • Gradient Descent and Variants
  • Stochastic Calculus Primer

Machine Learning Fundamentals

  • Supervised vs Unsupervised
  • Bias–Variance Trade-off
  • Cross-Validation
  • Tree-Based Methods
  • Support Vector Machines
  • Clustering and Dimensionality Reduction
  • Classification Metrics

Deep Learning

  • Feedforward Networks
  • Backpropagation
  • Optimizers and Schedules
  • Regularization in DL
  • Architectures for Finance
  • Loss Functions

Options Pricing

  • Payoffs and Put–Call Parity
  • Risk-Neutral Valuation
  • Binomial Trees
  • Black–Scholes
  • The Greeks
  • Volatility Smile and Surface
  • Exotic Options

Portfolio Theory

  • Mean–Variance Optimization
  • CAPM and Factor Models
  • Sharpe, Sortino, and Information Ratio
  • Black–Litterman
  • Risk Parity

Trading & Risk Applications

  • Value-at-Risk
  • Expected Shortfall
  • Backtesting
  • Market Making Basics
  • Execution and Market Microstructure
  • Statistical Arbitrage
Study Guide/Machine Learning Fundamentals
Section 19 · Lesson 19.84

Supervised vs Unsupervised

Learning from labels versus learning structure from data alone.

Supervised learning fits a function f:X→Yf: X \to Yf:X→Y from labeled training data (xi,yi)(x_i, y_i)(xi​,yi​). The labels yyy might be discrete (classification: spam vs not, default vs not) or continuous (regression: predicted return, predicted volatility).

Unsupervised learning has no labels — only inputs xix_ixi​. The goal is to find structure: clusters of similar points, low-dimensional manifolds, anomalous outliers, or generative distributions.

Semi-supervised mixes the two: a few labels and many unlabeled points. Self-supervised learning, dominant in modern NLP, creates labels from the data itself (predict the next word given the previous ones).

In quant work, supervised methods predict returns, default risk, and execution slippage. Unsupervised methods find regime clusters, factor structures, and trade-pattern anomalies.

You have a dataset of 10,00010{,}00010,000 daily stock movements with no labels and want to find groups of similarly-behaving stocks. Which type of learning?

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Stochastic Calculus Primer
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Bias–Variance Trade-off