Advanced Topics in Probability and Statistics

Understand how to model and manipulate probabilities involving multiple random variables.

A key distribution in high-dimensional analysis with applications in finance and machine learning.

Use covariance matrices to study the relationships between multiple variables and how they combine.

Distinguish between correlated behavior and actual causal effects using data and reasoning.

Extend Markov Chains to cases where the states are hidden but generate observable outputs.

Model arrivals and events over time using renewal theory and Poisson processes.

Study systems that evolve randomly over time, including Brownian motion.

Martingales model fair processes and are fundamental in probability theory and finance.

Use random sampling to estimate complex probabilistic outcomes.

A variance-reducing technique for more efficient simulations of rare events.

A powerful resampling method to estimate confidence intervals and distributions.

An approximate Bayesian method that scales to large and complex models.

Explore MLE's theory, regularity conditions, and properties in depth.

Understand the limits of parameter estimation precision.

Compare models using likelihood-based hypothesis tests.

Improve estimation in high dimensions using regularization or Bayesian shrinkage.

Model nested or multi-level data with hierarchical priors.

Use conjugate priors to simplify Bayesian updates analytically.

Make optimal decisions under uncertainty using expected utility.

Use the posterior to make predictions about future data.

Model time-dependent data using autoregressive and moving average techniques.

Diagnose and transform time series data for meaningful analysis.

A recursive estimator for dynamic linear systems with noise.

Measure uncertainty or information content in a probability distribution.

Quantify the difference between two probability distributions.

Measure the amount of information shared between two random variables.

Apply entropy principles to optimize decisions and compress information.

Measure and manage portfolio risk using quantile-based methods.

Understand option valuation using probabilistic and stochastic techniques.

Change probability measure to price derivatives under no-arbitrage conditions.

Evaluate model predictions and assess their performance over time.

Compare uncertain outcomes using stochastic orderings and utility theory.

Break down tough problems by conditioning on smart sub-events.

Use clever variable substitutions and indicators to simplify calculations.

Use probabilistic bounds to analyze rare events and tail risks.