Introduction: Bayesian Statistics with Applications in Econometrics, Finance, and Machine Learning using Python
Welcome to the study of Bayesian statistics. This textbook establishes a rigorous mathematical and computational framework for analyzing decision-making and quantitative modeling under uncertainty.
In classical frequentist statistics, parameters governing population distributions are treated as fixed, immutable, yet unknown constants. Inference under that paradigm relies on long-run asymptotic frequencies derived from hypothetical, infinite repetitions of an identical experiment.
Conversely, the Bayesian paradigm reframes the foundational philosophy of statistical inference by treating all unknown parameters as latent random variables. Rather than searching for a singular true coordinate in a parameter space, we assign a joint probability distribution to the parameters, reflecting an initial state of knowledge or degree of belief.
As empirical observations accumulate, this probability space updates systematically via probability theory. This unified perspective provides a coherent mathematical architecture for statistical inference, dynamic decision theory, and out-of-sample predictive modeling.
The pedagogical design of this course portfolio is split into three distinct phases of quantitative mastery:
We establish mathematical statistics frameworks by analyzing exact inverse probability derivations. This section builds structural comprehension through canonical conjugate families (such as the Beta-Binomial and Normal-Normal paradigms), optimal point estimation via customized loss fields, and the mathematical boundaries of credible regions.
When high-dimensional integration renders analytical solutions intractable, we introduce advanced computational simulation tools. Students will study the mechanics of Markov Chain Monte Carlo (MCMC) algorithms—moving from basic random-walk Metropolis-Hastings setups to gradient-directed transition kernels like Hamiltonian Monte Carlo (HMC) and the No-U-Turn Sampler (NUTS)—alongside deterministic optimizations like Evidence Lower Bound (ELBO) maximization in Variational Inference (VI).
The final section applies these theoretical constructs directly to high-dimensional empirical datasets without oversimplification:
- Econometrics: Regularizing dense parameter matrices through Bayesian Vector Autoregressions (BVAR) and dynamic linear state-space models.
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Quantitative Finance: Mitigating parameter risk and input sensitivity loops within asset allocation frameworks and predictive Value-at-Risk (
$\text{VaR}$ ) calculation. -
Machine Learning: Structuring regularization via the Bayesian Lasso, mapping epistemic uncertainty in Bayesian Neural Networks (BNNs), and implementing non-parametric functional estimation with Gaussian Processes (
$\mathcal{GP}$ ).
To connect theoretical foundations with empirical application, this curriculum uses hands-on scientific computation implemented in Python. We will prioritize the use of PyMC, a modern probabilistic programming framework that compiles complex generative models into optimized C or JAX backends to execute gradient-based sampling and variational optimization.
By utilizing these tools alongside diagnostics from the ArViZ ecosystem, students will learn to construct, validate, and critique complex statistical architectures. By the conclusion of this curriculum, you will possess a deep, intuitive understanding of both the mathematical properties and the computational execution of modern Bayesian workflows.