AdvancedQuantitative MethodsPython

Run this module

cd "Quantitative Methods - Bayesian Inference"
python "bayesian_inference.py"

Quantitative Methods — Bayesian Inference¶

A strategy wins 7 of its first 10 trades. Is its true win rate 70%? Almost certainly not — ten trades is barely any evidence. Bayesian inference is the disciplined way to answer questions like this: start with a prior belief, observe data, and combine them into a posterior that captures what you now believe and how uncertain you still are.

posterior  ∝  likelihood × prior

The payoff is honest uncertainty. Instead of a single point estimate you get a whole distribution — and a credible interval you can actually act on.

Functions¶

Function	Description
`beta_binomial_update(prior_alpha, prior_beta, successes, failures)`	Conjugate update for a probability
`beta_mean(alpha, beta)`	Posterior mean of a Beta distribution
`beta_credible_interval(alpha, beta, level)`	Equal-tailed credible interval for `p`
`probability_greater_than(alpha, beta, threshold)`	Posterior `P(p > threshold)`
`normal_known_variance_update(prior_mean, prior_var, data, data_var)`	Bayesian update of an unknown mean

Conjugate models¶

This module focuses on conjugate priors, where the posterior has the same form as the prior and the update collapses to simple arithmetic:

Beta-Binomial — estimating a probability (a win rate, a default rate). A Beta(α, β) prior plus s successes and f failures gives a Beta(α + s, β + f) posterior. Beta(1, 1) is the flat "I know nothing" prior.
Normal-Normal (known variance) — estimating an unknown mean. The posterior mean is a precision-weighted average of the prior mean and the sample mean — shrinkage made precise.

Example¶

from bayesian_inference import (
    beta_binomial_update, beta_mean, beta_credible_interval, probability_greater_than,
)

# 7 wins, 3 losses from a flat prior.
a, b = beta_binomial_update(1, 1, successes=7, failures=3)
print(beta_mean(a, b))                       # 0.667 — not 0.70
print(beta_credible_interval(a, b, 0.95))    # (0.39, 0.89) — wide! only 10 trades
print(probability_greater_than(a, b, 0.5))   # 0.887 — probably beats a coin flip

Frequentist vs. Bayesian intervals¶

A 95% credible interval means exactly what people wish a confidence interval meant: given the prior and the data, there is a 95% probability the parameter lies inside it. That is a statement about the parameter, not about hypothetical repeated experiments — which is why it is so natural for decision making.

Shrinkage — taming noisy estimates¶

Sample means of returns are notoriously noisy. A Normal-Normal update pulls a raw estimate toward your prior in proportion to how little data you have:

posterior mean = (prior_precision · prior_mean + data_precision · sample_mean)
                 / (prior_precision + data_precision)

This is the same instinct behind James-Stein estimators and the Black-Litterman model, which blends market-implied returns (the prior) with an investor's views (the data).

Practical notes¶

The strength of a Beta prior is α + β — read it as a number of "pseudo trades". Beta(20, 20) says "I'd need ~40 real trades to be talked out of 50/50".
Credible intervals here are equal-tailed (the same probability in each tail); highest-posterior-density intervals differ for skewed posteriors.
For non-conjugate models you would sample the posterior (MCMC) instead — but the intuition you build here carries over directly.
Estimating uncertainty in a backtest metric? See Quantitative Methods - Bootstrap for the resampling counterpart.

Continue in Quantitative Methods¶

Quantitative Methods - Bootstrap

The bootstrap estimates the sampling distribution of any statistic by resampling the observed data with replacement — no normality assumption required. It is the honest way to put confidence intervals around backtest metrics like Sharpe ratio, mean return, or maximum drawdown.
Quantitative Methods - Cointegration

Cointegration: two non-stationary series whose linear combination is stationary. Backbone of statistical arbitrage and pairs trading.
Quantitative Methods - Copulas

This module demonstrates the concept of Copulas, specifically the Gaussian Copula, used in quantitative finance to model the dependency structure between multivariate random variables.
Quantitative Methods - Extreme Value Theory

Most risk models assume returns are normally distributed. They are not —
Quantitative Methods - Factor Models

Factor models explain asset returns as a linear combination of systematic factors plus a stock-specific residual. The Fama-French 3-Factor Model (1992) extended CAPM by adding two well-documented risk premia: the Size premium (SMB) and the Value premium (HML), dramatically improving the explanation of cross-sectional stock returns.
Quantitative Methods - GARCH

GARCH (Generalized Autoregressive Conditional Heteroskedasticity) captures volatility clustering — high-volatility days tend to follow high-volatility days. Used for risk forecasting, option pricing, and VaR.

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