Statistics Advanced

You flip a coin three times and get three heads. What should you now believe about the coin — and precisely how much? "Probably still fair" is an instinct; a posterior distribution is an answer. It tells you exactly how far three heads should move you from wherever you started, what a fourth head would add, and why the answer changes if the coin came from a bank rather than a magician. Statistics Core taught you to ask whether an effect is real; this track teaches you to hold degrees of belief as numbers and move them by the rules.

Four modules build that machinery. Advanced Probability formalises the toolbox — random variables, expectation, the classical distributions and the limit theorems that explain why so much of the world looks Gaussian. Estimation Theory then turns the telescope on your own numbers: an average is an estimate, so what makes one estimator better than another — bias, variance, and the maximum-likelihood recipe that underlies half of machine learning? Bayesian Inference is the coin question at full strength: priors, posteriors, and updating as a discipline rather than a metaphor. The closing module reopens the hypothesis tests you already ran in Statistics Core and shows the engine room — power, error trade-offs, and what a p-value actually promises (less than most people think).

This is the proofs-behind-the-instincts track, and it earns its place the first time a result surprises you. Loss functions are negative log-likelihoods; regularisation is a prior wearing a work jacket; every confidence interval you will ever report has an estimation-theory pedigree. None of it is required to keep moving through the platform — but readers who take the detour stop treating those formulas as furniture. Expect real mathematics at a deliberate pace: every derivation is walked, not waved at, and each module ends in a lab where the theory has to survive contact with simulated data.