Advanced Topics in Probability and Statistics
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This picks up where a first course in probability stops, and it is aimed at the roles where the maths is the job: quant research, risk, and the desks that build their own models.
It runs through multivariate distributions and covariance structure, then the process models used to describe things that evolve, including Markov and hidden Markov models, martingales and Poisson processes. Simulation follows, from plain Monte Carlo to importance sampling and the bootstrap. The inference sections go deep on maximum likelihood, Fisher information and the Cramer-Rao bound, likelihood ratio tests, and shrinkage, then into hierarchical and decision-theoretic Bayes. Time series, information theory and a set of direct trading applications close it out, including the difference between Value at Risk and expected shortfall that a risk limit written on the first alone will not notice.
It assumes Fundamentals of Probability and Statistics and comfort with integrals and matrix notation. The final section is a problem-solving toolkit: conditioning, indicator variables and the inequalities that turn an intractable question into a bound.
Joint, Marginal, and Conditional DistributionsDescribing several random variables at once, and the two operations that reduce a joint distribution to something usable.Free previewMultivariate Normal DistributionThe workhorse of portfolio mathematics, its closure properties, and the tail behaviour that makes it dangerous.Free previewCovariance Matrices and Linear CombinationsThe matrix that turns a set of positions into one risk number, and why estimating it is harder than it looks.Free previewCorrelation vs. CausationWhy the distinction is a practical trading concern rather than a statistics-class slogan.
Hidden Markov ModelsInferring an unobserved regime from what it produces, and the three questions an HMM answers.Renewal and Poisson ProcessesModelling arrivals in continuous time, and the clustering that makes the simplest model wrong for order flow.Introduction to Stochastic ProcessesRandomness evolving over time, and the handful of processes that describe most of finance.Martingales and Fair GamesFair games formalised, the optional stopping theorem, and why no betting system can beat one.
Monte Carlo SimulationEstimating by sampling, the convergence rate that limits it, and the dimension where it wins.Importance SamplingSampling where the action is, then correcting for the bias, which is how rare events become estimable.BootstrappingGetting a confidence interval for any statistic without a formula, and the one assumption it makes.Variational InferenceTurning inference into optimisation, and the systematic error you accept in exchange for speed.
Maximum Likelihood Estimation: Deep DiveThe general-purpose estimator, its asymptotic guarantees, and the conditions under which those guarantees evaporate.Fisher Information and the Cramer-Rao BoundHow much a dataset can possibly tell you about a parameter, and the floor that puts under any estimator's variance.Likelihood Ratio TestsComparing nested models properly, and why a better fit is not automatically a better model.Shrinkage Estimators and Empirical BayesDeliberately biasing estimates toward a common value, and why that reliably beats the obvious estimator.
Hierarchical Bayesian ModelsLetting groups differ while borrowing strength from each other, which is the right answer to evaluating many traders.Conjugate Priors in Exponential FamiliesPrior families that make Bayesian updating arithmetic, and the pseudo-count interpretation that makes them intuitive.Bayesian Decision TheoryTurning a posterior into an action, and why the loss function determines which summary of the posterior you should use.Posterior Predictive DistributionsPredicting the next observation while accounting for the fact that you do not know the parameters.
AR, MA, and ARIMA ModelsLinear models for serially dependent data, and why they work better on volatility than on returns.Stationarity and AutocorrelationThe precondition for almost every time series method, and the spurious regressions that follow from ignoring it.Introduction to Kalman FiltersOptimal recursive estimation of a hidden state, and the gain that decides how much to trust each new observation.
Shannon EntropyMeasuring uncertainty in bits, and why maximum uncertainty sits exactly where a market is hardest to trade.Kullback-Leibler DivergenceThe cost of using the wrong distribution, and why its asymmetry is a feature you have to choose deliberately.Mutual InformationDependence that correlation cannot see, and the estimation problem that limits its use.Applications of Entropy in InferenceMaximum entropy as the principle of assuming nothing extra, and where information theory shows up across modelling.
Value-at-Risk and Expected ShortfallThe industry's two risk numbers, why VaR fails a basic coherence test, and what expected shortfall fixes.Option Pricing via Probability ModelsThe three pricing methods, what each can and cannot handle, and how to choose between them.Risk-neutral MeasuresA change of probability that removes preferences from pricing, and why it does not mean anyone is risk neutral.Backtesting ForecastsEvaluating predictions honestly, and the four ways a backtest lies to you.
Stochastic Dominance and UtilityComparing risky prospects without committing to a utility function, and what to do when neither dominates.Strategic ConditioningChoosing what to condition on, which is the single most useful problem-solving move in interview probability.Indicator Variables and Trick ExpectationsThe single most powerful shortcut in interview probability, because it ignores dependence entirely.Inequalities: Chernoff, Hoeffding, and MoreExponential tail bounds, and the enormous improvement they give over Chebyshev when you know more.
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