This module develops the fundamental inferential theory necessary for applying statistical methods in the field of data science. Topics covered are: parameter estimation, properties of estimators, likelihood theory, uncertainty quantification, and hypothesis testing. Students will learn how to formulate real-world problems in terms of statistical models and answer scientific questions by applying the models to data using optimising procedures; they will also learn to assess the adequacy of a given model and choose among several competing models.

1. Statistical modelling - formulating the real world problem using an appropriate statistical model. 2. Properties of estimators - bias, variance, bias-variance trade-off, mean-squared error, efficiency, Cramer-Rao Lower Bound (CRLB). 3. Estimation - method of moments, likelihood function, the Bayesian paradigm. 4. Optimisation - least squares, Newton-Raphson, Markov Chain Monte Carlo (MCMC). 5. Uncertainty quantification - confidence intervals, classical normal data (t-test for mean and difference between means, F-test), Central Limit Theorem (CLT), asymptotic theory, the information matrix, MCMC samples, parametric and non-parametric bootstrapping, the delta method. 6. Hypothesis testing - formally posing a scientific question mathematically, properties of a test (type 1 and 2 errors, power), likelihood ratio test, Neyman-Pearson Lemma, p-value calculation, multiple testing and false discovery rate. 7. Model assessment and selection - chi-squared goodness-of-fit test, prediction, cross-validation, Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC), multi-model inference.

On successful completion of this module, students will be able to: 1. Formulate a real world problem in terms of a statistical model to answer a scientific question. 2. Develop a statistical estimation procedure using software (e.g., R, Python, Matlab, Stan) including uncertainty quantification (i.e., confidence interval). 3. Calculate the bias and variability associated with an estimation procedure. 4. Assess the adequacy of a statistical model relative to other candidate models and also in absolute terms. 5. Evaluate a hypothesis test using common metrics such as the false discovery rate and testing power.

On successful completion of this module, students will be able to: 1. Appreciate the role of statistical inference in scientific inquiry. 2. Acknowledge that models approximate reality, i.e., "all models are wrong".

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The module will be taught using lectures and tutorials with formal statistical theory complimented by its application to problems arising in the real world. Having learned a variety of modern inferential techniques, students will be able to apply and interpret methods for existing problems and develop their own methods for new problems encountered in practice. This module fosters independent problem solvers who will be comfortable in tackling various scientific challenges from first principles. Thus, they will be knowledgeable in the area of the fundamental statistical theory, creative in that they will be able to develop inferential approaches for new problems, articulate in explaining the role and potential pitfalls associated with its use in scientific inquiry, and responsible in their own applications of statistics for decision making.