Course pages 2019–20

**Subsections**

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Foundations of Data Science

*Lecturer: Dr D.J. Wischik*

*No. of lectures and practical classes:* 12 + 4

*Suggested hours of supervisions:* 3

*Prerequisite courses: *
either Mathematics for Natural Sciences, or the equivalent from the
Maths Tripos

*This course is a prerequisite for:*
Part II Machine Learning and Bayesian Inference, Information Retrieval,
Quantum Computing, Natural Language Processing.

### Aims

This course introduces fundamental tools for describing and reasoning about data. There are two themes: describing the behaviour of random systems; and making inferences based on data generated by such systems. The course will survey a wide range of models and tools, and it will emphasize how to design a model and what sorts of questions one might ask about it.

### Lectures

**Likelihood.**Random variables. Random samples. Maximum likelihood estimation, likelihood profile.**Random variables.**Rules for expectation and variance. Generating random variables. Empirical distribution. Monte Carlo estimation; law of large numbers. Central limit theorem.**Inference.**Estimation, confidence intervals, hypothesis testing, prediction. Bootstrap. Bayesianism. Logistic regression, natural parameters.**Feature spaces.**Vector spaces, bases, inner products, projection. Model fitting as projection. Linear modeling. Choice of features.**Random processes.**Markov chains. Stationarity and convergence. Drift models. Examples, including estimation and memory.**Probabilistic modelling.**Independence; joint distributions. Descriptive, discriminative, and causal models. Latent variable models. Random fields.

### Objectives

At the end of the course students should

- be able to formulate basic probabilistic models, including discrete time Markov chains and linear models
- be familiar with common random variables and their uses, and with the use of empirical distributions rather than formulae
- be able to use expectation and conditional expectation, limit theorems, equilibrium distributions
- understand different types of inference about noisy data, including model fitting, hypothesis testing, and making predictions
- understand the fundamental properties of inner product spaces and orthonormal systems, and their application to model representation

### Recommended reading

* F.M. Dekking, C. Kraaikamp, H.P. Lopuhaä, L.E. Meester (2005).
*A modern introduction to probability and statistics: understanding why and how.*
Springer.

S.M. Ross (2002).
*Probability models for computer science.*
Harcourt / Academic Press.

M. Mitzenmacher & E. Upfal (2005).
*Probability and computing: randomized algorithms and probabilistic analysis.*
Cambridge University Press.

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