Applied Probability. School of Mathematics and Statistics, University of Sheffield. (University of Sheffield) Applied Probability / 8

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1 Applied Probability School of Mathematics and Statistics, University of Sheffield (University of Sheffield) Applied Probability / 8

2 Introduction You will have seen probability models. (University of Sheffield) Applied Probability / 8

3 Introduction You will have seen probability models. For example, discrete variables Y 1,..., Y n form a (discrete time) Markov chain if P(Y k = y k Y k 1 = y k 1 ) = p yk 1,y k (University of Sheffield) Applied Probability / 8

4 Introduction You will have seen probability models. For example, discrete variables Y 1,..., Y n form a (discrete time) Markov chain if P(Y k = y k Y k 1 = y k 1 ) = p yk 1,y k where p yk 1,y k matrix P. is the (y k 1, y k ) th element of a transition probability (University of Sheffield) Applied Probability / 8

5 This module This module builds on that in a number of ways. (University of Sheffield) Applied Probability / 8

6 This module This module builds on that in a number of ways. It will introduce further types of probability model: it will extend the idea of a Markov chain to continuous time; (University of Sheffield) Applied Probability / 8

7 This module This module builds on that in a number of ways. It will introduce further types of probability model: it will extend the idea of a Markov chain to continuous time; it will look at models for random scatterings of points. (University of Sheffield) Applied Probability / 8

8 Inference It will also consider the idea of how we might carry out inference for our probability models. (University of Sheffield) Applied Probability / 8

9 Inference It will also consider the idea of how we might carry out inference for our probability models. A simple example is where we have a discrete time Markov chain with two states, but with an unknown transition matrix ( ) θ1 1 θ 1, 1 θ 2 θ 2 where we have unknown parameters θ = (θ 1, θ 2 ). (University of Sheffield) Applied Probability / 8

10 Inference It will also consider the idea of how we might carry out inference for our probability models. A simple example is where we have a discrete time Markov chain with two states, but with an unknown transition matrix ( ) θ1 1 θ 1, 1 θ 2 θ 2 where we have unknown parameters θ = (θ 1, θ 2 ). We can ask the question of how to learn about these parameters θ from observations Y 1,..., Y n. (University of Sheffield) Applied Probability / 8

11 Likelihood There are lots of special cases, but for most purposes we use the likelihood L(θ y 1,..., y n ) = P(Y 1 = y 1,..., Y n = y n θ). (University of Sheffield) Applied Probability / 8

12 Likelihood There are lots of special cases, but for most purposes we use the likelihood L(θ y 1,..., y n ) = P(Y 1 = y 1,..., Y n = y n θ). or the corresponding log-likelihood l = log L. (University of Sheffield) Applied Probability / 8

13 Likelihood There are lots of special cases, but for most purposes we use the likelihood L(θ y 1,..., y n ) = P(Y 1 = y 1,..., Y n = y n θ). or the corresponding log-likelihood l = log L. We will also see applied problems in which probability models and statistical inference for their parameters are useful. Examples of modelling will be drawn from meteorology, epidemic studies, manpower planning, seismology, disease mapping and elsewhere. (University of Sheffield) Applied Probability / 8

14 Likelihood There are lots of special cases, but for most purposes we use the likelihood L(θ y 1,..., y n ) = P(Y 1 = y 1,..., Y n = y n θ). or the corresponding log-likelihood l = log L. We will also see applied problems in which probability models and statistical inference for their parameters are useful. Examples of modelling will be drawn from meteorology, epidemic studies, manpower planning, seismology, disease mapping and elsewhere. We will also look at the wider question of how to develop models to address particular questions, including issues of model fit and model comparison. (University of Sheffield) Applied Probability / 8

15 Contents Theory, examples, and some computation. (University of Sheffield) Applied Probability / 8

16 Contents Theory, examples, and some computation. Basic Poisson process, and review of Markov chains (University of Sheffield) Applied Probability / 8

17 Contents Theory, examples, and some computation. Basic Poisson process, and review of Markov chains Likelihood-based inference (University of Sheffield) Applied Probability / 8

18 Contents Theory, examples, and some computation. Basic Poisson process, and review of Markov chains Likelihood-based inference Inference for Markov chains (University of Sheffield) Applied Probability / 8

19 Contents Theory, examples, and some computation. Basic Poisson process, and review of Markov chains Likelihood-based inference Inference for Markov chains Continuous time Markov chains (University of Sheffield) Applied Probability / 8

20 Contents Theory, examples, and some computation. Basic Poisson process, and review of Markov chains Likelihood-based inference Inference for Markov chains Continuous time Markov chains Extensions of the Poisson process (University of Sheffield) Applied Probability / 8

21 Contents Theory, examples, and some computation. Basic Poisson process, and review of Markov chains Likelihood-based inference Inference for Markov chains Continuous time Markov chains Extensions of the Poisson process Examples, including epidemics and queues (University of Sheffield) Applied Probability / 8

22 Assessment Exam at the end of the module. (University of Sheffield) Applied Probability / 8

23 Assessment Exam at the end of the module. Standard closed book exam. (University of Sheffield) Applied Probability / 8

24 Assessment Exam at the end of the module. Standard closed book exam. Note that this is a change from previous years, when the exam has been open book. (University of Sheffield) Applied Probability / 8

25 Feedback Three sets of exercises to hand in, in weeks 4, 7 and 10. (University of Sheffield) Applied Probability / 8

26 Feedback Three sets of exercises to hand in, in weeks 4, 7 and 10. These will be marked for feedback. (University of Sheffield) Applied Probability / 8

27 Feedback Three sets of exercises to hand in, in weeks 4, 7 and 10. These will be marked for feedback. Contacting me: jonathan.jordan@sheffield.ac.uk or room I9. (University of Sheffield) Applied Probability / 8

28 Feedback Three sets of exercises to hand in, in weeks 4, 7 and 10. These will be marked for feedback. Contacting me: jonathan.jordan@sheffield.ac.uk or room I9. Discussion board on MOLE. (University of Sheffield) Applied Probability / 8

29 Feedback Three sets of exercises to hand in, in weeks 4, 7 and 10. These will be marked for feedback. Contacting me: jonathan.jordan@sheffield.ac.uk or room I9. Discussion board on MOLE. Course website uk/applprob/index.html (University of Sheffield) Applied Probability / 8

Statistics & Data Sciences: First Year Prelim Exam May 2018

Statistics & Data Sciences: First Year Prelim Exam May 2018 Instructions: 1. Do not turn this page until instructed to do so. 2. Start each new question on a new sheet of paper. 3. This is a closed book