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Emulation, Elicitation and Calibration UQ12 Minitutorial Presented by: Tony O’Hagan, Peter Challenor, Ian Vernon UQ12 minitutorial - session 11

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Outline of the minitutorial Three sessions of about 2 hours each Session 1: Monday, 2pm – 4pm, State C Overview of UQ ; total UQ; introduction to emulation; elicitation Session 2: Tuesday, 2pm – 4pm, State C Building and using an emulator; sensitivity analysis Session 3: Wednesday, 2pm – 4pm, State C Calibration and history matching; galaxy formation case study Intended to introduce the applied maths/engineering UQ people to UQ methods developed in the statistics community 2UQ12 minitutorial - session 1

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Session 1 Introduction and elicitation

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Outline Introduction UQ and Total UQ Managing uncertainty A brief case study Emulators Elicitation Elicitation principles Elicitation practice UQ12 minitutorial - session 14

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UQ and Total UQ UQ12 minitutorial - session 15

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What is UQ? Uncertainty quantification A term that seems to have been devised by engineers Faced with uncertainty in some particular kinds of analyses Characterising how uncertainty about inputs to a complex computer model induces uncertainty about outputs Large body of work in engineering and applied maths Uncertainty quantification What statisticians do! And have always done In every field of application, for all kinds of analyses In particular, statisticians have developed methods for propagating and quantifying output uncertainty And lots more relating to the use of complex simulation models UQ12 minitutorial - session 16

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Simulators In almost all fields of science, technology, industry and policy making, people use mechanistic models For understanding, prediction, control Huge variety A model simulates a real-world, usually complex, phenomenon as a set of mathematical equations Models are usually implemented as computer programs We will refer to a computer implementation of a model as a simulator UQ12 minitutorial - session 17

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Why worry about uncertainty? Simulators are increasingly being used for decision-making Taking very seriously the implied claim that the simulator represents and predicts reality How accurate are model predictions? There is growing concern about uncertainty in model outputs Particularly where simulator predictions are used to inform scientific debate or environmental policy Are their predictions robust enough for high stakes decision- making? UQ12 minitutorial - session 18

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For instance … Models for climate change produce different predictions for the extent of global warming or other consequences Which ones should we believe? What error bounds should we put around these? Are simulator differences consistent with the error bounds? Until we can answer such questions convincingly, why should anyone have faith in the science? UQ12 minitutorial - session 19

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The simulator as a function In order to talk about the uncertainty in model predictions we need some simple notation Using computer language, a simulator takes a number of inputs and produces a number of outputs We can represent any output y as a function y = f(x) of a vector x of inputs UQ12 minitutorial - session 110

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Where is the uncertainty? How might the simulator output y = f(x) differ from the true real-world value z that the simulator is supposed to predict? Error in inputs x Initial values Forcing inputs Model parameters Error in model structure or solution Wrong, inaccurate or incomplete science Bugs, solution errors UQ12 minitutorial - session 111

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Quantifying uncertainty UQ12 minitutorial - session 1 12 The ideal is to provide a probability distribution p(z) for the true real-world value The centre of the distribution is a best estimate Its spread shows how much uncertainty about z is induced by uncertainties on the previous slide How do we get this? Input uncertainty: characterise p(x), propagate through to p(y) Structural uncertainty: characterise p(z–y)

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More uncertainties It is important to recognise two more uncertainties that arise when working with simulators 1.The act of propagating input uncertainty is imprecise Approximations are made Introducing additional code uncertainty 2.A key task in managing uncertainty is to use observations of the real world to tune or calibrate the model We need to acknowledge uncertainty due to measurement error UQ12 minitutorial - session 113

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Code uncertainty – Monte Carlo The simplest way to propagate uncertainty is Monte Carlo Take a large random sample of realisations from p(x) Run the simulator at each sampled x to get a sample of outputs This is a random sample from p(y) E.g. sample mean estimates E(Y) Even with a very large sample, MC computations are not exact Sample is an approximation of the population Standard error of sample mean is population s.d. over root n This is code uncertainty MC has a built-in statistical quantification of code uncertainty UQ12 minitutorial - session 114

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Code uncertainty – alternatives to MC MC is impractical for simulators that require significant resources, so other methods have been developed Polynomial chaos methods PC expansions are always truncated The truncation error is where the main code uncertainty lies Also in solving Galerkin equations Surrogate models (e.g. emulators) Approximations to the true f(.) Code uncertainty lies in the approximation error UQ12 minitutorial - session 115

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How to quantify uncertainty To quantify uncertainty in the true real world value that the simulator is trying to predict we need the following steps Quantify uncertainty in inputs, p(x) Propagate to uncertainty in output, p(y) Quantify and account for code uncertainty Quantify and account for model discrepancy uncertainty Engineering/applied maths UQ apparently only deals with the second step Ironically, this is the one step that doesn’t actually involve quantifying uncertainty! UQ12 minitutorial - session 116

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Total UQ Here are my key demands 1.UQ for any quantity of interest must quantify all components of uncertainty 2.All UQ must be in the form of explicit, quantified probability distributions 3.All quantifications of uncertainty should be credible representations of what is, and is not, known None of this is easy but we should at least try I call these aspirations the Total UQ Manifesto UQ12 minitutorial - session 117

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Managing uncertainty UQ12 minitutorial - session 118

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UQ is not enough The presence of uncertainty creates several important tasks Engineering/applied maths UQ addresses only one of these Managing uncertainty Uncertainty analysis – how much uncertainty do we have? This is the basic UQ task Sensitivity analysis – which sources of uncertainty drive overall uncertainty, and how? Understanding the system, prioritising research Calibration – how can we reduce uncertainty? Use of observations –Tuning, data assimilation, history matching, inverse problems Experimental design UQ12 minitutorial - session 119

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Decision-making under uncertainty – can we cope with uncertainty? Robust engineering design Optimisation under uncertainty UQ12 minitutorial - session 120

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MUCM Managing Uncertainty in Complex Models Large 4-year UK research grant June 2006 to September 2010 7 postdoctoral research associates, 4 project PhD students Objective to develop BACCO methods into a basic technology, usable and widely applicable MUCM2: New directions for MUCM Smaller 2-year grant to September 2012 Scoping and developing research proposals UQ12 minitutorial - session 121

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Primary MUCM deliverables Methodology and papers moving the technology forward Papers both in statistics and application area journals The MUCM toolkit Documentation of the methods and how to use them With emphasis on what is found to work reliably across a range of modelling areas Web-based Case studies Three substantial case studies Showcasing methods and best practice Linked to toolkit Events Workshops – conceptual and hands-on Short courses Conferences – UCM 2010 and UCM 2012 (July 2-4) UQ12 minitutorial - session 122

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Focus on the The toolkit is a ‘recipe book’ The good sort that encourages you to experiment There are recipes (procedures) but also lots of explanation of concepts and discussion of choices It is not a software package Software packages are great if they are in your favourite language But it probably wouldn’t be! Packages are dangerous without basic understanding Th e purpose of the toolkit is to build that understanding And it enables you to easily develop your own code UQ12 minitutorial - session 123

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Resources Introduction to emulators O'Hagan, A. (2006). Bayesian analysis of computer code outputs: a tutorial. Reliability Engineering and System Safety 91, 1290-1300. The MUCM website http://mucm.ac.uk The MUCM toolkit http://mucm.ac.uk/toolkit The UCM 2012 conference http://mucm.ac.uk/UCM2012.html UQ12 minitutorial - session 124

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This minitutorial This minitutorial covers the key elements of Total UQ and uncertainty management Emulators Surrogate models that include quantification of code uncertainty Brief outline in this session then details in session 2 Elicitation Tools for rigorous quantification of fundamental uncertainties Introduction to this big field in this session Management tools Sensitivity analysis in session 2 Calibration and history matching in session 3 UQ12 minitutorial - session 125

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A brief case study Complex emulation and expert elicitation were essential components of this exercise UQ12 minitutorial - session 126

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Example: UK carbon flux in 2000 Vegetation model predicts carbon exchange from each of 700 pixels over England & Wales in 2000 Principal output is Net Biosphere Production Accounting for uncertainty in inputs Soil properties Properties of different types of vegetation Land usage Also code uncertainty But not structural uncertainty Aggregated to England & Wales total Allowing for correlations Estimate 7.46 Mt C (± 0.54 Mt C) UQ12 minitutorial - session 127

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Maps UQ12 minitutorial - session 128 Mean NBPStandard deviation

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England & Wales aggregate UQ12 minitutorial - session 129 PFT Plug-in estimate (Mt C) Mean (Mt C) Variance (Mt C 2 ) Grass5.284.370.2453 Crop0.850.430.0327 Deciduous2.131.800.0221 Evergreen0.800.860.0048 Covariances-0.0081 Total9.067.460.2968

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Emulators UQ12 minitutorial - session 130

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So far, so good, but In principle, Total UQ is straightforward In practice, there are many technical difficulties Formulating uncertainty on inputs Elicitation of expert judgements Propagating input uncertainty Modelling structural error Anything involving observational data! The last two are intricately linked And computation UQ12 minitutorial - session 131

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The problem of big models Tasks like uncertainty propagation and calibration require us to run the simulator many times Uncertainty propagation Implicitly, we need to run f(x) at all possible x Monte Carlo works by taking a sample of x from p(x) Typically needs thousands of simulator runs Calibration Traditionally done by searching x space for good fits to the data Both become impractical if the simulator takes more than a few seconds to run 10,000 runs at 1 minute each takes a week of computer time We need a more efficient technique UQ12 minitutorial - session 132

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More efficient methods This is what UQ theory is mostly about Engineering/Applied Maths UQ Polynomial chaos expansions of random variables Approximate by truncating Thereby build an expansion of outputs Compute by Monte Carlo etc. using this surrogate representation Statistics UQ Gaussian process emulation of the simulator A different kind of surrogate Propagate input uncertainty through surrogate By Monte Carlo or analytically UQ12 minitutorial - session 133

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Gaussian process representation More efficient approach First work in early 1980s (DACE) Represent the code as an unknown function f(.) becomes a random process We generally represent it as a Gaussian process (GP) Or its second-order moment version Training runs Run simulator for sample of x values Condition GP on observed data Typically requires many fewer runs than Monte Carlo And x values don’t need to be chosen randomly UQ12 minitutorial - session 134

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Emulation Analysis is completed by prior distributions for, and posterior estimation of, hyperparameters The posterior distribution is known as an emulator of the computer simulator Posterior mean estimates what the simulator would produce for any untried x (prediction) With uncertainty about that prediction given by posterior variance Correctly reproduces training data Gets its UQ right! An essential requirement of credible quantification UQ12 minitutorial - session 135

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2 code runs UQ12 minitutorial - session 136 Consider one input and one output Emulator estimate interpolates data Emulator uncertainty grows between data points

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3 code runs UQ12 minitutorial - session 137 Adding another point changes estimate and reduces uncertainty

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5 code runs UQ12 minitutorial - session 138 And so on

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Then what? Given enough training data points we can in principle emulate any simulator output accurately So that posterior variance is small “everywhere” Typically, this can be done with orders of magnitude fewer model runs than traditional methods At least in relatively low-dimensional problems Use the emulator to make inference about other things of interest E.g. uncertainty analysis, sensitivity analysis, calibration The key feature that distinguishes an emulator from other kinds of surrogate Code uncertainty is quantified naturally And credibly UQ12 minitutorial - session 139

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Elicitation principles UQ12 minitutorial - session 140

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Where do probabilities come from? Consider the probability distribution for a model input Like the hydraulic conductivity K in a geophysical model Suppose we ask an expert, Mary Mary gives a probability distribution for K We might be particularly interested in one probability in that distribution Like the probability that K exceeds 10 -3 (cm/sec) Mary’s distribution says Pr(K > 10 -3 ) = 0.2 UQ12 minitutorial - session 141

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How can K have probabilities? Almost everyone learning probability is taught the frequency interpretation The probability of something is the long run relative frequency with which it occurs in a very long sequence of repetitions How can we have repetitions of K? It’s a one-off, and will only ever have one value It’s that unique value we’re interested in Mary’s distribution can’t be a probability distribution in that sense So what do her probabilities actually mean? And does she know? UQ12 minitutorial - session 142

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Mary’s probabilities Mary’s probability 0.3 that K > 10 -3 is a judgement She thinks it’s more likely to be below 10 -3 than above So in principle she would bet even money on it In fact she would bet $2 to win $1 (because 0.7 > 2/3) Her expectation of around 10 -3.5 is a kind of best estimate Not a long run average over many repetitions Her probabilities are an expression of her beliefs They are personal judgements You or I would have different probabilities We want her judgements because she’s the expert! We need a new definition of probability UQ12 minitutorial - session 143

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Subjective probability The probability of a proposition E is a measure of a person’s degree of belief in the truth of E If they are certain that E is true then Pr(E) = 1 If they are certain it is false then Pr(E) = 0 Otherwise Pr(E) lies between these two extremes Exercise UQ12 minitutorial - session 144

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Subjective includes frequency The frequency and subjective definitions of probability are compatible If the results of a very long sequence of repetitions are available, they agree Frequency probability equates to the long run frequency All observers who accept the sequence as comprising repetitions will assign that frequency as their (personal or subjective) probability for the next result in the sequence Subjective probability extends frequency probability But also seamlessly covers propositions that are not repeatable It’s also more controversial UQ12 minitutorial - session 145

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It doesn’t include prejudice etc! The word “subjective” has derogatory overtones Subjectivity should not admit prejudice, bias, superstition, wishful thinking, sloppy thinking, manipulation... Subjective probabilities are judgements but they should be careful, honest, informed judgements As “objective” as possible without ducking the issue Using best practice Formal elicitation methods Bayesian analysis Probability judgements go along with all the other judgements that a scientist necessarily makes And should be argued for in the same careful, honest and informed way UQ12 minitutorial - session 146

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UQ12 minitutorial - session 147 But people are poor probability judges Our brains evolved to make quick decisions Heuristics are short-cut reasoning techniques Allow us to make good judgements quickly in familiar situations Judgement of probability is not something that we evolved to do well The old heuristics now produce biases Anchoring and adjustment Availability Representativeness The range-frequency compromise Overconfidence

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Anchoring and adjustment When asked to make two related judgements, the second is affected by the first The second is judged relative to the first By adjustment away from the first judgement The first is called the anchor Adjustment is typically inadequate Second response too close to the first (anchor) Anchoring can be strong even when obviously not really relevant to the second question 48UQ12 minitutorial - session 1

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The probability of an event is judged more likely if we can quickly bring to mind instances of it Things that are more memorable are deemed more probable High profile train accidents in the UK lead people to imagine rail travel is more risky than it really is My judgement of the risk of dying from a particular disease will be increased if I know (of) people who have the disease or have died from it Availability 49UQ12 minitutorial - session 1

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Representativeness An event is considered more probable if the components of its description fit together Even when the juxtaposition of many components is actually improbable “Linda is 31, single, outspoken and very bright. She studied philosophy at university and was deeply concerned with issues of discrimination and social justice. Is Linda … “A bank teller? “A bank teller and active in the feminist movement?” The second is often judged more probable than the first We are a story-telling species This is also called the conjunction fallacy 50UQ12 minitutorial - session 1

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?? Range-frequency compromise Probability judgements are affected by how many alternatives are presented An example with a quantitative variable Elicitation for X = cost of building project (in £k) Ask for probabilities for ranges (0, 10), (10, 20 ), (20, 100) (0, 10), (10, 20 ), (20, 50), (50, 100) Probabilities in first 2 ranges smaller in second case Even if (20,100) is really unlikely We tend to spread probabilities evenly between whatever options we are given May be a kind of anchoring UQ12 minitutorial - session 151

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Overconfidence It is generally said that experts are overconfident When asked to give 95% interval, say, then far fewer than 95% contain the true value Several possible explanations Wish to demonstrate expertise Anchoring to a central estimate Difficulty of judging extreme events Not thinking ‘outside the box’ Expertise often consists of specialist heuristics Situations we elicit judgements on are not typical UQ12 minitutorial - session 152

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Probably over-stated as a general phenomenon Experts can be under-confident if afraid of consequences A matter of personality and feeling of security Some evidence that people are not over-confident if asked for intervals of moderate probability E.g. 66% or 50% Evidence of over-confidence is not from real experts making judgements on serious questions Students and ‘almanac’ questions Good elicitation practice needs to recognise these problems Answers depend on how the questions are posed Protocol should avoid or minimise biases UQ12 minitutorial - session 153

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Elicitation practice UQ12 minitutorial - session 154

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Why elicit distributions? Occasionally, we want expert opinion about a discrete proposition The Democrats will win the next US presidential election There is, or has at one time been, life on Mars Then a single probability needs to be elicited Mostly, though, we are interested in opinion about an uncertain quantity The mean response of patients to a new drug The increase in global temperature caused by a doubling of atmospheric CO 2 Then we need to elicit a probability distribution In fact, we are often interested in several quantities 55UQ12 minitutorial - session 1

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Too many probabilities! We’ll stick to a single quantity for now One way to think of a distribution for a quantity X is as a set of probabilities Pr(X < x) for all possible x values That’s a lot of probabilities to elicit! If we sat down to elicit them one by one, the interrogation would never finish! And we’d have serious anchoring problems! We need a pragmatic approach UQ12 minitutorial - session 156

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A pragmatic approach Animal welfare – what proportion of a herd is diseased? X = incidence/1000 of a parasite Expert says Pr(X 30) = 0.2 Facilitator fits an inverse gamma distribution to the two given probabilities Check expert agrees Pr(X < 20) 0.68 The usual approach has two steps 1. Elicit a few probabilities or other ‘summaries’ 2. Fit a distribution to those summaries UQ12 minitutorial - session 157 0.20.4 01030

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Elicit a few summaries We can just elicit a few probabilities As in the last example Other possible ‘summaries’: Mean, median mode Often expert is just asked for an ‘estimate’ But that begs the question of what kind of estimate The mean and mode are not recommended Asking for the median is OK (value with probability 0.5 on either side) UQ12 minitutorial - session 158 Mode = 6 Median = 13 Mean = 30

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More summaries Not variances! Fixed probability intervals But not for extreme probability values Shape E.g. Unimodal – or bimodal!! Skewed Shape is important Often overlooked Should at least be checked by feedback (showing the fitted distribution) UQ12 minitutorial - session 159

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Then fit a distribution Any convenient distribution As long as it fits the elicited summaries adequately At this point, the choice should not matter The idea is that we have elicited enough Any reasonable choice of distribution will be similar to any other Elicitation can never be exact The elicited summaries are only approximate anyway If the choice does matter i.e. different fitted distributions give different answers to the problem for which we are doing the elicitation then we need to elicit more summaries UQ12 minitutorial - session 160

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UQ12 minitutorial - session 161 The SHELF system SHELF is a package of documents and simple software to aid elicitation General advice on conducting the elicitation Templates for recording the elicitation Suitable for several different basic methods Annotated versions of the templates with detailed guidance Some R functions for fitting distributions and providing feedback SHELF is freely available and we welcome comments and suggestions for additions Developed by Jeremy Oakley and myself R functions by Jeremy http://tonyohagan.co.uk/shelf

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Contents SHELF Overview_v2.0 SHELF Pre-elicitation Briefing SHELF Pre-elicitation Form + version with notes SHELF 1 (Context) + version with notes SHELF 2 (Distribution) Q + version with notes SHELF 2 (Distribution) QP + version with notes SHELF 2 (Distribution) R + version with notes SHELF 2 (Distribution) RP + version with notes SHELF 2 (Distribution) T + version with notes SHELF 2 (Distribution) TP + version with notes SHELF2 Distribution fitting instructions shelf2.R UQ12 minitutorial - session 162

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UQ12 minitutorial - session 163

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UQ12 minitutorial - session 164

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Example We will now look at a hypothetical example of using SHELF Using a simplified form for a single expert 65UQ12 minitutorial - session 1

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Where to next Two more sessions to come in this minitutorial Tomorrow: Peter Challenor will present Session 2 on building and using emulators Wednesday I will begin Session 3 on using observations of the real world process – calibration and other tasks Ian Vernon will finish with a case study on a grand scale – history matching a galaxy formation model to observations of the universe UQ12 minitutorial - session 166

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Another conference UCM 2012 Still open for poster abstracts Early bird registration deadline 30 th April http://mucm.ac.uk/ucm2012 67 UQ12 minitutorial - session 1

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