MARLAP Manual Volume III: Chapter 19, Measurement Uncertainty
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What Is the Balance Between Certainty and Uncertainty?
What is the balance between certainty and uncertainty? Susan Griffin, Centre for Health Economics, University of York Overview • Decision analysis – Objective function – The decision to invest/disinvest in health care programmes – The decision to require further evidence • Characterising uncertainty – Epistemic uncertainty, quality of evidence, modelling uncertainty • Possibility for research Theory Objective function • Methods can be applied to any decision problem – Objective – Quantitative index measure of outcome • Maximise health subject to budget constraint – Value improvement in quality of life in terms of equivalent improvement in length of life → QALYs • Maximise health and equity s.t budget constraint – Value improvement in equity in terms of equivalent improvement in health The decision to invest • Maximise health subject to budget constraint – Constrained optimisation problem – Given costs and health outcomes of all available programmes, identify optimal set • Marginal approach – Introducing new programme displaces existing activities – Compare health gained to health forgone resources Slope gives rate at C A which resources converted to health gains with currently funded programmes health HGA HDA resources Slope gives rate at which resources converted to health gains with currently funded programmes CB health HDB HGB Requiring more evidence • Costs and health outcomes estimated with uncertainty • Cost of uncertainty: programme funded on basis of estimated costs and outcomes might turn out not to be that which maximises health -
2.1 Introduction Civil Engineering Systems Deal with Variable Quantities, Which Are Described in Terms of Random Variables and Random Processes
2.1 Introduction Civil Engineering systems deal with variable quantities, which are described in terms of random variables and random processes. Once the system design such as a design of building is identified in terms of providing a safe and economical building, variable resistances for different members such as columns, beams and foundations which can sustain variable loads can be analyzed within the framework of probability theory. The probabilistic description of variable phenomena needs the use of appropriate measures. In this module, various measures of description of variability are presented. 2. HISTOGRAM AND FREQUENCY DIAGRAM Five graphical methods for analyzing variability are: 1. Histograms, 2. Frequency plots, 3. Frequency density plots, 4. Cumulative frequency plots and 5. Scatter plots. 2.1. Histograms A histogram is obtained by dividing the data range into bins, and then counting the number of values in each bin. The unit weight data are divided into 4- kg/m3 wide intervals from to 2082 in Table 1. For example, there are zero values between 1441.66 and 1505 Kg / m3 (Table 1), two values between 1505.74 kg/m3 and 1569.81 kg /m3 etc. A bar-chart plot of the number of occurrences in each interval is called a histogram. The histogram for unit weight is shown on fig.1. Table 1 Total Unit Weight Data from Offshore Boring Total unit Total unit 2 3 Depth weight (x-βx) (x-βx) Depth weight Number (m) (Kg / m3) (Kg / m3)2 (Kg / m3)3 (m) (Kg / m3) 1 0.15 1681.94 115.33 -310.76 52.43 1521.75 2 0.30 1906.20 2050.36 23189.92 2.29 1537.77 -
Would ''Direct Realism'' Resolve the Classical Problem of Induction?
NOU^S 38:2 (2004) 197–232 Would ‘‘Direct Realism’’ Resolve the Classical Problem of Induction? MARC LANGE University of North Carolina at Chapel Hill I Recently, there has been a modest resurgence of interest in the ‘‘Humean’’ problem of induction. For several decades following the recognized failure of Strawsonian ‘‘ordinary-language’’ dissolutions and of Wesley Salmon’s elaboration of Reichenbach’s pragmatic vindication of induction, work on the problem of induction languished. Attention turned instead toward con- firmation theory, as philosophers sensibly tried to understand precisely what it is that a justification of induction should aim to justify. Now, however, in light of Bayesian confirmation theory and other developments in epistemology, several philosophers have begun to reconsider the classical problem of induction. In section 2, I shall review a few of these developments. Though some of them will turn out to be unilluminating, others will profitably suggest that we not meet inductive scepticism by trying to justify some alleged general principle of ampliative reasoning. Accordingly, in section 3, I shall examine how the problem of induction arises in the context of one particular ‘‘inductive leap’’: the confirmation, most famously by Henrietta Leavitt and Harlow Shapley about a century ago, that a period-luminosity relation governs all Cepheid variable stars. This is a good example for the inductive sceptic’s purposes, since it is difficult to see how the sparse background knowledge available at the time could have entitled stellar astronomers to regard their observations as justifying this grand inductive generalization. I shall argue that the observation reports that confirmed the Cepheid period- luminosity law were themselves ‘‘thick’’ with expectations regarding as yet unknown laws of nature. -
Stat 345 - Homework 3 Chapter 4 -Continuous Random Variables
STAT 345 - HOMEWORK 3 CHAPTER 4 -CONTINUOUS RANDOM VARIABLES Problem One - Trapezoidal Distribution Consider the following probability density function f(x). 8 > x+1 ; −1 ≤ x < 0 > 5 > <> 1 ; 0 ≤ x < 4 f(x) = 5 > 5−x ; 4 ≤ x ≤ 5 > 5 > :>0; otherwise a) Sketch the PDF. b) Show that the area under the curve is equal to 1 using geometry. c) Show that the area under the curve is equal to 1 by integrating. (You will have to split the integral into pieces). d) Find P (X < 3) using geometry. e) Find P (X < 3) with integration. Problem Two - Probability Density Function Consider the following function f(x), where θ > 0. 8 < c (θ − x); 0 < x < θ f(x) = θ2 :0; otherwise a) Find the value of c so that f(x) is a valid PDF. Sketch the PDF. b) Find the mean and standard deviation of X. p c) If nd , 2 and . θ = 2 P (X > 1) P (X > 3 ) P (X > 2 − 2) c) Compute θ 3θ . P ( 10 < X < 5 ) Problem Three - Cumulative Distribution Functions For each of the following probability density functions, nd the CDF. The Median (M) of a distribution is dened by the property: 1 . Use the CDF to nd the Median of each distribution. F (M) = P (X ≤ M) = 2 a) (Special case of the Beta Distribution) 8 p <1:5 x; 0 < x < 1 f(x) = :0; otherwise b) (Special case of the Pareto Distribution) 8 < 1 ; x > 1 f(x) = x2 :0; otherwise c) Take the derivative of your CDF for part b) and show that you can get back the PDF. -
Climate Scepticism, Epistemic Dissonance, and the Ethics of Uncertainty
SYMPOSIUM A CHANGING MORAL CLIMATE CLIMATE SCEPTICISM, EPISTEMIC DISSONANCE, AND THE ETHICS OF UNCERTAINTY BY AXEL GELFERT © 2013 – Philosophy and Public Issues (New Series), Vol. 3, No. 1 (2013): 167-208 Luiss University Press E-ISSN 2240-7987 | P-ISSN 1591-0660 [THIS PAGE INTENTIONALLY LEFT BLANK] A CHANGING MORAL CLIMATE Climate Scepticism, Epistemic Dissonance, and the Ethics of Uncertainty Axel Gelfert Abstract. When it comes to the public debate about the challenge of global climate change, moral questions are inextricably intertwined with epistemological ones. This manifests itself in at least two distinct ways. First, for a fixed set of epistemic standards, it may be irresponsible to delay policy- making until everyone agrees that such standards have been met. This has been extensively discussed in the literature on the precautionary principle. Second, key actors in the public debate may—for strategic reasons, or out of simple carelessness—engage in the selective variation of epistemic standards in response to evidence that would threaten to undermine their core beliefs, effectively leading to epistemic double standards that make rational agreement impossible. The latter scenario is aptly described as an instance of what Stephen Gardiner calls “epistemic corruption.” In the present paper, I aim to give an explanation of the cognitive basis of epistemic corruption and discuss its place within the moral landscape of the debate. In particular, I argue that epistemic corruption often reflects an agent’s attempt to reduce dissonance between incoming scientific evidence and the agent’s ideological core commitments. By selectively discounting the former, agents may attempt to preserve the coherence of the latter, yet in doing so they threaten to damage the integrity of evidence-based deliberation. -
Measurement, Uncertainty, and Uncertainty Propagation 205
Measurement, Uncertainty, and Uncertainty Propagation 205 Name _____________________ Date __________ Partners ________________ TA ________________ Section _______ ________________ Measurement, Uncertainty, and Uncertainty Propagation Objective: To understand the importance of reporting both a measurement and its uncertainty and to address how to properly treat uncertainties in the lab. Equipment: meter stick, 2-meter stick DISCUSSION Understanding nature requires measuring things, be it distance, time, acidity, or social status. However, measurements cannot be “exact”. Rather, all measurements have some uncertainty associated with them.1 Thus all measurements consist of two numbers: the value of the measured quantity and its uncertainty2. The uncertainty reflects the reliability of the measurement. The range of measurement uncertainties varies widely. Some quantities, such as the mass of the electron me = (9.1093897 ± 0.0000054) ×10-31 kg, are known to better than one part per million. Other quantities are only loosely bounded: there are 100 to 400 billion stars in the Milky Way. Note that we not talking about “human error”! We are not talking about mistakes! Rather, uncertainty is inherent in the instruments and methods that we use even when perfectly applied. The goddess Athena cannot not read a digital scale any better than you. Significant Figures The electron mass above has eight significant figures (or digits). However, the measured number of stars in the Milky Way has barely one significant figure, and it would be misleading to write it with more than one figure of precision. The number of significant figures reported should be consistent with the uncertainty of the measurement. In general, uncertainties are usually quoted with no more significant figures than the measured result; and the last significant figure of a result should match that of the uncertainty. -
Uncertainty 2: Combining Uncertainty
Introduction to Mobile Robots 1 Uncertainty 2: Combining Uncertainty Alonzo Kelly Fall 1996 Introduction to Mobile Robots Uncertainty 2: 2 1.1Variance of a Sum of Random Variables 1 Combining Uncertainties • The basic theory behind everything to do with the Kalman Filter can be obtained from two sources: • 1) Uncertainty transformation with the Jacobian matrix • 2) The Central Limit Theorem 1.1 Variance of a Sum of Random Variables • We can use our uncertainty transformation rules to compute the variance of a sum of random variables. Let there be n random variables - each of which are normally distributed with the same distribution: ∼()µσ, , xi N ,i=1n • Let us define the new random variabley as the sum of all of these. n yx=∑i i=1 • By our transformation rules: σ2 σ2T y=JxJ • where the Jacobian is simplyn ones: J=1…1 • Hence: Alonzo Kelly Fall 1996 Introduction to Mobile Robots Uncertainty 2: 3 1.2Variance of A Continuous Sum σ2 σ2 y = n x • Thus the variance of a sum of identical random variables grows linearly with the number of elements in the sum. 1.2 Variance of A Continuous Sum • Consider a problem where an integral is being computed from noisy measurements. This will be important later in dead reckoning models. • In such a case where a new random variable is added at a regular frequency, the expression gives the development of the standard deviation versus time because: t n = ----- ∆t in a discrete time system. • In this simple model, uncertainty expressed as standard deviation grows with the square root of time. -
STAT 345 Spring 2018 Homework 5 - Continuous Random Variables Name
STAT 345 Spring 2018 Homework 5 - Continuous Random Variables Name: Please adhere to the homework rules as given in the Syllabus. 1. Trapezoidal Distribution. Consider the following probability density function. 8 x+1 ; −1 ≤ x < 0 > 5 > 1 < 5 ; 0 ≤ x < 4 f(x) = 5−x > ; 4 ≤ x ≤ 5 > 5 :>0; otherwise a) Sketch the PDF. b) Show that the area under the curve is equal to 1 using geometry. c) Show that the area under the curve is equal to 1 by integrating. (You will have to split the integral into peices). d) Find P (X < 3) using geometry. e) Find P (X < 3) with integration. 2. Let X have the following PDF, for θ > 0. ( c (θ − x); 0 < x < θ f(x) = θ2 0; otherwise a) Find the value of c which makes f(x) a valid PDF. Sketch the PDF. b) Find the mean and standard deviation of X. θ 3θ c) Find P 10 < X < 5 . 3. PDF to CDF. For each of the following PDFs, find and sketch the CDF. The Median (M) of a continuous distribution is defined by the property F (M) = 0:5. Use the CDF to find the Median of each distribution. a) (Special case of Beta Distribution) p f(x) = 1:5 x; 0 < x < 1 b) (Special case of Pareto Distribution) 1 f(x) = ; x > 1 x2 4. CDF to PDF. For each of the following CDFs, find the PDF. a) 8 0; x < 0 <> F (x) = 1 − (1 − x2)2; 0 ≤ x < 1 :>1; x ≥ 1 b) ( 0; x < 0 F (x) = 1 − e−x2 ; x ≥ 0 5. -
Measurements and Uncertainty Analysis
Measurements & Uncertainty Analysis Measurement and Uncertainty Analysis Guide “It is better to be roughly right than precisely wrong.” – Alan Greenspan Table of Contents THE UNCERTAINTY OF MEASUREMENTS .............................................................................................................. 2 TYPES OF UNCERTAINTY .......................................................................................................................................... 6 ESTIMATING EXPERIMENTAL UNCERTAINTY FOR A SINGLE MEASUREMENT ................................................ 9 ESTIMATING UNCERTAINTY IN REPEATED MEASUREMENTS ........................................................................... 9 STANDARD DEVIATION .......................................................................................................................................... 12 STANDARD DEVIATION OF THE MEAN (STANDARD ERROR) ......................................................................... 14 WHEN TO USE STANDARD DEVIATION VS STANDARD ERROR ...................................................................... 14 ANOMALOUS DATA ................................................................................................................................................. 15 FRACTIONAL UNCERTAINTY ................................................................................................................................. 15 BIASES AND THE FACTOR OF N–1 ...................................................................................................................... -
Investment Behaviors Under Epistemic Versus Aleatory Uncertainty
Investment Behaviors Under Epistemic versus Aleatory Uncertainty 2 Daniel J. Walters, Assistant Professor of Marketing, INSEAD Europe, Marketing Department, Boulevard de Constance, 77300 Fontainebleau, 01 60 72 40 00, [email protected] Gülden Ülkümen, Gülden Ülkümen, Associate Professor of Marketing at University of Southern California, 701 Exposition Boulevard, Hoffman Hall 516, Los Angeles, CA 90089- 0804, [email protected] Carsten Erner, UCLA Anderson School of Management, 110 Westwood Plaza, Los Angeles, CA, USA 90095-1481, [email protected] David Tannenbaum, Assistant Professor of Management, University of Utah, Salt Lake City, UT, USA, (801) 581-8695, [email protected] Craig R. Fox, Harold Williams Professor of Management at the UCLA Anderson School of Management, Professor of Psychology and Medicine at UCLA, 110 Westwood Plaza #D511 Los Angeles, CA, USA 90095-1481, (310) 206-3403, [email protected] 3 Abstract In nine studies, we find that investors intuitively distinguish two independent dimensions of uncertainty: epistemic uncertainty that they attribute to missing knowledge, skill, or information, versus aleatory uncertainty that they attribute to chance or stochastic processes. Investors who view stock market uncertainty as higher in epistemicness (knowability) are more sensitive to their available information when choosing whether or not to invest, and are more likely to reduce uncertainty by seeking guidance from experts. In contrast, investors who view stock market uncertainty as higher -
The Flaw of Averages)
THE NEED FOR GREATER REFLECTION OF HETEROGENEITY IN CEA or (The fLaw of averages) Warren Stevens PhD CONFIDENTIAL © 2018 PAREXEL INTERNATIONAL CORP. Shimmering Substance Jackson Pollock, 1946. © 2018 PAREXEL INTERNATIONAL CORP. / 2 CONFIDENTIAL 1 Statistical summary of Shimmering Substance Jackson Pollock, 1946. Average color: Brown [Beige-Black] Average position of brushstroke © 2018 PAREXEL INTERNATIONAL CORP. / 3 CONFIDENTIAL WE KNOW DATA ON EFFECTIVENESS IS FAR MORE COMPLEX THAN WE LIKE TO REPRESENT IT n Many patients are better off with the comparator QALYS gained © 2018 PAREXEL INTERNATIONAL CORP. / 4 CONFIDENTIAL 2 A THOUGHT EXPERIMENT CE threshold Mean CE Mean CE treatment treatment #1 #2 n Value of Value of Cost per QALY treatment treatment #2 to Joe #1 to Bob © 2018 PAREXEL INTERNATIONAL CORP. / 5 CONFIDENTIAL A THOUGHT EXPERIMENT • Joe is receiving a treatment that is cost-effective for him, but deemed un cost-effective by/for society • Bob is receiving an treatment that is not cost-effective for him, but has been deemed cost-effective by/for society • So who’s treatment is cost-effective? © 2018 PAREXEL INTERNATIONAL CORP. / 6 CONFIDENTIAL 3 HYPOTHETICALLY THERE ARE AS MANY LEVELS OF EFFECTIVENESS (OR COST-EFFECTIVENESS) AS THERE ARE PATIENTS What questions or uncertainties stop us recognizing this basic truth in scientific practice? 1. Uncertainty (empiricism) – how do we gauge uncertainty in a sample size of 1? avoiding false positives 2. Policy – how do we institute a separate treatment policy for each and every individual? 3. Information (data) – how we do make ‘informed’ decisions without the level of detail required in terms of information (we know more about populations than we do about people) © 2018 PAREXEL INTERNATIONAL CORP. -
Estimating and Combining Uncertainties 8Th Annual ITEA Instrumentation Workshop Park Plaza Hotel and Conference Center, Lancaster Ca
Estimating and Combining Uncertainties 8th Annual ITEA Instrumentation Workshop Park Plaza Hotel and Conference Center, Lancaster Ca. 5 May 2004 Howard Castrup, Ph.D. President, Integrated Sciences Group www.isgmax.com [email protected] Abstract. A general measurement uncertainty model we treat these variations as statistical quantities and say is presented that can be applied to measurements in that they follow statistical distributions. which the value of an attribute is measured directly, and to multivariate measurements in which the value of an We also acknowledge that the measurement reference attribute is obtained by measuring various component we are using has some unknown bias. All we may quantities. In the course of the development of the know of this quantity is that it comes from some model, axioms are developed that are useful to population of biases that also follow a statistical uncertainty analysis [1]. Measurement uncertainty is distribution. What knowledge we have of this defined statistically and expressions are derived for distribution may derive from periodic calibration of the estimating measurement uncertainty and combining reference or from manufacturer data. uncertainties from several sources. In addition, guidelines are given for obtaining the degrees of We are likewise cognizant of the fact that the finite freedom for both Type A and Type B uncertainty resolution of our measuring system introduces another estimates. source of measurement error whose sign and magnitude are unknown and, therefore, can only be described Background statistically. Suppose that we are making a measurement of some These observations lead us to a key axiom in quantity such as mass, flow, voltage, length, etc., and uncertainty analysis: that we represent our measured values by the variable x.