Bridging uncertain and ambiguous knowledge with imprecise probabilities

Bridging uncertain and ambiguous knowledge with imprecise probabilities

  • Simon L. Rinderknechta, b, Corresponding author contact information,
  • Mark E. Borsukc,
  • Peter Reicherta, b, E-mail the corresponding author
  • a Eawag, Swiss Federal Institute of Aquatic Science and Technology, Department of Systems Analysis, Integrated Assessment and Modelling, CH-8600 Dübendorf, Switzerland
  • b ETH Zurich, Department of Environmental Sciences, Institute of Biogeochemistry and Pollutant Dynamics (IBP), CH-8092 Zürich, Switzerland
  • c Dartmouth College, Thayer School of Engineering, Hanover, NH 03755-8000, USA
  • Received 31 July 2010. Revised 25 July 2011. Accepted 27 July 2011. Available online 13 September 2011.

View full text


Model-based environmental decision support requires that uncertainty be rigorously evaluated. Whether uncertainty is aleatory or epistemic, we argue that probability is the natural mathematical construct for describing uncertainty in predictions used for decision-making. If expert knowledge is elicited using stated preferences between lotteries, and the experts are rational in the sense of avoiding sure loss, then the resulting knowledge quantifications will be consistent with the axiomatic foundation of probability theory. This idea can be extended to the description of intersubjective knowledge when the intent is to characterize the state of knowledge of the scientific community. Many methods for probability elicitation have been reported, but there is nearly always some degree of ambiguity in translating elicited quantities into probabilistic description. This would include: any lack of fit of a particular distributional form to elicited data; incertitude in the elicited data themselves; and/or disagreement in the elicited data across multiple experts. By replacing a precise probability distribution by a set of distributions, the mathematical concept of imprecise probabilities provides a means for representing this ambiguity. In this way, imprecise probabilities can form a bridge between total ignorance and precisely characterized risk by allowing for a continuous degree of imprecision to represent ambiguity. We introduce three metrics to describe the relative ambiguity of important attributes of probability distributions, namely their width, shape, and mode. These metrics are applicable to sets of distributions characterized by using any available method, and we derive the specific forms of these metrics for the Density Ratio Class, which we have found to have many desirable properties. Based on these metrics and on elicitation data from the literature, we use three examples to demonstrate the wide variety of ambiguity that can be present in elicited knowledge. Imprecise probabilities allow us to quantify this ambiguity and consider it in environmental decision-making. Our examples were implemented using a package we recently developed and made freely available for the R statistical programming environment.


  • Expert elicitation;
  • Subjective probabilities;
  • Intersubjective knowledge;
  • Interval probabilities;
  • Qualitative expertise;
  • Quantitative expertise;
  • Robust Bayesian inference;
  • Robust Bayesian statistics;
  • Quantile elicitation;
  • Imprecise probabilities;
  • Probability box;
  • Quantile class;
  • Density Ratio Class

Eliciting density ratio classes

  • Simon L. Rinderknechta, b,
  • Mark E. Borsukc,
  • Peter Reicherta, b, Corresponding author contact information, E-mail the corresponding author
  • a Eawag, Swiss Federal Institute of Aquatic Science and Technology, CH-8600 Dübendorf, Switzerland
  • b ETH Zürich, Institute of Biogeochemistry and Pollutant Dynamics (IBP), CH-8092 Zürich, Switzerland
  • c Dartmouth College, Thayer School of Engineering, Hanover, NH 03755-8000, USA
  • Received 22 April 2010. Revised 8 October 2010. Accepted 11 February 2011. Available online 16 February 2011.

View full text


The probability distributions of uncertain quantities needed for predictive modelling and decision support are frequently elicited from subject matter experts. However, experts are often uncertain about quantifying their beliefs using precise probability distributions. Therefore, it seems natural to describe their uncertain beliefs using sets of probability distributions. There are various possible structures, or classes, for defining set membership of continuous random variables. The Density Ratio Class has desirable properties, but there is no established procedure for eliciting this class. Thus, we propose a method for constructing Density Ratio Classes that builds on conventional quantile or probability elicitation, but allows the expert to state intervals for these quantities. Parametric shape functions, ideally also suggested by the expert, are then used to bound the nonparametric set of shapes of densities that belong to the class and are compatible with the stated intervals. This leads to a natural metric for the size of the class based on the ratio of the total areas under upper and lower bounding shape functions. This ratio will be determined by the characteristics of the shape functions, the scatter of the elicited values, and the explicit expert imprecision, as characterized by the width of the stated intervals. We provide some examples, both didactic and real, and conclude with recommendations for the further development and application of the Density Ratio Class.


  • Probability assessment;
  • Expert elicitation of vague knowledge;
  • Imprecise probabilities;
  • Robust Bayesian statistics;
  • Quantile elicitation;
  • Density Ratio Class
Corresponding author at: Eawag, Swiss Federal Institute of Aquatic Science and Technology, CH-8600 Dübendorf, Switzerland. Tel.: +41 587655281.

Copyright © 2011 Elsevier Inc. All rights reserved.

Kenneth Arrow’s Impossibility Theorem | Unmöglichkeitstheorem bei Gruppenentscheiden

(English below)

Gibt es eine echte Regel für das Konstruieren einer sozialen Präferenz aus individuellen Präferenzen? Mit anderen Worten: Existiert überhaupt eine faire Prozedur für Gruppenentscheide?

Kenneth Arrow untersuchte diese Frage, indem er Kriterien aufstellte, welche berücksichtigt werden sollten, um ein akzeptables Regelwerk beim identifizieren von sozialen Präferenzen aus individuellen Präferenzen zu konstruieren:

1 Soziale Präferenzen sollten komplett sein. D.h., zwischen zwei Alternativen, A und B, können drei Fälle auftreten: A wird B vorgezogen (oder umgekehrt) oder es besteht Indifferenz in der Wahl.
2 Soziale Präferenzen sollten transitiv sein. D.h., wenn A gegenüber B vorgezogen wird und B gegenüber C, dann wird auch A gegenüber C vorgezogen.
3 Wenn jedes einzelne Individuum die Alternative A gegenüber B bevorzugt, dann sollte die gesamte Gesellschaft A favorisieren (schwaches Pareto Prinzip).
4 Soziale Präferenzen sollten unabhängig von einzelnen Individuen sein (Kein Diktator).
5 Soziale Präferenzen sollten unabhängig von irrelevanten Alternativen sein. D.h., Wenn die Gesellschaft die Alternative A gegenüber B vorzieht, dann sollte das unabhängig aller anderen Alternativen sein.

Ok, Kenneth Arrow hat mathematisch bewiesen, dass es kein Regelwerk geben kann, welches eine gesellschaftliche Präferenz aus individuellen Präferenzen, unter den aufgestellten Kriterien, herleitet. Mit anderen Worten: Sowie das Majorz-System, als auch das Proporz-Prinzip, oder irgendein-Prinzip, verletzt mindestens, unter gegebenen aber möglichen Umständen, eine der obigen Regeln, wenn eine Gruppe zu einer Entscheidungspräferenz kommen will.

Übrigens hat sich Kenneth Arrow hauptsächlich mit dieser Erkenntnis den Nobel Preis in Ökonomie geholt.

Den Ausweg gibt es nur durch den politischen Prozess. Wenn die individuellen Meinungen Gemeinsamkeiten aufweisen, können durchwegs Gruppenentscheide getroffen werden.

Trotzdem, Arrows „General Impossibility Theorem“ stellt nach wie vor für Sozialphilosophien ein Problem dar, bei denen – wie in der Demokratie – die sozialen Entscheidungen aus dem Willen der Einzelnen abgeleitet werden.

Ist es nicht so, dass wir ebensoviel Zeit brauchen um das Wahlsystem zu bestimmen, wie die eigentliche Wahl zu treffen?


Is there a rule for constructing social preferences from individual preferences? In other words, does a fair procedure for group decisions even exist?

Kenneth Arrow examined the problem rigorously by specifying a set of requirements that should be satisfied by an acceptable rule for constructing socially preferences from individual preferences; i.e.,

* Social preferences should be complete in that given a choice between alternatives A and B it should say whether A is preferred to B, or B is preferred to A or that their is a social indifference between A and B.
* Social preferences should be transitive; i.e., if A is preferred to B and B is preferred to C then A is also preferred to C.
* If every individual prefers A to B then socially A should be preferred to B.
* Socially preferences should not depend only upon the preferences of one individual; i.e., the dictator.
* Social preferences should be independent of irrelevant alternatives; i.e., the social preference of A compared to B should be independent of preferences for other alternatives.

What Kenneth Arrow was able to prove mathematically is that there is no method for constructing social preferences from arbitrary individual preferences. In other words, there is no rule, majority voting or otherwise, for establishing social preferences from arbitrary individual preferences.

This was a major result and for it and other work Kenneth Arrow received the Nobel prize in economics.

There is one way out of this impasse for making social decisions through the political process. If the individual preferences have some commonality then social preferences can be constructed. If the alternatives can be represented as being elements of a spectrum and the preferences of the individuals exhibit single peakedness then social preferences can be constructed.

2nd IBP PhD Congress | Eawag Dübendorf | Forum Chriesbach | Friday 17th of April 2009 Start: 8.30 a.m. | Program

8.00-8.30   Registration
8.30-8.40   Janet Hering Eawag Director –  Welcome speech
8.40-9.00   Simon-Lukas Rinderknecht Eawag Siam – Eliciting density ratio class priors
9.00-9.20   Anne Dietzel Eawag Siam – Effects of changing anthropogenic pressures on water quality and plankton dynamics in three Swiss lakes – Long-term simulations with the biogeochemical-ecological lake model BELAMO
9.20-9.40   Ilaria Stendardo ETH Environmental Physics – Long-term oxygen trends in the North Atlantic
9.40-10.00  Tonya Del Sontro Eawag Surf – Quantifying extreme methane emissions from a Swiss reservoir
10.00-11.00 Coffee break and poster session 1
11.00-12.00 EAWAG FRIDAY SEMINAR – Johan Rockström, Stockholm Environment Institute and Stockholm Resilience Centre, Sweden – Building water resilience in the face of Global Environmental Change: The need for a green-blue water paradigm
12.00-13.40 Lunch
13.40-14.00 Guido Bronner ETH Environmental Chemistry – Sorption of multifunctional and polar compounds (incl.pesticides) to peat and soils: Experimental findings and LFER-modeling
14.00-14.20 Michael Aeschbacher ETH Environmental Chemistry – Redox properties of humic substances: Electrochemical characterization
14.20-14.40 Michael Madlinger ETH Environmental Chemistry – Adsorption of transgenic Cry proteins to mineral and organic soil surfaces: Effects of soil composition and solution chemistry
14.40-15.50 Coffee break and poster session 2
15.50-16.10 Jakob Frommer ETH Soil Chemistry – Chromium in the environment: an X-ray absorption study
16.10-16.30 Irene Wittmer Eawag Utox – Biocide and pesticide inputs to surface waters
16.30-16.50 Claire Farnsworth Eawag W&T – A hydrous manganese oxide doped gel probe sampler for measuring in situ reductive dissolution rates
17.00       Apéro
18.00       Dinner

1 Jafet Andersson Eawag Siam – SWAT capable of simulating smallholder food production in the Thukela River Basin, South Africa
2 Tobias Bergmiller ETH Molecular Microbial Ecology – Replacement of conserved essential functions of E. coli
3 Robert Brankatschk Eawag Umik – Succession of bacterial nitrogen transformation processes in a glacier forefield
4 Dörte Carstens & Krista Köllner Eawag Surf – Degradation and transformation of lacustrine organic nitrogen compounds: microbiology and biogeochemistry
5 Rang Cho ETH Environmental Microbiology – Methane turnover in the rice root zone: A novel quantification concept
6 Olivier Eugster ETH Environmental Physics – Should the magnitude of water column denitrification be revised downward?
7 Claudine Hauri ETH Environmental Physics – Changes of the aragonite saturation horizon in eastern boundary upwelling systems
8 Anke Hofacker ETH Soil Chemistry – How does temperature affect colloidal trace metal release from a submerged riparian soil?
9 Susanne Kern Eawag Uchem – Identification of transformation products of organic contaminants in natural waters by computer-aided prediction and high-resolution mass spectrometry
10 Andreas Kretschmann Eawag Uchem – Time resolved effect model for Daphnia magna – Measurement and modeling of the toxicokinetic of Diazinon
11 Claudia Lorrai Eawag Surf – Aquatic eddy correlation
12 Danielle Madureira Eawag Utox – Strategy to analyze gene expression profiles in Hepa cells exposed to BaP as a prerequisite for a cell-wide understanding of BaP-cell interactions
13 Holger Nestler Eawag Utox – Profiling the proteome of Chlamydomonas reinhardtii exposed to herbicides
14 Judith Neuwöhner Eawag Utox – Physiological modes of action of fluoxetine and its human metabolites in algae
15 Nela Nikolic ETH Molecular Microbial Ecology – Phenotypic variation of genetically identical bacteria growing in various sugars
16 Simone Peter Eawag Surf – Restoration of riverine floodplains: Effect of increased environmental heterogeneity on transformations of organic matter and nutrients
17 Flavio Piccapietra Eawag Utox – Physicochemical characterization of silver nanoparticles: effect of pH and ionic strength on aggregation
18 Maaike Ramseier Eawag W&T – Formation of assimilable organic carbon by different oxidants
19 Christian Scheidegger Eawag Utox – Chracterization of metal-phytochelatin complexes induced by lead in the green alga Chlamydomonas reinhardtii
20 Yvonne Scheidegger Eawag W&T – Air and water inclusions in stalagmites as new climate proxies
21 Marita Skarpeli-Liati ETH Environmental Chemistry – Nitrogen isotope fractionation during the oxidation of substituted anilines at manganese oxide surfaces
22 Friedhelm Steinhilber Eawag Surf – Reconstruction of solar activity during the Holocene
23 Kay Steinkamp ETH Environmental Physics – Oceanic constraints on terrestrial carbon fluxes
24 Tobias Vogt Eawag W&T – Investigation of bank filtration in gravel and sand aquifers using time-series analysis
25 Jannis Wenk Eawag W&T – Inhibition of triplet-induced oxidation reactions by different types of dissolved organic matter
26 Roland Zurbrügg Eawag Surf – Exploring dam impacts on tropical floodplain biogeochemistry

Note: Talks will be held in the room Forum Chriesbach C20
Registration, coffee breaks, poster sessions and lunch will take place on the ground floor (Forum Chriesbach, B-floor)
Apéro and dinner will be served at aqa (Forum Chriesbach, A-floor)