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.

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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.

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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.

Tribute to Marlilyn Monroe


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Warhol made this painting the year screen legend Marilyn Monroe committed suicide. He painted the canvas an iridescent gold and silkscreened the star’s face in the center of the composition. Like other paintings by Warhol that feature Monroe’s likeness, this … Continue reading

How to Dance 1953

Go Goofy Go

1. Listen to the rhythm. With most music that people dance to on a casual basis, you can pick up the beat by counting to four repeatedly (one, two, three, four, one, two, three, four…). Do this out loud. Sometimes it’s very slow, sometimes it’s very fast, but the elements of the song usually repeat or change every time you start the count again. Practice listening to the music on the radio or, if possible, the music you’ll probably dance to. If the rhythm doesn’t fit, you may find yourself counting to three rather than four, but unless you’re dancing the waltz, it’s probably actually four counts with the last one silent (one, two, three, [pause], one, two, three, [pause]). Sometimes you’ll be tempted to listen to the drums, which in some kinds of music (such as salsa) will sound off in groups of three. Try to listen beyond this and find the count of four. If you can’t, go to another song.

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2. Bob your head to the rhythm. If the beat is on the slower side, you can bob your head so that your chin is down at every count. If the beat is faster, your chin will be down on two of the counts and up on the other two. Do this gently, this isn’t headbanging. The idea is to start getting a part of your body physically moving with the rhythm. As your movements get more complicated, you might “lose” the rhythm, and you should always return to this step if that happens.

3. Shift your weight between your feet. You can continue bobbing your head, if it helps, but don’t stop counting yet. Shift all of your weight to one foot (you can lift the other foot slightly off the ground to make sure all your weight is off of it). At every other count (preferably 1 and 3) shift your weight completely to the other foot. You can also shift your weight at every single count, but starting out slow will help you get comfortable before you start dancing fast. Keep your legs “loose” and bend your knees slightly; there should be just a little bit of “bounce” to your weight shift, and a subtle bounce (in place) on the counts when you aren’t shifting your weight as well.

4. Move your feet. Once you’re shifting your weight to the rhythm, practice moving your feet. Right before you shift your weight to a foot, move it slightly, even just an inch or two from where it was before. When you dance with someone else, you’ll need to be more careful to move around in a way that accommodates your partner without stepping on them; but for now, just make very small movements. Keep the music on, keep counting, keep moving a little bit with each time you shift your weight (at every other count). When you move your foot, keep it close to the ground. You can kick your feet up in the air later, when you’re letting loose.

5. Move your hips. When you put your weight on a foot, move your hips (and your body) slightly in the direction of that foot. If you shift your weight onto your right foot, for example, move your hips to the right. You can twist your body slightly to add a little more movement: when you move to the right, put your right shoulder forward a little and left shoulder back, and vice versa for when you move to the left. Note that this range of movement (hip movement, twisting, swiveling) is usually exaggerated by women to emphasize the female form.

6. Throw your hands in the air. This is the part that indicates to people that you’re having fun. If you’re uncomfortable, the tendency is to keep your arms close, or let them hang limp. Instead, move your arms around. Keep your hands open or in very loose fists, but never stiff (unless you’re doing the robot). You can put your arms in the air for 8 or 12 counts, then hold them straight out in front of you for 8 or 12 counts, then at 90 degree angles at your sides (like when you’re running) for another 8 or 12 counts. Keep switching it up. If you’re dancing with someone, you can put your hands on their shoulders, waist, or hips.

7. Lose yourself in the music. Let your body move naturally to the rhythm. You want it to look smooth rather than jerky and stiff. As you relax and get into the music, you might find your feet moving a little further, your head swinging a little wider, and an involuntary clap shining through. And sometimes, you’ll lose the rhythm (especially when songs slow down right in the middle)–just go back to step one.

8. Practice dancing whenever you have the chance. Build your confidence by dancing in the kitchen, in the bedroom—anywhere you feel comfortable. Not only will this help to improve your general fitness, but you’ll start to feel more comfortable in your body. Watch yourself in a mirror to get an idea of what looks good and what doesn’t.

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School in a mesh wire fence

When I was in Montréal I discovered a French School – a so called “Collège Français”. It was not very seen that I take pictures. However I could shoot at least one shot before I was sent away. And still the main subject on the picture makes the mesh wire fence. Somehow I think it is quite sad, that kids (in uniforms) are put in mesh wire fences… Furthermore, mesh wire fences cause in any case a huge amount of problems… see once more the famous Maschendrahtzaun on Youtube.

School in a mesh wire fence in Montréal

School in a mesh wire fence in Montréal

The Red Chair

Sometimes a picture tells more than 1000 words. Sometimes 1000 words may help to tell more about a picture. This is obviously the case for that “Red Chair Picture” shown aside. Unfortunately, the real true story was lost. Please help to reconstruct the nearly surely crazy life of the Red Chair. Hints which are useful to learn more (or even everything – who knows?) about the Red Chair are welcomed posts below. I’am just impressed about the entire appearance of his Red Majesty and wonder what people may know about it. Thanks.

words 1000 picture

Word = Picture / 1000

Financial Crisis | How easy can it be explained?

However, you still wonder why we are in a financial crisis? You can not understand the reason why? It is terribly easy to see how the market worked this out. Have a careful look at this video:

w^3. The Crisis of Credit Visualized . internet

I found it excellent. Just great. Any questions?