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On triangular norms, metric spaces and a general formulation of the discrete inverse problem or starting to think logically about uncertainty On triangular norms, metric spaces and a general formulation of the discrete inverse problem or starting to think logically about uncertainty
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Formulating a scientific question Such definition does not use notions such as Probability, Sampling, Filtering, Distance, McMC, Density etc.. How to formulate such quantification problem? quantifying uncertainty = quantifying lack of (human) understanding
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An uncertainty quantification problem ? Two independent vagueness statements are given about X in terms of two pdfs What is the “combined” pdf ? what is uncertainty about X? x x pdf
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An example photo weight scale A set of similar chairs
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A question of conjunction: logical “and” AND10 110 000 0.60.4 0.8?? 0.2?? p1p1 p2p2 p2p2 p1p1
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Axioms for conjunctions t-norms Which functions follow these axioms?
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Examples of t-norms p1p1 p2p2
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The impact of choice of conjunction p1p1 p2p2 p1p1 AND0.250.50.75 0.50.1250.250.375 AND0.250.50.75 0.50.210.370.47 AND0.250.50.75 0.50.250.5 p1p1 strong weak
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What does the weight of a chair have to do with my reservoir? Model m: the unobservable parameters describing part of a physical system Data d: the observable parameters describing part of a physical system Reservoir: various sources of information on (d,m) Expert interpretation Physical models, rules, empirical relationships,…. The reservoir data d obs
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Formulating the discrete inverse problem (Tarantola & Valette, 1982)
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A third source of information: the metric density on (d,m) Cartesian variables yy xx Non-Cartesian variables rr ss D=metric tensor
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Formulating discrete inverse problems (Tarantola) How do we combine these three probability densities into a single probability density as a quantification of uncertainty ?
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Application of conjunctions to densities The metric density acts as a neural element in conjunction of densities
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Application of conjunctions to density ratios
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Formulating the problem in metric spaces All real problems involve non-Cartesian variables with non-Euclidean distances Approximate non-Euclidean distances with Euclidean distances How? use multi-dimensional scaling
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Recall MDS Water-cut Time, days d obs d eigen-component 1 eigen-component 2 d obs Warning common confusion MDS ≠ PCA
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What does multi-dimensional scaling achieve?
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Back to the problem formulation
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Illustration of concepts Show that uncertainty quantification is dependent on the choice of t-norm Show how metric space formulation works NOT YET: show any practical methodology based on it
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The t-norm used
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Example by Celine Scheidt m1m1 m2m2 m 2000......
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Metric space of model parameters Prior of model parameters
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Metric space of data parameters Prior of data parameters
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Modeling the “pre-theory”
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Modeling the “theory” The “theory” is assumed exact Given a uniform distributed m what is the predicted d ?
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Modeling the “theory”
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Comparison of PDFs calculated from different t-norms Weaker conjunctions lead to increased uncertainty
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Some statistics on remaining uncertainty of sum and product t-normP50 Std. dev. Product48.28.6 w = 0.7547.79.3 w = 0.547.210.5 w = 0.2547.811.9 Minimum47.713.0 t-normP50 Std. dev. Product21.72.0 w = 0.7521.82.1 w = 0.522.12.4 w = 0.2522.02.6 Minimum22.32.7 Sum Product
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Comparison with Bayesian theory Formulations neglecting metric densities lead to paradoxes (Mosegaard, 2009) lead to inconsistencies, e.g. negative saturations (EnKf) Bayesian theory is a special case of conjunction theory Allows only limited non-linearity (Tarantola, 1982) Conditional probabilities are ill-defined (Tarantola, 1987) Often requires linear vector spaces for d and m (EnKf) Confusion with “prior”, “belief”, “updating vs revision”
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Bayesian theory Implicit vs explicit updating models the relationship between m and some background knowledge B 0 at time t 0
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Simple example on Bayesianism Exploration setting: Data: 2D seismic sections Do we have Deltaic (heterogeneous sand) or Aeolian (homogeneous sand) ? “A” = the reservoir is deltaic “B” = the 2D seismic data (e.g. a bright spot) Expert 1 knowledge expert provides assessment Expert 2 data expert : numerical modeling to get
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What is the problem? Inconsistent with implicit conditioning model of Bayesianism but knowledge expert and data expert look at the same data, → their assessments are related somewhat → too small uncertainty Solution
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Explicit conditioning
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The or -model ?
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Formulating using conjunctions Formulate the problem as a conjunction problem, not as an updating problem Updating can be part of sub-problems such as with expert 2 Conjunctions model explicitly interdependency between E 1 and E 2
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Opinions (for debate) The most critical element of an uncertainty quantification question lies in the formulation of the problem, not necessarily its sampling The choice of conjunction = defining a way of reasoning about terms such as independence (Bayesian theory provides only one way of reasoning)
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Final quote as far as the propositions of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality. Albert Einstein as far as the propositions of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality. Albert Einstein
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Opinions (for debate) The most critical element of an uncertainty quantification question lies in the formulation of the problem, not necessarily its sampling The choice of conjunction = defining a way of reasoning about terms such as independence (Bayesian theory provides only one way of reasoning)
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What conjunction to choose? Choosing a conjunction = choosing a way of reasoning/thinking How to deal with contradiction ? What does it mean if “something is true”, “something is false” ? For example: the minimum = weak conjunction = choosing for intuitionistic logic A negation does not mean something is false, but is refutable You can only say something is false if you have a counterexample Applies to any interpretative science (geosciences)
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Example What is the proper Euclidean distance ?
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Proof of axiom 4
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The notion of conditional probability
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All Bayesian theory is a special case of conjunction theory
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Bayesian theory cannot deal with most non-linear problems Bayesian theory can handle this Bayesian theory cannot handle this
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Ensemble Kalman filter
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Additional limitation of Bayes’ rule Background knowledge can be probabilistic, but any new data cannot New concepts cannot be assimilated/integrated Anything with probability 0/1 cannot be updated
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