MinNorm approximation of MaxEnt/MinDiv problems for probability tables Patrick Bogaert and Sarah Gengler
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Rebuilding probability tables
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Rebuilding probability tables • Limited number of samples Poor estimates
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Rebuilding probability tables • Limited number of samples Poor estimates
• How to integrate experts opinion ?
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Rebuilding probability tables • Limited number of samples Poor estimates
• How to integrate experts opinion ?
Rewriting information as equality / inequality constraints
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Rebuilding probability tables • Limited number of samples Poor estimates
• How to integrate experts opinion ?
Rewriting information as equality / inequality constraints
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Rebuilding probability tables • Limited number of samples Poor estimates
• How to integrate experts opinion ?
Rewriting information as equality / inequality constraints • Equality constraints MaxEnt • Inequality constraints Minimum divergence (MinDiv)
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Rebuilding probability tables • Limited number of samples Poor estimates
• How to integrate experts opinion ?
Rewriting information as equality / inequality constraints • Equality constraints MaxEnt • Inequality constraints Minimum divergence (MinDiv) Need for an efficient methodology to rebuild probability tables from both equality and inequality constraints
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The MaxEnt problem
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The MaxEnt problem • Equality constraints
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The MaxEnt problem • Equality constraints
• Entropy maximized subject to the equality constraints
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The MaxEnt problem • Equality constraints
• Entropy maximized subject to the equality constraints
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The MaxEnt problem • Equality constraints
• Entropy maximized subject to the equality constraints
Sequence of MinNorm problems for solving the MaxEnt problem
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MinNorm as an approximation of MaxEnt
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MinNorm as an approximation of MaxEnt • Taylor series of ln pi around pi = ki
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MinNorm as an approximation of MaxEnt • Taylor series of ln pi around pi = ki
• Truncating at degree one and summing over i
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MinNorm as an approximation of MaxEnt • Taylor series of ln pi around pi = ki
• Truncating at degree one and summing over i
• In particular, if ki = 1/n
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MinNorm as an approximation of MaxEnt • For any other choice of the ki ‘s, by completing the square
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MinNorm as an approximation of MaxEnt • For any other choice of the ki ‘s, by completing the square
• Summing over i
Where
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The MinDiv problem
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The MinDiv problem • Divergence or Kullback-Leibler distance
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The MinDiv problem • Divergence or Kullback-Leibler distance
• Equality constraints =0
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Maximizing
The MinDiv problem • Divergence or Kullback-Leibler distance
• Equality constraints =0
Maximizing
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The MinDiv problem • Divergence or Kullback-Leibler distance
• Equality constraints =0
Maximizing
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The MinDiv problem • Divergence or Kullback-Leibler distance
• Equality constraints =0
Maximizing
Sequence of MinNorm problems for solving the MinDiv problem
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The MinDiv problem • Divergence or Kullback-Leibler distance
• Equality constraints =0
Maximizing
Sequence of MinNorm problems for solving the MinDiv problem Both Equality and Inequality constraints can be processed together by MinNorm approximations
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MinNorm as an approximation of MinDiv
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MinNorm as an approximation of MinDiv • Taylor series around pi = ki and completing the square
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MinNorm as an approximation of MinDiv • Taylor series around pi = ki and completing the square
• Summing over i
Where
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Application in drainage classes mapping
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Application in drainage classes mapping • Categorical data are found in a wide variety of applications
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Application in drainage classes mapping
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Application in drainage classes mapping • Categorical data are found in a wide variety of applications • 90 % of variables collected in soil surveys are categorical • Soil drainage, an important criterion in rating soils for various uses
degree of cross correlation (see examples of speech signals in Caiafa et al. .... the vector (CMB) by using the assumption A2 instead of using the Maximum ...
has been registered as the paper number 90 and is accepted for presentation ... if you are allowed to present your work during the workshop, its inclusion in the.
Mar 31, 2006 - Fax: 33-1-60 19 04 86 or 33-1-69 41 30 60. E-mail: ... Purdue University, Indiana, USA ... Abstracts (one page of about 400 words) of the ...
Page 1 ... Bayesian Inference and Maximum Entropy Methods in Science ... INTEGRATED DATA ANALYSIS: NON-PARAMETRIC PROFILE GRADIENT ESTIMATION by ... has been registered as the paper number 66 and is accepted for ...
Jul 8, 2006 - Poster sizes will be determined and poster presenters will be notified in a future email. It is expected that the poster will not exceed 60 inches ...
Oct 30, 2015 - Publishes 3 periodical publications (REE, 3E.I, e-STA). ⢠Publishes 3 .... mathematical formalizations of ideas coming from biology, ... http://www.yann-ollivier.org/rech/publs/hdr_intro.pdf. Y. Ollivier ... form for q-bits are given
Electrode selection for non-invasive Fetal Electrocardiogram Extraction using Mutual Information. Criteria by. F. Vrin, R. Sameni, F. Parmentier, Ch. HÃrail, ...
The specific assignment to either an oral or poster presentation can be found in the list that follows the closing registration deadline. The precise schedule is ...
has been registered as the paper number 48 and is accepted for presentation during the workshop ... proceedings of the conference is subject to the result of the peer review during the conference and to the ... This support can be in the form of.
the norm closure of the Gibbs family fails due to discontinuities of the maximum-entropy inference. The current .... closure operator cl. rI .... w for the inner product.
topology, and is idempotent [22] but the rI-topology is strictly finer than the norm topol- ogy. In the classical case (of finite support) the rI-topology equals the norm ...
An amended MaxEnt formulation for deriving Tsallis factors, and associated issues. Background. Power laws. When maximized under mean constraint, it leads ...
Jul 5, 2010 - POS & Parser. Spelling. Corrector. Tokenizer. Model including. Ontology. The body could raise tempratuer. The body could raise temperature.
where the conditional score functions Ï. (k,l) si of the estimated ... The final version of the paper will contain the theoretical details and the experimental results ...
entropy is maximised subject to âobservableâ constraints on the mean values of certain flow rates and/or potential differences, and also âphysicalâ constraints ...
Welcome to this 34th edition and 5th edition in France of ... A few worlds about this years edition ... great scientific discussions, questions and answers, all with.
Feb 28, 2006 - An amended MaxEnt formulation for deriving Tsallis factors, and associated issues. J.-F. Bercher. Ãquipe Signal et Information, MOSIM, ESIEE, ...
Feb 28, 2006 - This work applies the differential equation method developed by Chiu ... Vennard (1996) Elementary Fluid Mechanics, 7th ed., John Wiley, NY. 5. H. Schlichting, K. Gersten (2001), Boundary Layer Theory, 8th ed., Springer, NY.
A method for handling data in presence of uncertainty with qualitative values is the theory of Dempster-Shafer (DS). The DS model includes the Bayesian.
increase the degree of truth for which a hypothesis is definitely either confirmed ... of aggregation techniques include arithmetic averages, geometric averages, ...
Jul 5, 2010 - free-text, English-language responses (few words to ... C1: An isosceles triangle has (at least) two angles the same size .... 150 (114). 1.00.
