MinNorm approximation of MaxEnt/MinDiv problems for probability - PowerPoint PPT Presentation

MinNorm approximation of MaxEnt/MinDiv problems for probability tables Patrick Bogaert and Sarah Gengler UCL

Rebuilding probability tables UCL

Rebuilding probability tables • Limited number of samples  Poor estimates UCL

Rebuilding probability tables • Limited number of samples  Poor estimates • How to integrate experts opinion ? UCL

Rebuilding probability tables • Limited number of samples  Poor estimates • How to integrate experts opinion ? Rewriting information as equality / inequality constraints  UCL

Rebuilding probability tables • Limited number of samples  Poor estimates • How to integrate experts opinion ? Rewriting information as equality / inequality constraints  UCL

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 )  UCL

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 UCL

The MaxEnt problem UCL

The MaxEnt problem • Equality constraints UCL

The MaxEnt problem • Equality constraints • Entropy maximized subject to the equality constraints UCL

The MaxEnt problem • Equality constraints • Entropy maximized subject to the equality constraints UCL

The MaxEnt problem • Equality constraints • Entropy maximized subject to the equality constraints  Sequence of MinNorm problems for solving the MaxEnt problem UCL

MinNorm as an approximation of MaxEnt UCL

MinNorm as an approximation of MaxEnt • Taylor series of ln p i around p i = k i UCL

MinNorm as an approximation of MaxEnt • Taylor series of ln p i around p i = k i • Truncating at degree one and summing over i UCL

MinNorm as an approximation of MaxEnt • Taylor series of ln p i around p i = k i • Truncating at degree one and summing over i • In particular, if k i = 1/n UCL

MinNorm as an approximation of MaxEnt • For any other choice of the k i ‘s, by completing the square UCL

MinNorm as an approximation of MaxEnt • For any other choice of the k i ‘s, by completing the square • Summing over i Where UCL

The MinDiv problem UCL

The MinDiv problem • Divergence or Kullback-Leibler distance UCL

The MinDiv problem • Divergence or Kullback-Leibler distance • Equality constraints = 0 Maximizing UCL

The MinDiv problem • Divergence or Kullback-Leibler distance • Equality constraints = 0 Maximizing  UCL

The MinDiv problem • Divergence or Kullback-Leibler distance • Equality constraints = 0 Maximizing   UCL

The MinDiv problem • Divergence or Kullback-Leibler distance • Equality constraints = 0 Maximizing    Sequence of MinNorm problems for solving the MinDiv problem UCL

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 UCL

MinNorm as an approximation of MinDiv UCL

MinNorm as an approximation of MinDiv • Taylor series around p i = k i and completing the square UCL

MinNorm as an approximation of MinDiv • Taylor series around p i = k i and completing the square • Summing over i Where UCL

Application in drainage classes mapping UCL

Application in drainage classes mapping • Categorical data are found in a wide variety of applications UCL

Application in drainage classes mapping UCL

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 UCL

Application in drainage classes mapping UCL

Application in drainage classes mapping UCL

Application in drainage classes mapping UCL

Application in drainage classes mapping UCL

Application in drainage classes mapping UCL

Application in drainage classes mapping UCL

Application in drainage classes mapping UCL

Application in drainage classes mapping UCL

Integrating the lithological map : 4 cases UCL

Integrating the lithological map : 4 cases UCL

Integrating the lithological map : 4 cases UCL

Spatial prediction UCL

Integrating the lithological map : 4 cases UCL

Spatial prediction UCL

Integrating the lithological map : 4 cases UCL

Conclusions UCL

Conclusions UCL

Conclusions  MinNorm Approximations UCL

Conclusions  MinNorm Approximations UCL

Conclusions  MinNorm Approximations UCL

Conclusions  MinNorm Approximations UCL

Conclusions  MinNorm Approximations UCL

Conclusions  MinNorm Approximations UCL

Conclusions  MinNorm Approximations UCL

Thank you for your attention UCL

References UCL