Discussion of "A Gibbs Sampler for a Class of Random Convex Polytopes".

In particular, the mapping Graph HT <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="script">F</mi><mo>:</mo><mi mathvariant="bold">u</mi><mo>-></mo><mi> </mi></mrow></math> ht is the inverse mapping of the data-generating Equation (1). The elegantly constructed I i in Proposition 3.2 has components proportional to the inverse of the exponentiated directed graph path "value", minimized over ratios of components Graph HT <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mfrac><mrow><msub><mrow><mi>u</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi mathvariant="script">l</mi></mrow></msub></mrow><mrow><msub><mrow><mi>u</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>k</mi></mrow></msub></mrow></mfrac></mrow></math> ht over all Graph HT <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>n</mi><mo> </mo><mo>{</mo><mn>1</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>N</mi><mo>}</mo></mrow></math> ht . The exposition of the development of the ideas is a series of geometric arguments that directly relate ratios of components of I i to ratios of the values of components of the Graph HT <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>u</mi></mrow><mn>1</mn></msub><mo>,</mo><mo>...</mo><mo>,</mo><msub><mrow><mi>u</mi></mrow><mi>N</mi></msub></mrow></math> ht via Graph HT <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi> </mi></mrow><mrow><mi>k</mi><mo>-></mo><mi mathvariant="script">l</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>u</mi></mrow><mi>n</mi></msub><mo stretchy="false">)</mo></mrow></math> ht . [Extracted from the article]