I’m a bit curious about what job “dimension” is doing here. Given that I can map an arbitrary vector in Rn to some point in R via a bijective measurable map (https://en.wikipedia.org/wiki/Standard_Borel_space#Kuratowski’s_theorem), it would seem that the KPD theorem is false. Is there some other notion of “sufficient statistic complexity” hiding behind the idea of dimensionality, or am I missing something?
There’s a smoothness assumption. I assumed differentiability, although that could be weakened somewhat. (The assumption is hidden in the sister post to this one, The Additive Summary Equation.)
I’m a bit curious about what job “dimension” is doing here. Given that I can map an arbitrary vector in Rn to some point in R via a bijective measurable map (https://en.wikipedia.org/wiki/Standard_Borel_space#Kuratowski’s_theorem), it would seem that the KPD theorem is false. Is there some other notion of “sufficient statistic complexity” hiding behind the idea of dimensionality, or am I missing something?
There’s a smoothness assumption. I assumed differentiability, although that could be weakened somewhat. (The assumption is hidden in the sister post to this one, The Additive Summary Equation.)