pleuviri — LessWrong

More freedom does not necessarily mean less modeling burden (and certainly not less burden of choice). Making 300 decisions in sequence is a greater burden than making the first 10 and having the others automatically handled by context. Low-level programming languages are burdensome despite (or perhaps because of) the freedom that they expose to the modeler. And universal constructions are appealing in part because they allow you to specify important information arguably without making any (unjustified) choices at all (i.e., very little modeling burden)! ... (read more)

Imprecise beliefs: a tiny introduction

pleuviri22d*10

Not all credal sets satisfy the inclusion/exclusion rule that characterizes DS belief functions. Also, they arguably combine differently (Dempster's rule vs conditioning pointwise).

Definitely the right way of turning a DS function into a belief is not the way you capture credal sets, since they are notoriously not idempotent (i.e., for most basic DS mass functions ).

Imprecise beliefs: a tiny introduction

pleuviri22d30

Fair enough, although, if we're modeling a continuum of variables anyway, it seems unfair to restrict to finite numbers of nodes. Even so, if we make use of function-valued variables, I'm not yet 100% sure it's true that is not the semantics of any PDG with finite and . My efforts to find a counterexample were thwarted by not having a good way to express derivatives, but I'm not sure if this is due to some fundamental limit, or because of my own short-sightedness.

Thanks again for the two nice concrete examples!

Imprecise beliefs: a tiny introduction

pleuviri22d30

Thank you for your generous recognition and changes to the post.

To connect some dots, recall that is the condition under which certain PDGs (those with ) are equivalent to factor graphs. This is the boundary of the region , or equivalently where inference is known to be fixed-treewidth tractable (on par with factor graphs and BNs, but the precise complexity is unknown with a gap of vs for PDGs of variables), and is solvable precisely because it can be compiled to a convex problem. In the original PDG paper, Joe insisted that we quarantine discussion... (read more)

Imprecise beliefs: a tiny introduction

pleuviri25d32

fwiw Dempster-Shafer belief functions are a very nice special class of PDGs :)

Imprecise beliefs: a tiny introduction

pleuviri25d*Ω8110

Thanks for putting this together! Permit me to promote PDGs a bit, which I think are under-sold here :)

First: motivating a "degree of inconsistency" scoring-function semantics, valued in and additive, was a primary innovation of the original PDG paper, which I think deserves some credit for this high-level picture of beliefs. Furthermore, my sense is that PDGs already capture the important elements of these other formalisms (many of them actually embed as nice sub-classes of PDGs), and the extra structure of the PDG formalism yields significant benefits... (read more)