pleuviri

• pleuviri 5 Aug 2026 0:22 UTC

  3 points

  0

  in reply to: Erez Abrams’s comment on: Im­pre­cise be­liefs: a tiny introduction

  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)! As davidad points out, this burden is a matter of syntax, i.e., the interface for constructing valid objects. Do the constraints of that interface provide helpful, simplifying guidance? Or does working within the constraints impose (computational) difficulties on the modeler? My experience is that the structure of PDGs does both: it makes specifying “natural” beliefs easier and “unnatural” ones more difficult.

  My understanding is that @davidad wants to sidestep syntax and focus on delivering an IR (intermediate representation) for beliefs: a simple common compilation target that retains some useful structure (in this case, additive combination). This effectively deflects my point about the modeler’s burden, since his aim is instead to reduce the burden of a compiler that targets the IR. This allows him to endorse PDGs as one particular (nice) language for articulating inconsistency functionals, casting PDG semantics as the way to compile them. This makes a lot of sense to me, and the mathematical simplicity is definitely an appealing advantage over PDGs for this purpose.

  However, I’m not 100% convinced that this is a good idea. The class of PDGs is itself already a useful IR, since it unifies so much, and it intentionally occupies a different point in the design space. The structure can indeed create some burden for certain modelers (e.g., for one who wants to arrive at a certain preconceived belief functional, but now, annoyingly, has to figure out how to write it down in terms of local conditional probabilities and confidences), but this exercise also has nice side-effects (e.g., gives you justification for your loss function), often yielding significant interpretability benefits. As you point out, the map from PDGs to belief functionals is non-injective; in my view, that is because the mapping loses something that is worth tracking in a belief state. There’s technically a difference between specifying P(X) and P(Y|X) vs P(Y) and P(X|Y) (all with the same confidence), even though we all agree they are semantically equivalent. Making this distinction (and others in the same class) can be relevant for revising your beliefs (e.g., if you determine that your mechanism for forming conditional beliefs had a flaw). My concern is that direct specification of an inconsistency functional could short-circuit the modeling process that PDGs are designed to elicit.

• pleuviri 3 Aug 2026 2:42 UTC

  1 point

  0

  in reply to: davidad’s comment on: Im­pre­cise be­liefs: a tiny introduction

  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 ).

• pleuviri 2 Aug 2026 23:32 UTC

  3 points

  0

  in reply to: davidad’s comment on: Im­pre­cise be­liefs: a tiny introduction

  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!

• pleuviri 2 Aug 2026 23:16 UTC

  3 points

  0

  in reply to: davidad’s comment on: Im­pre­cise be­liefs: a tiny introduction

  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 of , so we assumed (in the definition of a PDG) that outside of §4.3; hence claim in §3.3. I apologize for the confusion that can easily result.

  I sympathize with your suspicion of negative belief, and I advise anyone who isn’t 100% sure of their footing to assume and move on. I see other reasons to call it an anti-pattern; a negative KL term on its own creates problems, for example—perhaps because it is not lower semi-continuous (on its own). Similarly, while we have the training wheels on, it’s best to stick to confidences when we aggregate beliefs as . But notice that negative confidence does not necessarily lead to loss of convexity (or ) overall. In Fig 1 of “Loss as the Inconsistency of a PDG”, for example, the divergence construction has confidence on one arc and on the another, yet the problem remains within a broader convex region where the PDG inconsistency can be solved in closed form (although it lies outside the the “canonical” convex region ). The minimum value is negative in this case, but the semantics of this PDG are bounded below, so we can ensure with additional context. Negative confidences also play an intuitive role in modeling generative adversarial networks (GANs), in the LIR framework. Most importantly, the structural weights themselves can be regarded as negative confidences relative to the base measure! (optional: reflect again on the condition .) To summarize: negative confidences can be useful, don’t necessarily result in negative overall belief in context. Here is a general interpretation of possibly-negative beliefs : view them as “pseudo”-beliefs that are only valid in contexts in which they result in a proper posterior belief .

  Finally: I like your answer to my challenge :)
  I think there might be a PDG angle, but I haven’t fully worked it out yet. Here are my thoughts so far:

  • To model “bob’s belief is probably wrong”, I think there are a lot of different different modeling choices we can make; my gut reaction with a PDG is just to assign to , which can work provided there’s enough context to stabilize—but I agree this can be problematic in isolation. Your definition of is definitely better-behaved on its own, but the distance metric and radius are not present in the gloss “bob’s belief is probably wrong”, so in my view this particular functional requires a more complex story to motivate. (I was hoping to get you to tell such a story so I could try to follow it using the PDG formalism.) Still, I am inclined to agree that your functional is worth expressing, and I also agree that seems impossible to capture the underlying exclusion exclusion ball with the semantics of a fixed finite PDG. That said, I strongly suspect it is expressible via a minimax game over PDGs, and would not be surprised if it can be expressed via local inconsistency resolution (LIR) on a PDG with two foci (which I’m tempted to call a “bi-focal” PDG) using negative confidence, like how we model GANs with PDGs. Another relevant fact: the (-regularized entropic) optimal transport distance is itself the (-)inconsistency of a PDG that has high-confidence marginal beliefs and together with a soft “cost” implemented by the belief . I will respond again if I finish working out the details in a way that yields deeper insight.

• pleuviri 30 Jul 2026 21:40 UTC

  3 points

  2

  in reply to: Mateusz Bagiński’s comment on: Im­pre­cise be­liefs: a tiny introduction

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

• pleuviri 30 Jul 2026 20:59 UTC

  LW: 18 AF: 10

  0

  AF

  on: Im­pre­cise be­liefs: a tiny introduction

  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 (e.g., a visual calculus that yields insight into key facts of information theory and algorithms in machine learning). Further generalization is definitely possible (c.f., Ch3 of Halpern’s Reasoning about Uncertainty), but, in practice, I see little reason to generalize further than PDGs.

  Why not simply declare that everything of import is a set/​category/​TM? In tension with generality is the desire to expose minimal knobs and buttons to the modeler. A common complaint about probability is that it’s too difficult for an agent to maintain a joint probability distribution about all variables of interest—and the burden of selecting a lower semi-continuous function on distributions (over which variables?) is WAY higher, since there are far more choices to make, and less standard guidance on how to make them. PDG semantics are arguably the natural way to answer these questions and produce a belief starting from probabilistic information and confidences.

  A challenge for @davidad: can you motivate a scenario that really requires stepping outside of the sub-class of beliefs generated by PDGs, and is clearly better-modeled by a different lower semi-continuous function?

  Finally, a couple of quick technical points/​corrections:

  • PDG semantics are convex for many parameter values of interest, but they can also be non-convex, e.g., when qualitative/​structural beliefs are strong (the PDG that simply claims that variables X and Y are independent, for example, has a non-convex set of optima), or when some confidences become negative, as occurs when we model distrust. In these extended parameter regimes, technically there are also cases when the PDG scoring function can be negative—and, while we can avoid these regimes or clip the score to zero to fit into this picture, I think something (related to the principle of maximum entropy) would be lost.

  • Lower semi-continuity was conceptually key in proving some of the theorems, but indeed it’s not explicit.

  • I have been working on a categorical picture of PDGs for some time. While I endorse the characterization of PDGs as forming a monoidal category, some details still need to be worked out. Furthermore, I strongly suspect there’s a beautiful theory lurking here that interacts with Baez and Fritz’s characterization of relative entropy and Leinster’s information loss. If there are any readers interested helping to develop this theory, please don’t hesitate to get in touch!

  What links here?

  • Im­pre­cise be­liefs: a tiny introduction by davidad (29 Jul 2026 22:17 UTC; 92 points)