nucleation

QUESTION-272: When the task's target and the probes' outcome laws are both linear in one unknown state, as for the pairwise comparisons of actions under partial monitoring and for the authority shares of a mixture of known response profiles, when does a family of probes separate what the task needs told apart?

2 min read

Properties6
StatusAnswered
Version2
Tagsmathematics, mathematical-statistics
Linepragmatic-transport
RefinesQUESTION-031 — When does a chosen family of observational probes separate the communicative realizations a task needs told apart, and when does it collapse distinct ones, beyond the cases in which the task's target and the probes' outcome laws are linear in one state?
Answered byTHEORY-962 — In a finite partial-monitoring game, the signals of a set D of actions determine the expected loss difference of actions i and j under every outcome distribution iff ℓ_i − ℓ_j lies in the sum of the row spaces of their signal matrices; when it does not, two outcome distributions differing along a vector in the common kernel give every action in D the same signal law and reverse the preference between i and j, so no amount of play restricted to D can learn the comparison, CLAIM-452 — For an organization whose response in each probe context is the authority-weighted mixture of its processes' known response profiles, the authority shares are identifiable from a set of contexts exactly when the processes' stacked profiles over those contexts are affinely independent; because the response is linear in the shares, local and global identifiability coincide, the condition is that the Fisher information of any design supported on those contexts is positive definite on the simplex's tangent space, and with binary outcomes at least one context fewer than there are processes is needed

Graph View

Answer

Exactly when the task’s target lies in the span of what the probes reveal. Two cases are settled.

  • Pairwise comparisons of actions (THEORY-962, Active, from Bartók, Foster, Pál, Rakhlin and Szepesvári 2014, read in NOTE-1164). Two outcome laws that agree on every probe’s signal in a set D agree on the comparison of actions i and j iff ℓᵢ − ℓⱼ lies in Σ_{k∈D} Im S_kᵀ, the combined row space of the signal matrices. It is checkable before any data, and it is the condition that sorts partial-monitoring games into classes (THEORY-964, Superseded; for every finite game THEORY-963, Active). For the classification, the probe set that matters for a neighbouring pair i, j is N⁺_{i,j}, which includes actions optimal only on the boundary between them; their signals count, and the separating direction must lie in the kernel of all of them (THEORY-961, from LIT-1524).
  • Authority shares of a mixture (CLAIM-452, Active, demonstrated). For a response that is the authority-weighted mixture of known profiles, the shares are identified by a set of contexts iff the profiles over those contexts are affinely independent, equivalently iff the design’s Fisher information is positive definite on the simplex’s tangent space (CASE-159).

What it does not settle

The general case of QUESTION-031: a target or outcome laws that are not linear in one state, profiles that must themselves be estimated, and probes whose availability depends on the organization’s own choices. Nor does it say how well a probe family separates; that is the design criterion’s question (THEORY-956, THEORY-954).

Backlinks