Topos-TheoreticProbability

Two thermometers in adjacent rooms will happily disagree about the doorway they share.

The usual fix is to fit one global model and treat the disagreement as noise, which throws away exactly the structure that says who should win where. This paper keeps every belief local instead: each sensor, region, or time window holds its own distribution, overlaps carry an explicit compatibility price, and descent — the sheaf-theoretic gluing condition — does the work a global model used to do. Bayesian updating becomes “make the locals compatible, then glue.”

• Base space: Choose your cover (e.g., rooms in a house × time windows, or sensor fields of view).
• Local sections (data): On each patch of the cover, store a belief object: a Gaussian, Dirichlet parameters, a credal set, a subjective-logic opinion.
• Restrictions: Maps that push beliefs down to overlaps: marginalize out variables, project coordinates, convert units.
• Compatibility loss: Measure mismatch on overlaps: symmetrized KL divergence KL(P ‖ Q) + KL(Q ‖ P), a Wasserstein distance, or an interval/credal distance.
• Inference = sheaf gluing: Minimize total incompatibility while each patch stays faithful to its own likelihood and prior. In the linear-Gaussian case this is a sheaf-Laplacian least-squares problem; in general it is the optimization analogue of gluing, and the minimizer is the fused belief.
• Bayesian updates: A new observation updates one patch locally; then re-solve the coherence problem, warm-starting from the previous solution.
• Diagnostics: Overlap residuals that survive optimization point at bad sensors, drift, or miscalibration. In the cohomological picture, a nonzero obstruction class is a contradiction no local adjustment can remove.

where Rᵢ#Pᵢ is the pushforward of Pᵢ along the restriction map and P̃ᵢ is the locally updated posterior on patch i. The minimizer is a family Pᵢ* as compatible as the data allow; when every overlap residual vanishes, that family is exactly a global section.

• Transport: Beliefs move between contexts by restriction and pushforward; the cost of a move is your transport metric (Wasserstein or KL).
• Epistemic: Beliefs may be second-order — uncertainty about the parameters themselves. Encode that with richer local data: credal sets, intervals, or hyperpriors.
• Objective: Transport cost across overlaps plus data-fidelity terms; the gluing constraint is what forces the transported beliefs to agree.

# A sketch, not a library: build_cover, converged,
# update_beliefs, etc. are yours to supply.

# Represent the cover and restrictions (graph or poset)
cover = build_cover(rooms, time_windows)

# Pick a belief carrier
# NumPy/SciPy for Gaussians; Dirichlet for discrete
beliefs = {node: GaussianBelief(mu, cov) for node in cover}

# Build overlap list with restriction matrices
overlaps = compute_overlaps(cover)
restrictions = {(i, j): linear_restriction_map(i, j)
                for (i, j) in overlaps}

# Solver: ADMM over overlaps
def sheaf_glue(beliefs, overlaps, restrictions, lambda_reg):
    while not converged(beliefs):
        # Measure disagreement on every overlap
        mismatch = {}
        for (i, j) in overlaps:
            Ri, Rj = restrictions[(i, j)]
            mismatch[(i, j)] = compute_wasserstein(Ri @ beliefs[i],
                                                   Rj @ beliefs[j])
        # Proximal/gradient step toward compatibility
        beliefs = update_beliefs(beliefs, mismatch, lambda_reg)
    return beliefs