Measurement-Induced Decision Loss: A feasible-set theory of microbiome intervention under observation loss

Key results

1. ShownFor a scalar affine channel (true category kept with weight 1 − β, otherwise replaced by a reference draw), larger β is a Blackwell garbling of smaller β. Closure is proven for the convexified monotone policy class, so feasible sets nest and MIDL is nonnegative and nondecreasing in β.

   An explicit row-stochastic kernel turns T_β1 into T_β2 for 0 ≤ β1 ≤ β2 ≤ 1 (using R^2 = R). A garbling theorem gives feasible-set inclusion under post-processing closure. A proof shows that a nondecreasing policy composed with the scalar kernel is a mixture of the original policy and constant policies. Delivered benefit over an uninformative channel plus MIDL sums to H_P(I) − H_P(R). · 3.1 Scalar affine observation channel; 3.2 Blackwell ordering and feasible-set…; 3.3 Scope of the Blackwell ordering

2. ShownIn the binary-state model, the host-optimal monotone policy switches from signal-dependent to a constant rule at a closed-form threshold βc(α, η), where α is the high-state prior and η is the aggressive tier's normalized burden. MIDL grows linearly up to βc and is flat beyond it.

   The feasible set is the triangle conv{A, Bβ, C}: constant policies A and C stay fixed while the observation-dependent vertex Bβ moves affinely. MIDL is piecewise linear, concave, nonnegative and nondecreasing in β, with slope D(α, η)/2 before the switch and at most one interior kink. α = 1/2 gives βc = 2 min{η, 1 − η}; α = η gives βc = 1. Perfect observation yields host utility α(1 − η). · 4.1 Posterior policy regimes; 4.2 Feasible-set geometry and…; Measurement-induced decision-loss landscape; S1.2 Proof of feasible-set geometry for K=2

3. ShownWith K ordered states, the feasible set under randomized monotone policies is a compact convex polytope spanned by finitely many deterministic-policy outcomes. Under scalar blur, the host-benefit frontier is convex piecewise linear and MIDL is concave piecewise linear in β.

   The proof counts nondecreasing policies as multisets of size K drawn from K values (combinations with repetition) and shows the feasible set is the convex hull of their outcomes. Each outcome is affine in β, so the frontier is an upper envelope of finitely many lines. Synthetic K = 5 (prior 0.12, 0.20, 0.31, 0.23, 0.14) and K = 3, 5, 6 systems show set contraction and MIDL kinks at policy switches. · 5 K-state feasible-set theory; S2.2 Finite-state illustrations

4. ShownHeterogeneous information loss breaks simple orderings. Componentwise larger state-specific loss need not be a Blackwell garbling, and blurring the weighted-mean stool composition once differs from the heterogeneous multi-segment aggregate by a weighted blur-composition covariance.

   For K = 3 with uniform r, the kernel needed to garble one channel into its componentwise-blurrier partner has a negative entry, and so does the reverse kernel. The aggregation error equals a weighted covariance of segment βs and compositions zs, is zero iff that term vanishes (e.g., equal compositions or equal βs) and has a Cauchy-Schwarz norm bound. Illustrated with 4 segments, 4 guilds and βeff = 0.32. · 6 State-specific information loss; 7 Multi-segment stool aggregation; S1.1 State-specific information loss and…; S1.3 Proofs for the multi-segment aggregation…

5. ShownAn expected ecological-margin floor preserves convexity, and both the unconstrained and floor-constrained host-benefit frontiers weakly improve with information. Yet the difference between them (the ecological-screening gap) can shrink and then widen as information loss grows.

   Binary example with α = 0.60, η = 0.30, d = 1, τ = 0.228: Gτ(0) = 0.1596, Gτ(4/9) ≈ 0.06086, Gτ(0.55) ≈ 0.06782. A constructed pair of nested sets gives gaps of 0 and 0.45. A two-guild generalized Lotka-Volterra (gLV) model, used as motivation, moves from stable coexistence to a stable boundary state via a transcritical bifurcation at ucrit ≈ 0.281553. · 8.1 Constraints on attainable outcomes; 8.2 Ecological reserve under community dynamics; S2.4.1 Two-guild dynamical model