Threshold equivalence, p.ec3 (corrected: $r_i^\delta = 0$ case)

For real $A\ge0$, $B$, $C$: if the thresholds of $(A,B,C)$ exist, the lower one is at most the upper one; and for every $p\in[0,1]$,

if and only if the thresholds exist and $p$ lies between them. In particular, under the two weak inequalities of Assumption 4, for every state $\mathbf s$ and person $i$: if the thresholds exist then $\mathbb P_i^{\min}(\mathbf s)\le\mathbb P_i^{\max}(\mathbf s)$, and for every $p\in[0,1]$,

The thresholds are thus a property of the quadratic and not of the single value $\mathbb P_i(\mathbf s)$.

Differences from the source:

• correction: the thresholds divide by $2r_i^\delta$, and Assumption 4 allows $r_i^\delta=0$ → the printed formulas for $r_i^\delta>0$, and for $r_i^\delta=0$ the solution of the linear inequality $Bp-C\ge0$; reason: counterexample satisfying every assumption of the source: $\mathcal I=\{1\}$, $\mathbb P(G_1=0)=\mathbb P(G_1=1)=1/2$, $r^0(0)=1$, $r^1(1)=5$, $r^1(0)=r^0(1)=0$, $\bar r^0(1)=1$, $\bar r^1(1)=0$, $c^s=c^f=1$ has $r^\delta=0$ and the printed formulas are undefined, while testing yields $3/2$ against $1/2$ for stopping
• reading: a set of genes $\mathcal J$ in the model of §2; Assumption 4, Assumption 5 and Theorem 1's proof are single-gene → one gene: $g_i\in\{0,1,2\}$; reason: Theorem 1 compares $\mathbb P_i(\mathbf s)$ with two real thresholds, which needs a scalar; its proof, Assumption 4 and §4.3 are single-gene (pp.15, ec2), and no multi-gene threshold statement is printed
• generalization: the Mendelian prior (5) → an arbitrary mass function on $\{0,1,2\}^{\mathcal I}$; reason: the argument does not use the pedigree and p.11 says priors incorporate all prior information; (5) is the special case given by the theorem eq_5_mendelian_prior
• generalization: Assumption 4 in full, with Assumptions 1, 2, 3 and stationary, positive, bounded rewards → only $r_i^1(0)\le\bar r_i^1(1)$ and $\bar r_i^0(1)\le r_i^0(0)$; reason: the dropped conditions are not used, and every instance of the source satisfies the two kept inequalities
• generalization: "inequality (EC.6) is equivalent to", a statement at the value $\mathbb P_i(\mathbf s)$ → the equivalence for every $p\in[0,1]$; reason: it specializes to $p=\mathbb P_i(\mathbf s)$, and without it "between the thresholds" says nothing beyond (EC.3)
• encoding: a state as a partition $\{s_0,s_1,s_2\}$ of the tested individuals, state space $\mathcal S=\{\mathbf s:\exists g\in\mathcal G(\mathbf s)\}$ → State I := I → Option (Fin 3), with Feasible meaning $\mathbb P(G\in\mathcal G(\mathbf s))>0$; reason: same objects when $\mathcal G$ is the support of the prior; the family is a Fintype, possibly empty
• encoding: $\alpha_i(\mathbf s)$ is a ratio with denominator $\mathbb P_i(\mathbf s)$ → $\alpha_i(\mathbf s)=0$ when the denominator is $0$; reason: the page leaves this case undefined; every use of $\alpha_i$ is then multiplied by $0$
• encoding: $V^*$ as the maximum of (3), an expectation over sample paths with absorbing state $\Delta$ and action $C$, undefined at infeasible states → Vstar: the maximum over the finite type Policy I of the posterior average of a rolled-out return, with no $\Delta$ and no $C$; $V^*=0$ at infeasible states; reason: given $G=g$ and the rule the path is determined, so the two expectations agree; in $\Phi_i$ and $\mathrm{gap}_i$ the infeasible children enter only with coefficient $0$, and in the thresholds the value $V^*(\mathbf s_0\cup\{i\})=0$ at $\mathbb P_i(\mathbf s)=1$ fills a value the page leaves undefined