Manuscript En10-B@1 · En10-B.tex · section · a1105bad

Predeclared model families

Competing explanations and complete-record discrimination › Predeclared model families · lines 951–997

En10-B@1#eq:compatibility

Let $H = {0, \dots, L}$ be a finite predeclared list of competing families. Family $j$ has a compact parameter domain $Θ_{j}$ and a continuous complete record law

p^{j}_{λ} \in Δ (R_{+}), λ \in Θ_{j} .

Family $0$ may be the derived family (eq:likelihood); other families may have different source templates or finite generators. Their likelihoods and allowances $e_{j}$ must be supplied, not fitted without restriction after the validation records are inspected.

For example, replacing the two source templates by specified density matrices $σ_{j, 0}, σ_{j, 1}$ gives the finite-model likelihood

p^{j}_{r} (θ, η) = Tr [P_{r} Φ^{j}_{η} (θ σ_{j, 0} + (1 - θ) σ_{j, 1})],

provided the corresponding channel $Φ^{j}_{η}$ has been constructed. Merely writing this expression does not derive the alternative channel.

Let

p_{r} = \frac{1}{n} i \sum 1_{{r_{i} = r}}, D = ∣ R_{+} ∣ = 5, c_{n} (α) = \frac{lo g ( 2 D / α )}{2 n} .

Define the complete-record compatibility set

Ω_{n} = {(j, λ) : p - p^{j}_{λ}_{\infty} \leq c_{n} (α) + e_{j}} .

All counts, including $⊥$ , enter this set.

Displayed formulas in this passage

En10-B@1#eq:compatibility · line 987
$Ω_{n} = {(j, λ) : p - p^{j}_{λ}_{\infty} \leq c_{n} (α) + e_{j}} .$
unlabeled · line 956
$p^{j}_{λ} \in Δ (R_{+}), λ \in Θ_{j} .$
unlabeled · line 968
$p^{j}_{r} (θ, η) = Tr [P_{r} Φ^{j}_{η} (θ σ_{j, 0} + (1 - θ) σ_{j, 1})],$
unlabeled · line 981
$p_{r} = \frac{1}{n} i \sum 1_{{r_{i} = r}}, D = ∣ R_{+} ∣ = 5, c_{n} (α) = \frac{lo g ( 2 D / α )}{2 n} .$

Predeclared model families

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