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170 lines
4.5 KiB
Markdown
170 lines
4.5 KiB
Markdown
# PURE MATH FORMULA: Fisher Distance on Δ₇
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## Zero English inside formulas. Each equality justified. Verifiable numerically.
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---
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## THE CHAIN
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**Given:** p, q ∈ Δ₇ (probability simplex, 8 dimensions)
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**Step 0 — Chentsov's metric:**
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g_p(u,v) = Σᵢ₌₁⁸ (uᵢ vᵢ / pᵢ)
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**Justification:** Chentsov 1972, Amari 1985. Unique metric respecting sufficient statistics.
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---
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**Step 1 — The √p embedding:**
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φ : Δ₇ → S⁷, φ(p) = (√p₁, √p₂, ..., √p₈)
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**Lemma:** ‖φ(p)‖₂ = 1
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‖φ(p)‖₂² = Σᵢ₌₁⁸ (√pᵢ)² = Σᵢ₌₁⁸ pᵢ = 1
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**Justification:** p ∈ Δ₇ ⇒ Σpᵢ = 1 by definition.
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---
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**Step 2 — Pullback of round metric:**
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(φ* g_{S⁷})_p(u,v) = ¼ · g_p(u,v)
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**Proof sketch:**
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Let c(t) be a curve in Δ₇, c(0) = p, ċ(0) = v.
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γ(t) = φ(c(t)) = (√c₁(t), ..., √c₈(t))
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γ̇ᵢ(0) = vᵢ / (2√pᵢ)
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Round metric on S⁷:
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⟨γ̇, γ̇⟩_{S⁷} = Σᵢ γ̇ᵢ² = Σᵢ vᵢ² / (4pᵢ) = ¼ · Σᵢ vᵢ²/pᵢ = ¼ · g_p(v,v)
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**Justification:** Chain rule + direct computation.
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---
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**Step 3 — Geodesics are great circles:**
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S⁷ has round metric ⇒ geodesics are great circles.
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Great-circle distance:
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d_{S⁷}(a,b) = arccos(⟨a,b⟩)
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**Justification:** Standard Riemannian geometry of the sphere.
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---
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**Step 4 — Inner product on S⁷:**
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⟨φ(p), φ(q)⟩ = Σᵢ₌₁⁸ √(pᵢ qᵢ)
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**Justification:** Definition of φ + Euclidean inner product.
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---
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**Step 5 — Fisher distance (THE FORMULA):**
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d_F(p,q) = 2 · d_{S⁷}(φ(p), φ(q))
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= 2 · arccos(⟨φ(p), φ(q)⟩)
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= 2 · arccos( Σᵢ₌₁⁸ √(pᵢ qᵢ) )
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**Justification:** Steps 2+3+4 combined. The factor 2 comes from Step 2 (g = 4·φ*g_{S⁷}).
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---
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## THE CLOSED-FORM RESULT
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```
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┌─────────────────────────────────────────────────────┐
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│ │
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│ d_F(p,q) = 2 · arccos( Σᵢ₌₁ⁿ √(pᵢ qᵢ) ) │
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│ │
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│ Domain: p, q ∈ Δₙ (any dimension n ≥ 2) │
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│ Range: [0, π] │
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│ Equality: d_F(p,q) = 0 ⟺ p = q │
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│ Max: d_F(p,q) = π when p, q are antipodal │
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│ (e.g., p = (1,0,...,0), q = (0,1,0,...,0)) │
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│ │
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└─────────────────────────────────────────────────────┘
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```
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---
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## VERIFICATION INSTANCE (n=8)
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**Inputs:**
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p = (0.3, 0.1, 0.15, 0.05, 0.2, 0.08, 0.07, 0.05)
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q = (0.2, 0.2, 0.1, 0.1, 0.15, 0.1, 0.1, 0.05)
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**Step A — Compute √(pᵢqᵢ):**
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√(0.3×0.2) = 0.24494897
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√(0.1×0.2) = 0.14142136
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√(0.15×0.1) = 0.12247449
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√(0.05×0.1) = 0.07071068
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√(0.2×0.15) = 0.17320508
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√(0.08×0.1) = 0.08944272
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√(0.07×0.1) = 0.08366600
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√(0.05×0.05) = 0.05000000
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**Step B — Sum:**
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S = 0.97586930
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**Step C — Arccos:**
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arccos(0.97586930) = 0.22012896
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**Step D — Multiply by 2:**
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d_F(p,q) = 2 × 0.22012896 = 0.44025792
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**OUTPUT: d_F(p,q) ≈ 0.440258**
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---
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## PROPERTIES (all verifiable)
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**Symmetry:**
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d_F(p,q) = 2·arccos(Σ√(pᵢqᵢ)) = 2·arccos(Σ√(qᵢpᵢ)) = d_F(q,p) ✓
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**Identity:**
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d_F(p,p) = 2·arccos(Σ√(pᵢpᵢ)) = 2·arccos(Σpᵢ) = 2·arccos(1) = 0 ✓
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**Triangle inequality:**
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d_F(p,q) ≤ d_F(p,r) + d_F(r,q) for all p,q,r ∈ Δ₇
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Proof: Great-circle distance on S⁷ satisfies triangle inequality.
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Pullback by isometry preserves triangle inequality. ✓
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**Bound:**
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0 ≤ d_F(p,q) ≤ π
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Proof: arccos: [-1,1] → [0,π]. The argument Σ√(pᵢqᵢ) ∈ [0,1]
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by Cauchy-Schwarz: (Σ√(pᵢqᵢ))² ≤ (Σpᵢ)(Σqᵢ) = 1. ✓
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---
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## WHY THIS IS THE RIGHT FORMULA
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1. **Chentsov's theorem** says: any metric respecting sufficient statistics
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MUST be the Fisher metric (up to constant).
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2. **The √p embedding** maps Δ₇ → S⁷ isometrically (up to factor 4).
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3. **Geodesics on S⁷** are great circles with known distance formula.
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4. **Pulling back** gives the Fisher distance formula above.
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There is no choice in this formula. It is forced by the geometry of the
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probability simplex combined with Chentsov's uniqueness result.
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