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b317718 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 | import OAI.Probability.GammaPassage.FiniteTails
import OAI.Probability.FirstPassage.PathWitnesses
namespace OAI
namespace GammaFPP
open MeasureTheory ProbabilityTheory PlanarFPP Finset
open scoped ENNReal
lemma trace_edge_length (p : Vertex) (ds : List Step) :
((trace p ds).map traversalEdge).length = ds.length := by
induction ds generalizing p with
| nil => rfl
| cons d ds ih =>
simpa only [trace, List.map_cons, List.length_cons] using
(congrArg Nat.succ (ih (p + d.delta)))
lemma pathCost_trace_edge_sum (w : Edge → ℝ) (p : Vertex) (ds : List Step) :
pathCost w p ds = (((trace p ds).map traversalEdge).map w).sum := by
induction ds generalizing p with
| nil => rfl
| cons d ds ih => simp only [pathCost, trace, List.map_cons, List.sum_cons,
traversalEdge, ih]
/-- Only a trail's traversal sum can be replaced by its distinct-edge set sum. -/
lemma pathCost_eq_distinct_edgeSum (w : Edge → ℝ) (p : Vertex) (ds : List Step)
(hn : ((trace p ds).map traversalEdge).Nodup) :
pathCost w p ds = ∑ e ∈ ((trace p ds).map traversalEdge).toFinset, w e := by
classical
rw [pathCost_trace_edge_sum]
exact (List.sum_toFinset w hn).symm
variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
variable {τ : Edge → Ω → ℝ} {shape rate : ℝ}
/-- Genuine Gamma lower tail for any fixed trail, with arbitrary real threshold.
The positive parameter s uses the native MGF at theta=-s. -/
theorem GammaEnvironment.cheap_path_probability
(h : GammaEnvironment shape rate P τ) (hs : 0 < shape) (hr : 0 < rate)
(s : ℝ) (hsp : 0 < s) (p : Vertex) (ds : List Step)
(hn : ((trace p ds).map traversalEdge).Nodup) (t : ℝ) :
P {ω | pathCost (fun e => τ e ω) p ds ≤ t} ≤
(ENNReal.ofReal ((rate / (rate + s)) ^ shape)) ^ ds.length /
ENNReal.ofReal (Real.exp (-s * t)) := by
classical
have hc : ((trace p ds).map traversalEdge).toFinset.card = ds.length := by
rw [List.toFinset_card_of_nodup hn, trace_edge_length]
have he (ω : Ω) : edgeSum ((trace p ds).map traversalEdge).toFinset τ ω =
pathCost (fun e => τ e ω) p ds := by
exact (pathCost_eq_distinct_edgeSum (fun e => τ e ω) p ds hn).symm
have hb := h.lower_tail_edgeSum hs hr (neg_neg_of_pos hsp)
((trace p ds).map traversalEdge).toFinset t
simpa only [he, hc, sub_neg_eq_add] using hb
/-- Conversion keeps the upper bound finite before passing to real probability. -/
theorem GammaEnvironment.cheap_path_probability_real
(h : GammaEnvironment shape rate P τ) (hs : 0 < shape) (hr : 0 < rate)
(s : ℝ) (hsp : 0 < s) (p : Vertex) (ds : List Step)
(hn : ((trace p ds).map traversalEdge).Nodup) (t : ℝ) :
P.real {ω | pathCost (fun e => τ e ω) p ds ≤ t} ≤
((rate / (rate + s)) ^ shape) ^ ds.length * Real.exp (s * t) := by
have hbase : 0 ≤ (rate / (rate + s)) ^ shape :=
(Real.rpow_pos_of_pos (div_pos hr (add_pos hr hsp)) shape).le
have hden : ENNReal.ofReal (Real.exp (-s * t)) ≠ 0 :=
ENNReal.ofReal_ne_zero_iff.mpr (Real.exp_pos _)
have hfin : (ENNReal.ofReal ((rate / (rate + s)) ^ shape)) ^ ds.length /
ENNReal.ofReal (Real.exp (-s * t)) ≠ ⊤ :=
ENNReal.div_ne_top (ENNReal.pow_ne_top ENNReal.ofReal_ne_top) hden
have hb := ENNReal.toReal_mono hfin (h.cheap_path_probability hs hr s hsp p ds hn t)
simp only [ENNReal.toReal_div, ENNReal.toReal_pow,
ENNReal.toReal_ofReal hbase, ENNReal.toReal_ofReal (Real.exp_pos _).le] at hb
change P.real _ ≤ _ at hb
rw [show -s * t = -(s * t) by ring, Real.exp_neg, div_inv_eq_mul] at hb
exact hb
/-- The probability that a finite family contains a cheap trail is bounded by
its cardinality times the single-trail estimate. -/
theorem GammaEnvironment.cheap_path_family_probability
(h : GammaEnvironment shape rate P τ) (hs : 0 < shape) (hr : 0 < rate)
(s : ℝ) (hsp : 0 < s) (p : Vertex) (F : Finset (List Step)) (n : ℕ)
(hF : ∀ ds ∈ F, ((trace p ds).map traversalEdge).Nodup ∧ ds.length = n)
(t : ℝ) :
P {ω | ∃ ds ∈ F, pathCost (fun e => τ e ω) p ds ≤ t} ≤
(F.card : ℝ≥0∞) * ((ENNReal.ofReal ((rate / (rate + s)) ^ shape)) ^ n /
ENNReal.ofReal (Real.exp (-s * t))) := by
classical
have he : {ω | ∃ ds ∈ F, pathCost (fun e => τ e ω) p ds ≤ t} =
⋃ ds ∈ F, {ω | pathCost (fun e => τ e ω) p ds ≤ t} := by
ext ω
simp
rw [he]
refine (measure_biUnion_finset_le _ _).trans ?_
calc
_ ≤ ∑ _ds ∈ F, ((ENNReal.ofReal ((rate / (rate + s)) ^ shape)) ^ n /
ENNReal.ofReal (Real.exp (-s * t))) := by
apply Finset.sum_le_sum
intro ds hds
simpa only [(hF ds hds).2] using
h.cheap_path_probability hs hr s hsp p ds (hF ds hds).1 t
_ = _ := by simp
end GammaFPP
end OAI
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