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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 99 100 101 102 103 104 105 106 | import OAI.Probability.FreeIsing.Posterior.FinitePosterior
namespace OAI
noncomputable section
open scoped BigOperators
namespace Problem367.FinitePosterior
variable {S V : Type*} [Fintype S] [Fintype V]
def fieldEnergy (energy : S → ℝ) (spin : S → V → ℝ) (x : V → ℝ) (s : S) : ℝ :=
energy s + ∑ v, x v * spin s v
def fieldMean (energy : S → ℝ) (spin : S → V → ℝ) (observable : S → ℝ)
(x : V → ℝ) : ℝ :=
(∑ s, observable s * Real.exp (fieldEnergy energy spin x s)) /
(∑ s, Real.exp (fieldEnergy energy spin x s))
def fieldMass (energy : S → ℝ) (spin : S → V → ℝ) (x : V → ℝ) (s : S) : ℝ :=
Real.exp (fieldEnergy energy spin x s) / (∑ r, Real.exp (fieldEnergy energy spin x r))
lemma fieldMean_eq_sum_fieldMass (energy : S → ℝ) (spin : S → V → ℝ)
(observable : S → ℝ) (x : V → ℝ) :
fieldMean energy spin observable x = ∑ s, observable s * fieldMass energy spin x s := by
simp only [fieldMean, fieldMass, ← mul_div_assoc, ← Finset.sum_div]
lemma fieldMean_along_line (energy : S → ℝ) (spin : S → V → ℝ)
(observable : S → ℝ) (x direction : V → ℝ) (t : ℝ) :
fieldMean energy spin observable (fun v => x v + t * direction v) =
mean (fieldEnergy energy spin x) (fun s => ∑ v, direction v * spin s v)
observable t := by
have heq (s : S) :
fieldEnergy energy spin (fun v => x v + t * direction v) s =
fieldEnergy energy spin x s + t * ∑ v, direction v * spin s v := by
simp only [fieldEnergy, add_mul, Finset.sum_add_distrib,
Finset.mul_sum, mul_assoc]
ring
simp only [fieldMean, heq, mean, partition, weight]
lemma mean_fieldEnergy_zero (energy : S → ℝ) (spin : S → V → ℝ)
(signal observable : S → ℝ) (x : V → ℝ) :
mean (fieldEnergy energy spin x) signal observable 0 =
fieldMean energy spin observable x := by
simp only [mean, partition, weight, zero_mul, add_zero, fieldMean]
lemma fieldPartition_pos [Nonempty S] (energy : S → ℝ) (spin : S → V → ℝ)
(x : V → ℝ) : 0 < ∑ s, Real.exp (fieldEnergy energy spin x s) := by
exact Finset.sum_pos (fun s _ => Real.exp_pos _) Finset.univ_nonempty
lemma fieldMass_pos [Nonempty S] (energy : S → ℝ) (spin : S → V → ℝ)
(x : V → ℝ) (s : S) : 0 < fieldMass energy spin x s :=
div_pos (Real.exp_pos _) (fieldPartition_pos energy spin x)
theorem continuous_fieldMean [Nonempty S] (energy : S → ℝ) (spin : S → V → ℝ)
(observable : S → ℝ) : Continuous (fieldMean energy spin observable) := by
unfold fieldMean
apply Continuous.div
· unfold fieldEnergy
fun_prop
· unfold fieldEnergy
fun_prop
· intro x
exact ne_of_gt (fieldPartition_pos energy spin x)
lemma fieldMean_const [Nonempty S] (energy : S → ℝ) (spin : S → V → ℝ)
(x : V → ℝ) (c : ℝ) : fieldMean energy spin (fun _ => c) x = c := by
rw [← mean_fieldEnergy_zero energy spin (fun _ => 0), mean_const]
lemma fieldMean_spin_mem_Ioo [Nonempty S] (energy : S → ℝ) (spin : S → V → ℝ)
(observable : S → ℝ) (x : V → ℝ)
(hs : ∀ s, observable s = -1 ∨ observable s = 1)
(hn : ∃ s, observable s = -1) (hp : ∃ s, observable s = 1) :
fieldMean energy spin observable x ∈ Set.Ioo (-1 : ℝ) 1 := by
rw [← mean_fieldEnergy_zero energy spin (fun _ => 0)]
exact spin_mean_mem_Ioo _ _ _ _ hs hn hp
theorem hasDerivAt_fieldMean [Nonempty S] (energy : S → ℝ) (spin : S → V → ℝ)
(observable : S → ℝ) (x direction : V → ℝ) :
HasDerivAt (fun t => fieldMean energy spin observable (fun v => x v + t * direction v))
(fieldMean energy spin (fun s => observable s * ∑ v, direction v * spin s v) x -
fieldMean energy spin observable x *
fieldMean energy spin (fun s => ∑ v, direction v * spin s v) x) 0 := by
simp_rw [fieldMean_along_line]
simpa only [mean_fieldEnergy_zero] using
hasDerivAt_mean (fieldEnergy energy spin x)
(fun s => ∑ v, direction v * spin s v) observable 0
theorem hasDerivAt_fieldMean_coordinate [Nonempty S] [DecidableEq V]
(energy : S → ℝ) (spin : S → V → ℝ) (observable : S → ℝ)
(x : V → ℝ) (u : V) :
HasDerivAt
(fun t => fieldMean energy spin observable (fun v => x v + if v = u then t else 0))
(fieldMean energy spin (fun s => observable s * spin s u) x -
fieldMean energy spin observable x * fieldMean energy spin (fun s => spin s u) x) 0 := by
have h := hasDerivAt_fieldMean energy spin observable x (fun v => if v = u then 1 else 0)
simpa only [mul_ite, mul_one, mul_zero, ite_mul, one_mul, zero_mul,
Finset.sum_ite_eq', Finset.mem_univ, ite_true] using h
end Problem367.FinitePosterior
end
end OAI
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