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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 | import OAI.Probability.FreeIsing.Gaussian.FiniteGaussianPlanted
import OAI.Probability.FreeIsing.Posterior.Fields
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped NNReal ENNReal ProbabilityTheory
namespace Problem367
theorem gaussianPi_map_sqrt {I : Type*} [Fintype I]
(t : ℝ) (ht : 0 ≤ t) :
(Measure.pi (fun _ : I => gaussianReal 0 1)).map
(fun x i => Real.sqrt t * x i) =
Measure.pi (fun _ : I => gaussianReal 0 ⟨t, ht⟩) := by
rw [Measure.pi_map_pi (μ := fun _ : I => gaussianReal 0 1)
(f := fun _ (x : ℝ) => Real.sqrt t * x) (fun _ => by fun_prop)]
congr 1
funext i
rw [gaussianReal_map_const_mul]
congr 1
· simp
· apply NNReal.coe_injective
change (Real.sqrt t) ^ 2 * 1 = t
rw [mul_one, Real.sq_sqrt ht]
theorem finiteStandardGaussianPlanted_joint_law
{S I : Type*} [Fintype S] [MeasurableSpace S] [MeasurableSingletonClass S]
[Fintype I] (μ : Measure S) [IsProbabilityMeasure μ]
(t : ℝ) (ht : 0 ≤ t) (spin : S → I → ℝ) :
(μ.prod (Measure.pi (fun _ : I => gaussianReal 0 1))).map
(fun z : S × (I → ℝ) =>
(z.1, fun i => t * spin z.1 i + Real.sqrt t * z.2 i)) =
μ ⊗ₘ finiteGaussianObservationKernel ⟨t, ht⟩ spin := by
have : IsMarkovKernel (finiteGaussianObservationKernel ⟨t, ht⟩ spin) :=
finiteGaussianObservationKernel_isMarkov ⟨t, ht⟩ spin
apply map_planted_eq_compProd μ (Measure.pi (fun _ : I => gaussianReal 0 1))
(finiteGaussianObservationKernel ⟨t, ht⟩ spin)
(fun s x i => t * spin s i + Real.sqrt t * x i)
· exact Measurable.of_eval (fun index =>
(measurable_const.mul ((measurable_of_finite (fun state => spin state index)).comp
measurable_fst)).add (measurable_const.mul
((measurable_pi_apply index).comp measurable_snd)))
· intro s
have hscale : Measurable (fun x : I → ℝ => fun i => Real.sqrt t * x i) := by
fun_prop
have hshift : Measurable (fun x : I → ℝ => fun i => t * spin s i + x i) := by
fun_prop
rw [show (fun x : I → ℝ => fun i => t * spin s i + Real.sqrt t * x i) =
(fun x : I → ℝ => fun i => t * spin s i + x i) ∘
(fun x : I → ℝ => fun i => Real.sqrt t * x i) from rfl,
← Measure.map_map hshift hscale, gaussianPi_map_sqrt t ht]
exact gaussianPi_map_signal ⟨t, ht⟩ (spin s)
namespace FinitePosterior
theorem exponential_posterior_eq_fieldMean
{S I : Type*} [Fintype S] [Nonempty S] [Fintype I]
[MeasurableSpace S] (μ : Measure S) (energy : S → ℝ)
(hprior : ∀ s, μ.real {s} = Real.exp (energy s) / ∑ a, Real.exp (energy a))
(spin : S → I → ℝ) (q : S → ℝ) (x : I → ℝ) :
(∑ s, μ.real {s} * Real.exp (∑ i, spin s i * x i) * q s) /
(∑ s, μ.real {s} * Real.exp (∑ i, spin s i * x i)) =
fieldMean energy spin q x := by
let Z := ∑ s, Real.exp (energy s)
have hZ : Z ≠ 0 := ne_of_gt (Finset.sum_pos
(fun s _ => Real.exp_pos (energy s)) Finset.univ_nonempty)
have hw (s : S) :
μ.real {s} * Real.exp (∑ i, spin s i * x i) =
Real.exp (fieldEnergy energy spin x s) / Z := by
rw [hprior s]
have hsum : (∑ i, spin s i * x i) = ∑ i, x i * spin s i := by
apply Finset.sum_congr rfl
intro i _
ring
rw [hsum]
simp only [fieldEnergy, Real.exp_add, Z]
ring
simp_rw [hw, div_mul_eq_mul_div, ← Finset.sum_div]
rw [div_div_div_cancel_right₀ hZ]
unfold fieldMean
congr 1
apply Finset.sum_congr rfl
intro s _
ring
end FinitePosterior
theorem finiteStandardGaussianPlanted_condExp_fieldMean
{S I : Type*} [Fintype S] [MeasurableSpace S] [MeasurableSingletonClass S]
[StandardBorelSpace S] [Nonempty S] [Fintype I]
(μ : Measure S) [IsProbabilityMeasure μ] (energy : S → ℝ)
(hprior : ∀ s, μ.real {s} = Real.exp (energy s) / ∑ a, Real.exp (energy a))
(t : ℝ) (ht : 0 ≤ t) (spin : S → I → ℝ)
(hspin : ∀ s i, spin s i ^ 2 = 1) (q : S → ℝ) :
let ρ := μ.prod (Measure.pi (fun _ : I => gaussianReal 0 1))
let Y : S × (I → ℝ) → I → ℝ :=
fun z i => t * spin z.1 i + Real.sqrt t * z.2 i
ρ[fun z => q z.1 | (inferInstance : MeasurableSpace (I → ℝ)).comap Y] =ᵐ[ρ]
fun z => FinitePosterior.fieldMean energy spin q (Y z) := by
dsimp only
have : IsMarkovKernel (finiteGaussianObservationKernel ⟨t, ht⟩ spin) :=
finiteGaussianObservationKernel_isMarkov ⟨t, ht⟩ spin
apply FiniteBayes.condExp_state_eq_of_posterior_integral
(μ.prod (Measure.pi (fun _ : I => gaussianReal 0 1))) μ
(finiteGaussianObservationKernel ⟨t, ht⟩ spin) Prod.fst
(fun z i => t * spin z.1 i + Real.sqrt t * z.2 i) measurable_fst
(by exact Measurable.of_eval (fun index =>
(measurable_const.mul ((measurable_of_finite (fun state => spin state index)).comp
measurable_fst)).add (measurable_const.mul
((measurable_pi_apply index).comp measurable_snd))))
(finiteStandardGaussianPlanted_joint_law μ t ht spin) q
filter_upwards [finiteGaussianObservation_posterior_mean μ ⟨t, ht⟩ spin hspin q]
with x hx
exact hx.trans (FinitePosterior.exponential_posterior_eq_fieldMean
μ energy hprior spin q x)
theorem finiteStandardGaussianPlanted_integral_test_fieldMean
{S I : Type*} [Fintype S] [MeasurableSpace S] [MeasurableSingletonClass S]
[StandardBorelSpace S] [Nonempty S] [Fintype I]
(μ : Measure S) [IsProbabilityMeasure μ] (energy : S → ℝ)
(hprior : ∀ s, μ.real {s} = Real.exp (energy s) / ∑ a, Real.exp (energy a))
(t : ℝ) (ht : 0 ≤ t) (spin : S → I → ℝ)
(hspin : ∀ s i, spin s i ^ 2 = 1) (q : S → ℝ)
(h : (I → ℝ) → ℝ) (hh : Measurable h) (C : ℝ)
(hbound : ∀ x, ‖h x‖ ≤ C) :
let ρ := μ.prod (Measure.pi (fun _ : I => gaussianReal 0 1))
let Y : S × (I → ℝ) → I → ℝ :=
fun z i => t * spin z.1 i + Real.sqrt t * z.2 i
(∫ z, h (Y z) * q z.1 ∂ρ) =
∫ z, h (Y z) * FinitePosterior.fieldMean energy spin q (Y z) ∂ρ := by
let ρ := μ.prod (Measure.pi (fun _ : I => gaussianReal 0 1))
let Y : S × (I → ℝ) → I → ℝ :=
fun z i => t * spin z.1 i + Real.sqrt t * z.2 i
change (∫ z, h (Y z) * q z.1 ∂ρ) = _
have hY : Measurable Y := by
exact Measurable.of_eval (fun index =>
(measurable_const.mul ((measurable_of_finite (fun state => spin state index)).comp
measurable_fst)).add (measurable_const.mul
((measurable_pi_apply index).comp measurable_snd)))
have hm : (inferInstance : MeasurableSpace (I → ℝ)).comap Y ≤
(inferInstance : MeasurableSpace (S × (I → ℝ))) := hY.comap_le
have htest : StronglyMeasurable[(inferInstance : MeasurableSpace (I → ℝ)).comap Y]
(fun z => h (Y z)) :=
(hh.comp (comap_measurable Y)).stronglyMeasurable
have hq : Integrable (fun z : S × (I → ℝ) => q z.1) ρ := by
apply (integrable_const (∑ s, |q s|)).mono'
· exact ((measurable_of_finite q).comp measurable_fst).aestronglyMeasurable
· filter_upwards [] with z
rw [Real.norm_eq_abs]
exact Finset.single_le_sum (fun s _ => abs_nonneg (q s)) (Finset.mem_univ z.1)
have hpull := condExp_stronglyMeasurable_mul_of_bound hm htest hq C
(Filter.Eventually.of_forall (fun z => hbound (Y z)))
have hposterior := finiteStandardGaussianPlanted_condExp_fieldMean
μ energy hprior t ht spin hspin q
calc
(∫ z, h (Y z) * q z.1 ∂ρ) =
∫ z, ρ[(fun z => h (Y z)) * (fun z => q z.1) | (inferInstance : MeasurableSpace (I → ℝ)).comap Y] z ∂ρ :=
(integral_condExp hm).symm
_ = ∫ z, h (Y z) * ρ[(fun z => q z.1) | (inferInstance : MeasurableSpace (I → ℝ)).comap Y] z ∂ρ :=
integral_congr_ae hpull
_ = ∫ z, h (Y z) * FinitePosterior.fieldMean energy spin q (Y z) ∂ρ := by
apply integral_congr_ae
filter_upwards [hposterior] with z hz
rw [hz]
end Problem367
end
end OAI
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