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namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped ENNReal NNReal
namespace Problem367
theorem map_withDensity_comp_observation
{X Y : Type*} [MeasurableSpace X] [MeasurableSpace Y]
(mu : Measure X) (F : X → Y) (hF : Measurable F)
(L : Y → ℝ≥0∞) (hL : Measurable L) :
Measure.map F (mu.withDensity (L ∘ F)) =
(Measure.map F mu).withDensity L := by
ext s hs
rw [Measure.map_apply hF hs, withDensity_apply _ (hs.preimage hF),
withDensity_apply _ hs, Measure.restrict_map hF hs, lintegral_map hL hF]
rfl
theorem gaussianPi_variable_signal_withDensity {I : Type*} [Fintype I]
(t : I → ℝ≥0) (s : I → ℝ) :
Measure.pi (fun i => gaussianReal ((t i : ℝ) * s i) (t i)) =
(Measure.pi (fun i => gaussianReal 0 (t i))).withDensity
(fun x : I → ℝ => ENNReal.ofReal
(Real.exp (∑ i : I, (s i * x i - (t i : ℝ) * s i ^ 2 / 2)))) := by
classical
calc
Measure.pi (fun i => gaussianReal ((t i : ℝ) * s i) (t i)) =
(Measure.pi (fun i => gaussianReal 0 (t i))).withDensity
(fun x => ∏ i, ENNReal.ofReal (gaussianSignalLikelihood (t i) (s i) (x i))) := by
exact pi_withDensity_of_probability
(fun i => gaussianReal 0 (t i))
(fun i => gaussianReal ((t i : ℝ) * s i) (t i))
(fun i x => ENNReal.ofReal (gaussianSignalLikelihood (t i) (s i) x))
(fun i => by unfold gaussianSignalLikelihood; fun_prop)
(fun i => gaussianReal_signal_withDensity (t i) (s i))
_ = _ := by
apply withDensity_congr_ae
filter_upwards [] with x
rw [← ENNReal.ofReal_prod_of_nonneg (fun i hi => by
unfold gaussianSignalLikelihood; positivity)]
simp only [gaussianSignalLikelihood, ← Real.exp_sum]
theorem gaussianIncrements_signal_withDensity {I : Type*} [Fintype I]
(t : I → ℝ≥0) (s : ℝ) :
Measure.pi (fun i => gaussianReal ((t i : ℝ) * s) (t i)) =
(Measure.pi (fun i => gaussianReal 0 (t i))).withDensity
(fun x => ENNReal.ofReal
(gaussianSignalLikelihood (∑ i, t i) s (∑ i, x i))) := by
classical
rw [gaussianPi_variable_signal_withDensity]
congr 1
funext x
congr 1
unfold gaussianSignalLikelihood
congr 1
simp only [Finset.sum_sub_distrib, ← Finset.mul_sum, ← Finset.sum_div,
← Finset.sum_mul, NNReal.coe_sum]
def cumulativeObservation {n : ℕ} (x : Fin n → ℝ) (k : Fin (n + 1)) : ℝ :=
∑ i : Fin n, if i.val < k.val then x i else 0
theorem measurable_cumulativeObservation (n : ℕ) :
Measurable (@cumulativeObservation n) := by
apply Measurable.of_eval
intro k
apply Finset.measurable_sum
intro i hi
by_cases h : i.val < k.val
· simpa only [ite_eq_left h] using measurable_pi_apply i
· simp only [ite_eq_right h]
exact measurable_const
@[simp] theorem cumulativeObservation_last {n : ℕ} (x : Fin n → ℝ) :
cumulativeObservation x (Fin.last n) = ∑ i, x i := by
simp [cumulativeObservation, Fin.is_lt]
@[simp] theorem cumulativeObservation_zero {n : ℕ} (x : Fin n → ℝ) :
cumulativeObservation x 0 = 0 := by
simp [cumulativeObservation]
theorem gaussianSkeleton_signal_withDensity {n : ℕ}
(t : Fin n → ℝ≥0) (s : ℝ) :
Measure.map cumulativeObservation
(Measure.pi (fun i => gaussianReal ((t i : ℝ) * s) (t i))) =
(Measure.map cumulativeObservation
(Measure.pi (fun i => gaussianReal 0 (t i)))).withDensity
(fun y => ENNReal.ofReal
(gaussianSignalLikelihood (∑ i, t i) s (y (Fin.last n)))) := by
rw [gaussianIncrements_signal_withDensity]
have h := map_withDensity_comp_observation
(Measure.pi (fun i : Fin n => gaussianReal 0 (t i)))
(@cumulativeObservation n) (measurable_cumulativeObservation n)
(fun y : Fin (n + 1) → ℝ => ENNReal.ofReal
(gaussianSignalLikelihood (∑ i, t i) s (y (Fin.last n))))
(by unfold gaussianSignalLikelihood; fun_prop)
simpa only [Function.comp_def, cumulativeObservation_last] using h
def cumulativeVectorObservation {V : Type*} {n : ℕ}
(x : V × Fin n → ℝ) (v : V) : Fin (n + 1) → ℝ :=
cumulativeObservation (fun i => x (v, i))
theorem measurable_cumulativeVectorObservation (V : Type*) (n : ℕ) :
Measurable (@cumulativeVectorObservation V n) := by
apply Measurable.of_eval
intro v
exact (measurable_cumulativeObservation n).comp (by fun_prop)
@[simp] theorem cumulativeVectorObservation_last {V : Type*} {n : ℕ}
(x : V × Fin n → ℝ) (v : V) :
cumulativeVectorObservation x v (Fin.last n) = ∑ i, x (v, i) :=
cumulativeObservation_last _
theorem gaussianChannelIncrement_exponent {V : Type*} [Fintype V] {n : ℕ}
(t : Fin n → ℝ≥0) (s : V → ℝ) (x : V × Fin n → ℝ) :
(∑ p : V × Fin n, (s p.1 * x p - (t p.2 : ℝ) * s p.1 ^ 2 / 2)) =
(∑ v, s v * cumulativeVectorObservation x v (Fin.last n)) -
((∑ i, t i : ℝ≥0) : ℝ) / 2 * ∑ v, s v ^ 2 := by
classical
rw [Fintype.sum_prod_type]
simp only [cumulativeVectorObservation_last, Finset.sum_sub_distrib,
← Finset.mul_sum, ← Finset.sum_div, ← Finset.sum_mul, NNReal.coe_sum]
ring
theorem gaussianVectorSkeleton_signal_withDensity
{V : Type*} [Fintype V] {n : ℕ} (t : Fin n → ℝ≥0) (s : V → ℝ) :
Measure.map cumulativeVectorObservation
(Measure.pi (fun p : V × Fin n => gaussianReal ((t p.2 : ℝ) * s p.1) (t p.2))) =
(Measure.map cumulativeVectorObservation
(Measure.pi (fun p : V × Fin n => gaussianReal 0 (t p.2)))).withDensity
(fun y => ENNReal.ofReal
(gaussianVectorLikelihood (∑ i, t i) s (fun v => y v (Fin.last n)))) := by
rw [gaussianPi_variable_signal_withDensity]
have h := map_withDensity_comp_observation
(Measure.pi (fun p : V × Fin n => gaussianReal 0 (t p.2)))
(@cumulativeVectorObservation V n) (measurable_cumulativeVectorObservation V n)
(fun y : V → Fin (n + 1) → ℝ => ENNReal.ofReal
(gaussianVectorLikelihood (∑ i, t i) s (fun v => y v (Fin.last n))))
(by unfold gaussianVectorLikelihood; fun_prop)
simpa only [Function.comp_def, gaussianVectorLikelihood,
gaussianChannelIncrement_exponent] using h
theorem gaussianVectorSkeleton_spin_withDensity
{V : Type*} [Fintype V] {n : ℕ} (t : Fin n → ℝ≥0) (s : V → ℝ)
(hs : ∀ v, s v ^ 2 = 1) :
Measure.map cumulativeVectorObservation
(Measure.pi (fun p : V × Fin n => gaussianReal ((t p.2 : ℝ) * s p.1) (t p.2))) =
(Measure.map cumulativeVectorObservation
(Measure.pi (fun p : V × Fin n => gaussianReal 0 (t p.2)))).withDensity
(fun y => ENNReal.ofReal
(Real.exp ((∑ v, s v * y v (Fin.last n)) -
((∑ i, t i : ℝ≥0) : ℝ) * Fintype.card V / 2))) := by
simpa only [gaussianVectorLikelihood_of_sq_eq_one _ _ _ hs] using
gaussianVectorSkeleton_signal_withDensity t s
end Problem367
end
end OAI
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