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namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped ENNReal NNReal
namespace Problem367
def gaussianSignalLikelihood (t : ℝ≥0) (s x : ℝ) : ℝ :=
Real.exp (s * x - (t : ℝ) * s ^ 2 / 2)
theorem gaussianPDFReal_signal (t : ℝ≥0) (s x : ℝ) :
gaussianPDFReal ((t : ℝ) * s) t x =
gaussianPDFReal 0 t x * gaussianSignalLikelihood t s x := by
by_cases ht : t = 0
· simp [ht]
have ht' : (t : ℝ) ≠ 0 := by exact_mod_cast ht
have hexp : -(x - (t : ℝ) * s) ^ 2 / (2 * (t : ℝ)) =
-(x - 0) ^ 2 / (2 * (t : ℝ)) + (s * x - (t : ℝ) * s ^ 2 / 2) := by
field_simp
ring
simp only [gaussianPDFReal, gaussianSignalLikelihood, hexp, Real.exp_add]
ring
theorem gaussianPDF_signal (t : ℝ≥0) (s x : ℝ) :
gaussianPDF ((t : ℝ) * s) t x =
gaussianPDF 0 t x * ENNReal.ofReal (gaussianSignalLikelihood t s x) := by
rw [gaussianPDF, gaussianPDFReal_signal, ENNReal.ofReal_mul
(gaussianPDFReal_nonneg 0 t x)]
rfl
theorem gaussianReal_signal_withDensity (t : ℝ≥0) (s : ℝ) :
gaussianReal ((t : ℝ) * s) t =
(gaussianReal 0 t).withDensity
(fun x => ENNReal.ofReal (gaussianSignalLikelihood t s x)) := by
by_cases ht : t = 0
· simp [ht, gaussianSignalLikelihood, MeasureTheory.dirac_withDensity]
rw [gaussianReal_of_var_ne_zero _ ht, gaussianReal_of_var_ne_zero _ ht,
← withDensity_mul volume (measurable_gaussianPDF 0 t) (by
unfold gaussianSignalLikelihood
fun_prop)]
congr 1
funext x
exact gaussianPDF_signal t s x
theorem lintegral_pi_prod_of_measurable
{I : Type*} [Fintype I] {X : I → Type*}
[∀ i, MeasurableSpace (X i)]
(mu : ∀ i, Measure (X i)) [∀ i, IsProbabilityMeasure (mu i)]
(f : ∀ i, X i → ℝ≥0∞) (hf : ∀ i, Measurable (f i)) :
(∫⁻ x, ∏ i, f i (x i) ∂Measure.pi mu) = ∏ i, ∫⁻ x, f i x ∂mu i := by
classical
rw [lintegral_prod_eq_prod_lintegral_of_indepFun Finset.univ
(fun (i : I) (x : ∀ j, X j) => f i (x i))
(iIndepFun_pi (μ := mu) (X := f) (fun i => (hf i).aemeasurable))
(fun i => (hf i).comp (measurable_pi_apply i))]
apply Finset.prod_congr rfl
intro i hi
exact (measurePreserving_eval mu i).lintegral_comp (hf i)
theorem pi_withDensity_of_probability
{I : Type*} [Fintype I] {X : I → Type*}
[∀ i, MeasurableSpace (X i)]
(mu nu : ∀ i, Measure (X i))
[∀ i, IsProbabilityMeasure (mu i)] [∀ i, SigmaFinite (nu i)]
(f : ∀ i, X i → ℝ≥0∞) (hf : ∀ i, Measurable (f i))
(hnu : ∀ i, nu i = (mu i).withDensity (f i)) :
Measure.pi nu = (Measure.pi mu).withDensity (fun x => ∏ i, f i (x i)) := by
classical
apply Measure.pi_eq
intro s hs
have hpi : MeasurableSet (Set.pi Set.univ s) := MeasurableSet.pi Set.countable_univ
(fun i hi => hs i)
rw [withDensity_apply _ hpi, ← lintegral_indicator hpi]
have hind : (Set.pi Set.univ s).indicator (fun x => ∏ i, f i (x i)) =
fun x => ∏ i, (s i).indicator (f i) (x i) := by
funext x
by_cases hx : ∀ i, x i ∈ s i
· have hmem : x ∈ Set.pi Set.univ s := by simpa using hx
rw [Set.indicator_of_mem hmem]
exact Finset.prod_congr rfl (fun i hi => (Set.indicator_of_mem (hx i) _).symm)
· obtain ⟨i, hi⟩ := not_forall.mp hx
have hmem : x ∉ Set.pi Set.univ s := by simpa using not_forall.mpr ⟨i, hi⟩
rw [Set.indicator_of_notMem hmem]
exact (Finset.prod_eq_zero (Finset.mem_univ i) (Set.indicator_of_notMem hi _)).symm
rw [hind, lintegral_pi_prod_of_measurable mu _ (fun i => (hf i).indicator (hs i))]
apply Finset.prod_congr rfl
intro i hi
rw [hnu i, withDensity_apply _ (hs i), lintegral_indicator (hs i)]
def gaussianVectorLikelihood {I : Type*} [Fintype I]
(t : ℝ≥0) (s x : I → ℝ) : ℝ :=
Real.exp ((∑ i, s i * x i) - (t : ℝ) / 2 * ∑ i, s i ^ 2)
theorem prod_gaussianSignalLikelihood {I : Type*} [Fintype I]
(t : ℝ≥0) (s x : I → ℝ) :
(∏ i, gaussianSignalLikelihood t (s i) (x i)) = gaussianVectorLikelihood t s x := by
classical
simp only [gaussianSignalLikelihood, gaussianVectorLikelihood, ← Real.exp_sum]
congr 1
rw [Finset.sum_sub_distrib]
congr 1
rw [← Finset.sum_div, ← Finset.mul_sum]
ring
theorem gaussianPi_signal_withDensity {I : Type*} [Fintype I]
(t : ℝ≥0) (s : I → ℝ) :
Measure.pi (fun i => gaussianReal ((t : ℝ) * s i) t) =
(Measure.pi (fun _ : I => gaussianReal 0 t)).withDensity
(fun x => ENNReal.ofReal (gaussianVectorLikelihood t s x)) := by
classical
calc
Measure.pi (fun i => gaussianReal ((t : ℝ) * s i) t) =
(Measure.pi (fun _ : I => gaussianReal 0 t)).withDensity
(fun x => ∏ i, ENNReal.ofReal (gaussianSignalLikelihood t (s i) (x i))) := by
exact pi_withDensity_of_probability
(fun _ : I => gaussianReal 0 t)
(fun i : I => gaussianReal ((t : ℝ) * s i) t)
(fun i : I => fun x : ℝ => ENNReal.ofReal (gaussianSignalLikelihood t (s i) x))
(fun i => by unfold gaussianSignalLikelihood; fun_prop)
(fun i => gaussianReal_signal_withDensity t (s i))
_ = _ := by
apply withDensity_congr_ae
filter_upwards [] with x
calc
(∏ i, ENNReal.ofReal (gaussianSignalLikelihood t (s i) (x i))) =
ENNReal.ofReal (∏ i, gaussianSignalLikelihood t (s i) (x i)) :=
(ENNReal.ofReal_prod_of_nonneg (fun i hi => by
unfold gaussianSignalLikelihood
positivity)).symm
_ = ENNReal.ofReal (gaussianVectorLikelihood t s x) :=
congrArg ENNReal.ofReal (prod_gaussianSignalLikelihood t s x)
theorem gaussianVectorLikelihood_of_sq_eq_one {I : Type*} [Fintype I]
(t : ℝ≥0) (s x : I → ℝ) (hs : ∀ i, s i ^ 2 = 1) :
gaussianVectorLikelihood t s x =
Real.exp ((∑ i, s i * x i) - (t : ℝ) * Fintype.card I / 2) := by
classical
unfold gaussianVectorLikelihood
simp only [hs, Finset.sum_const, Finset.card_univ, nsmul_eq_mul, mul_one]
congr 1
ring
theorem gaussianPi_spin_withDensity {I : Type*} [Fintype I]
(t : ℝ≥0) (s : I → ℝ) (hs : ∀ i, s i ^ 2 = 1) :
Measure.pi (fun i => gaussianReal ((t : ℝ) * s i) t) =
(Measure.pi (fun _ : I => gaussianReal 0 t)).withDensity
(fun x => ENNReal.ofReal
(Real.exp ((∑ i, s i * x i) - (t : ℝ) * Fintype.card I / 2))) := by
simpa only [gaussianVectorLikelihood_of_sq_eq_one t s _ hs] using
gaussianPi_signal_withDensity t s
end Problem367
end
end OAI
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