| import OAI.Probability.FreeIsing.Brownian.Bridge |
| import OAI.Probability.FreeIsing.Gaussian.IncrementLikelihood |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| open MeasureTheory ProbabilityTheory |
| open scoped ENNReal NNReal |
|
|
| namespace Problem367 |
|
|
| theorem preBrownian_past_drift_withDensity |
| {Omega : Type*} [MeasurableSpace Omega] |
| {B : ℝ≥0 → Omega → ℝ} {mu : Measure Omega} |
| (hB : IsPreBrownianReal B mu) (hmB : ∀ t, Measurable (B t)) |
| {T : ℝ≥0} (hT : T ≠ 0) (s : ℝ) : |
| Measure.map (fun w (t : Set.Iic T) => (t.val : ℝ) * s + B t w) mu = |
| (Measure.map (fun w (t : Set.Iic T) => B t w) mu).withDensity |
| (fun y : Set.Iic T → ℝ => ENNReal.ofReal |
| (gaussianSignalLikelihood T s (y ⟨T, by simp⟩))) := by |
| let : IsProbabilityMeasure mu := hB.isGaussianProcess.isProbabilityMeasure |
| let R : Omega → Set.Iic T → ℝ := fun w t => brownianBridge B T t w |
| let E : Omega → (Set.Iic T → ℝ) × ℝ := fun w => (R w, B T w) |
| let Es : Omega → (Set.Iic T → ℝ) × ℝ := |
| fun w => (R w, (T : ℝ) * s + B T w) |
| let F : ((Set.Iic T → ℝ) × ℝ) → Set.Iic T → ℝ := |
| fun z t => z.1 t + ((t.val : ℝ) / (T : ℝ)) * z.2 |
| let L : ℝ → ℝ≥0∞ := fun x => ENNReal.ofReal (gaussianSignalLikelihood T s x) |
| have hR : Measurable R := by |
| apply Measurable.of_eval |
| intro t |
| exact (hmB t).sub ((hmB T).const_mul _) |
| have hE : Measurable E := hR.prodMk (hmB T) |
| have hEs : Measurable Es := hR.prodMk (measurable_const.add (hmB T)) |
| have hF : Measurable F := by unfold F; fun_prop |
| have hL : Measurable L := by unfold L gaussianSignalLikelihood; fun_prop |
| have hind : IndepFun R (B T) mu := preBrownian_bridge_indep_terminal hB hT |
| have hlaw : Measure.map E mu = (Measure.map R mu).prod (gaussianReal 0 T) := by |
| rw [IndepFun.map_prod_eq_prod_map_map hR.aemeasurable (hmB T).aemeasurable hind, |
| (hB.hasLaw_eval T).map_eq] |
| have hlaws : Measure.map Es mu = |
| (Measure.map R mu).prod (gaussianReal ((T : ℝ) * s) T) := by |
| have hshift : Measurable (fun x : ℝ => (T : ℝ) * s + x) := by fun_prop |
| have hbt : Measurable (fun w => (T : ℝ) * s + B T w) := by fun_prop |
| have hinds : IndepFun R (fun w => (T : ℝ) * s + B T w) mu := by |
| simpa only [Function.comp_def, id_eq] using |
| hind.comp measurable_id hshift |
| change Measure.map (fun w => (R w, (T : ℝ) * s + B T w)) mu = _ |
| rw [IndepFun.map_prod_eq_prod_map_map hR.aemeasurable |
| hbt.aemeasurable hinds] |
| congr 1 |
| simpa only [zero_add] using |
| (gaussianReal_const_add (hB.hasLaw_eval T) ((T : ℝ) * s)).map_eq |
| have hchange : Measure.map Es mu = |
| (Measure.map E mu).withDensity (fun z => L z.2) := by |
| rw [hlaws, hlaw, gaussianReal_signal_withDensity T s] |
| exact prod_withDensity_right hL |
| have hFE : F ∘ E = fun w (t : Set.Iic T) => B t w := by |
| funext w t |
| dsimp [F, E, R, brownianBridge] |
| ring |
| have hFEs : F ∘ Es = fun w (t : Set.Iic T) => (t.val : ℝ) * s + B t w := by |
| have hT' : (T : ℝ) ≠ 0 := by exact_mod_cast hT |
| funext w t |
| dsimp [F, Es, R, brownianBridge] |
| field_simp |
| ring |
| have hpast : Measurable (fun w (t : Set.Iic T) => B t w) := |
| Measurable.of_eval (fun t => hmB t) |
| calc |
| Measure.map (fun w (t : Set.Iic T) => (t.val : ℝ) * s + B t w) mu = |
| Measure.map F (Measure.map Es mu) := by rw [Measure.map_map hF hEs, hFEs] |
| _ = Measure.map F ((Measure.map E mu).withDensity (fun z => L z.2)) := by |
| rw [hchange] |
| _ = Measure.map F (Measure.map E (mu.withDensity (fun w => L (B T w)))) := by |
| congr 1 |
| exact (map_withDensity_comp_observation mu E hE (fun z => L z.2) |
| (hL.comp measurable_snd)).symm |
| _ = Measure.map (fun w (t : Set.Iic T) => B t w) |
| (mu.withDensity (fun w => L (B T w))) := by rw [Measure.map_map hF hE, hFE] |
| _ = _ := map_withDensity_comp_observation mu _ hpast |
| (fun y : Set.Iic T → ℝ => L (y ⟨T, by simp⟩)) (hL.comp (measurable_pi_apply _)) |
| |
| theorem preBrownian_past_vector_drift_withDensity |
| {Omega V : Type*} [MeasurableSpace Omega] [Fintype V] |
| {B : ℝ≥0 → Omega → ℝ} {mu : Measure Omega} |
| (hB : IsPreBrownianReal B mu) (hmB : ∀ t, Measurable (B t)) |
| {T : ℝ≥0} (hT : T ≠ 0) (s : V → ℝ) : |
| Measure.pi (fun v : V => |
| Measure.map (fun w (t : Set.Iic T) => (t.val : ℝ) * s v + B t w) mu) = |
| (Measure.pi (fun _ : V => |
| Measure.map (fun w (t : Set.Iic T) => B t w) mu)).withDensity |
| (fun y => ENNReal.ofReal |
| (gaussianVectorLikelihood T s (fun v => y v ⟨T, by simp⟩))) := by |
| classical |
| let : IsProbabilityMeasure mu := hB.isGaussianProcess.isProbabilityMeasure |
| have hpast : Measurable (fun w (t : Set.Iic T) => B t w) := |
| Measurable.of_eval (fun t => hmB t) |
| let : IsProbabilityMeasure |
| (Measure.map (fun w (t : Set.Iic T) => B t w) mu) := |
| (Measure.isProbabilityMeasure_map_iff hpast.aemeasurable).mpr inferInstance |
| have hshift (v : V) : |
| Measurable (fun w (t : Set.Iic T) => (t.val : ℝ) * s v + B t w) := by |
| apply Measurable.of_eval |
| intro t |
| exact measurable_const.add (hmB t) |
| let (v : V) : IsProbabilityMeasure |
| (Measure.map (fun w (t : Set.Iic T) => (t.val : ℝ) * s v + B t w) mu) := |
| (Measure.isProbabilityMeasure_map_iff (hshift v).aemeasurable).mpr inferInstance |
| rw [pi_withDensity_of_probability |
| (fun _ : V => Measure.map (fun w (t : Set.Iic T) => B t w) mu) |
| (fun v : V => Measure.map (fun w (t : Set.Iic T) => (t.val : ℝ) * s v + B t w) mu) |
| (fun v (y : Set.Iic T → ℝ) => ENNReal.ofReal |
| (gaussianSignalLikelihood T (s v) (y ⟨T, by simp⟩))) |
| (fun v => by unfold gaussianSignalLikelihood; fun_prop) |
| (fun v => preBrownian_past_drift_withDensity hB hmB hT (s v))] |
| apply withDensity_congr_ae |
| filter_upwards [] with y |
| rw [← ENNReal.ofReal_prod_of_nonneg (fun v hv => by |
| unfold gaussianSignalLikelihood; positivity), prod_gaussianSignalLikelihood] |
| |
| theorem preBrownian_past_spin_drift_withDensity |
| {Omega V : Type*} [MeasurableSpace Omega] [Fintype V] |
| {B : ℝ≥0 → Omega → ℝ} {mu : Measure Omega} |
| (hB : IsPreBrownianReal B mu) (hmB : ∀ t, Measurable (B t)) |
| {T : ℝ≥0} (hT : T ≠ 0) (s : V → ℝ) (hs : ∀ v, s v ^ 2 = 1) : |
| Measure.pi (fun v : V => |
| Measure.map (fun w (t : Set.Iic T) => (t.val : ℝ) * s v + B t w) mu) = |
| (Measure.pi (fun _ : V => |
| Measure.map (fun w (t : Set.Iic T) => B t w) mu)).withDensity |
| (fun y => ENNReal.ofReal |
| (Real.exp ((∑ v, s v * y v ⟨T, by simp⟩) - |
| (T : ℝ) * Fintype.card V / 2))) := by |
| simpa only [gaussianVectorLikelihood_of_sq_eq_one _ _ _ hs] using |
| preBrownian_past_vector_drift_withDensity hB hmB hT s |
| |
| end Problem367 |
| |
| end |
| |
| end OAI |
| |
