| import Mathlib | |
| import OAI.Probability.Perceptron.Variational.GaussianMarkRecursion | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory Filter Set | |
| open scoped Topology NNReal ENNReal | |
| namespace SphericalPerceptronFreeEnergy | |
| section | |
| variable {P X S : Type} [MeasurableSpace P] [MeasurableSpace X] [MeasurableSpace S] | |
| def parameterMarkStep (step : X×S → X) : (P×X)×S → P×X := | |
| fun a => (a.1.1,step (a.1.2,a.2)) | |
| lemma parameterMarkStep_measurable {step : X×S → X} (hs : Measurable step) : | |
| Measurable (parameterMarkStep (P := P) step) := | |
| (measurable_fst.comp measurable_fst).prodMk | |
| (hs.comp ((measurable_snd.comp measurable_fst).prodMk measurable_snd)) | |
| lemma decoratedTerminalTotalE_parameter (step : X×S → X) (H : P → X → ℝ) | |
| (n : ℕ) (p : P) (x : X) (η : DecoratedCascade S n) : | |
| decoratedTerminalTotalE (parameterMarkStep step) (Function.uncurry H) n ((p,x),η)= | |
| decoratedTerminalTotalE step (H p) n (x,η) := by | |
| induction n generalizing x with | |
| | zero => rfl | |
| | succ n ih => | |
| apply lintegral_congr | |
| intro q | |
| congr 1 | |
| exact ih (step (x,q.2.1)) q.2.2 | |
| lemma decoratedTerminalTotal_parameter (step : X×S → X) (H : P → X → ℝ) | |
| (n : ℕ) (p : P) (x : X) (η : DecoratedCascade S n) : | |
| decoratedTerminalTotal (parameterMarkStep step) (Function.uncurry H) n ((p,x),η)= | |
| decoratedTerminalTotal step (H p) n (x,η) := | |
| congrArg ENNReal.toReal (decoratedTerminalTotalE_parameter step H n p x η) | |
| end | |
| variable {P E : Type} [MeasurableSpace P] | |
| [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] | |
| [MeasurableSpace E] [BorelSpace E] | |
| lemma gaussianLinearCascadeLog_parameter_measurable | |
| (A : ℕ → E →L[ℝ] E) (R : E →L[ℝ] E) | |
| {H : P → E → ℝ} (hH : Measurable (Function.uncurry H)) (k : ℕ) (z : Fin k → ℝ) : | |
| Measurable (fun a : P × (E×DecoratedCascade E k) => | |
| gaussianLinearCascadeLog A R (H a.1) k a.2) := by | |
| let step := gaussianLinearMarkStep A | |
| let V := fun p : P => fun x : ℕ → E => H p (x 0) | |
| have hV : Measurable (Function.uncurry V) := hH.comp | |
| (measurable_fst.prodMk ((measurable_pi_apply 0).comp measurable_snd)) | |
| let T := fun a : P × (E×DecoratedCascade E k) => | |
| ((a.1,fun _ : ℕ => R a.2.1),a.2.2) | |
| have hT : Measurable T := | |
| (measurable_fst.prodMk (Measurable.of_eval (fun _ => | |
| R.measurable.comp (measurable_fst.comp measurable_snd)))).prodMk | |
| (measurable_snd.comp measurable_snd) | |
| let ν : ProbabilityMeasure E := gaussianMarkLaw | |
| have hs : Measurable step := gaussianLinearMarkStep_measurable A | |
| have h1 := (decoratedTerminalTotal_measurable ν (parameterMarkStep step) | |
| (parameterMarkStep_measurable hs) hV k z).comp hT | |
| have h0 := (decoratedTerminalTotal_measurable ν (parameterMarkStep step) | |
| (parameterMarkStep_measurable hs) (H := Function.uncurry (fun (_ : P) (_ : ℕ → E) => (0:ℝ))) | |
| measurable_const k z).comp hT | |
| have he (a : P × (E×DecoratedCascade E k)) : | |
| gaussianLinearCascadeLog A R (H a.1) k a.2= | |
| Real.log (decoratedTerminalTotal (parameterMarkStep step) (Function.uncurry V) k (T a)/ | |
| decoratedTerminalTotal (parameterMarkStep step) (Function.uncurry (fun (_ : P) (_ : ℕ → E) => (0:ℝ))) k (T a)) := by | |
| rw [decoratedTerminalTotal_parameter,decoratedTerminalTotal_parameter] | |
| rfl | |
| simp_rw [he] | |
| exact (h1.div h0).log | |
| end SphericalPerceptronFreeEnergy | |
| end | |
| end OAI | |