File size: 3,448 Bytes
a2ee19a | 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 | 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
|