openAI-math-dataset / lean /OAI /Probability /InvariantIsing /Gaussian /GaussianDiagonalQuadratic.lean
| import OAI.Probability.InvariantIsing.Gaussian.GaussianFourthMoment | |
| namespace OAI | |
| /-! Exact mean and variance of a diagonal Gaussian quadratic form. -/ | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory | |
| open scoped BigOperators | |
| namespace InvariantIsing | |
| def gaussianDiagonalQuadratic {N : ℕ} (a : Fin N → ℝ) (g : Fin N → ℝ) : ℝ := ∑ i, a i*(g i)^2 | |
| lemma gaussianDiagonalQuadratic_memLp {N : ℕ} (a : Fin N → ℝ) : | |
| MemLp (gaussianDiagonalQuadratic a) 2 (Measure.pi (fun _ : Fin N => gaussianReal 0 1)) := by | |
| exact memLp_finsetSum Finset.univ (fun i _ => | |
| (gaussianReal_square_memLp.comp_measurePreserving | |
| (measurePreserving_eval (fun _ : Fin N => gaussianReal 0 1) i)).const_mul (a i)) | |
| lemma gaussianDiagonalQuadratic_integrable {N : ℕ} (a : Fin N → ℝ) : | |
| Integrable (gaussianDiagonalQuadratic a) (Measure.pi (fun _ : Fin N => gaussianReal 0 1)) := by | |
| apply integrable_finsetSum | |
| intro i _ | |
| exact ((gaussianReal_square_memLp.comp_measurePreserving | |
| (measurePreserving_eval (fun _ : Fin N => gaussianReal 0 1) i)).const_mul (a i)).integrable (by norm_num) | |
| lemma gaussianDiagonalQuadratic_mean {N : ℕ} (a : Fin N → ℝ) : | |
| (∫ g, gaussianDiagonalQuadratic a g ∂Measure.pi (fun _ : Fin N => gaussianReal 0 1)) = ∑ i, a i := by | |
| unfold gaussianDiagonalQuadratic | |
| rw [integral_finsetSum] | |
| · apply Finset.sum_congr rfl | |
| intro i _ | |
| rw [integral_const_mul] | |
| have h := (measurePreserving_eval (fun _ : Fin N => gaussianReal 0 1) i).hasLaw.integral_comp | |
| (f := fun x : ℝ => x^2) (by fun_prop) | |
| simpa only [Function.comp_def,Function.eval,gaussianReal_second_moment,mul_one] using | |
| congrArg (fun x : ℝ => a i*x) h | |
| · intro i _ | |
| exact ((gaussianReal_square_memLp.comp_measurePreserving | |
| (measurePreserving_eval (fun _ : Fin N => gaussianReal 0 1) i)).const_mul (a i)).integrable (by norm_num) | |
| lemma gaussianDiagonalQuadratic_variance {N : ℕ} (a : Fin N → ℝ) : | |
| variance (gaussianDiagonalQuadratic a) (Measure.pi (fun _ : Fin N => gaussianReal 0 1)) = | |
| 2*∑ i, (a i)^2 := by | |
| have h := variance_sum_pi (μ := fun _ : Fin N => gaussianReal 0 1) | |
| (X := fun i x => a i*x^2) (fun i => gaussianReal_square_memLp.const_mul (a i)) | |
| have he : (∑ i : Fin N, fun g : Fin N → ℝ => a i*(g i)^2) = gaussianDiagonalQuadratic a := by | |
| funext g | |
| simp only [Finset.sum_apply,gaussianDiagonalQuadratic] | |
| rw [he] at h | |
| rw [h,Finset.mul_sum] | |
| apply Finset.sum_congr rfl | |
| intro i _ | |
| rw [variance_const_mul,gaussianReal_square_variance] | |
| ring | |
| end InvariantIsing | |
| end | |
| end OAI | |