| import OAI.Probability.PosteriorReplicas.Information.EntropyLossTest |
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| namespace OAI |
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| open MeasureTheory ProbabilityTheory InformationTheory Set Filter |
| open scoped ENNReal NNReal |
| noncomputable section |
| namespace PosteriorReplicas |
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| lemma log_tangent {x M : ℝ} (hx : 0 < x) (hM : 0 < M) : |
| Real.log x ≤ Real.log M + x/M - 1 := by |
| have h := Real.log_le_sub_one_of_pos (div_pos hx hM) |
| rw [Real.log_div hx.ne' hM.ne'] at h |
| linarith |
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| lemma log_replica_comparison {c Z J J' a a' v M : ℝ} |
| (hc : 0 < c) (hZ : 0 < Z) (hJ : 0 < J) (hJ' : 0 < J') |
| (ha : 0 < a) (ha' : 0 < a') (hv : 0 < v) (hM : 0 < M) (k m : ℕ) : |
| Real.log (c*J⁻¹^k*a⁻¹^m) - Real.log (c*Z⁻¹*J'⁻¹^k*a'⁻¹^m) ≤ |
| Real.log Z + (k:ℝ)*Real.log (J'/J) + (m:ℝ)*Real.log v + |
| (m:ℝ)*(Real.log M + (a'*a⁻¹*v⁻¹)/M - 1) := by |
| have hr := log_tangent (mul_pos (mul_pos ha' (inv_pos.mpr ha)) (inv_pos.mpr hv)) hM |
| simp only [Real.log_mul (mul_pos ha' (inv_pos.mpr ha)).ne' (inv_pos.mpr hv).ne', |
| Real.log_mul ha'.ne' (inv_pos.mpr ha).ne',Real.log_inv] at hr |
| have he : Real.log (c*J⁻¹^k*a⁻¹^m) - Real.log (c*Z⁻¹*J'⁻¹^k*a'⁻¹^m) = |
| Real.log Z + (k:ℝ)*Real.log (J'/J)+(m:ℝ)*(Real.log a'-Real.log a) := by |
| rw [Real.log_mul (mul_pos hc (pow_pos (inv_pos.mpr hJ) _)).ne' (pow_pos (inv_pos.mpr ha) _).ne', |
| Real.log_mul hc.ne' (pow_pos (inv_pos.mpr hJ) _).ne', |
| Real.log_mul (mul_pos (mul_pos hc (inv_pos.mpr hZ)) (pow_pos (inv_pos.mpr hJ') _)).ne' |
| (pow_pos (inv_pos.mpr ha') _).ne', |
| Real.log_mul (mul_pos hc (inv_pos.mpr hZ)).ne' (pow_pos (inv_pos.mpr hJ') _).ne', |
| Real.log_mul hc.ne' (inv_pos.mpr hZ).ne',Real.log_div hJ'.ne' hJ.ne'] |
| simp only [Real.log_pow,Real.log_inv] |
| ring |
| rw [he] |
| nlinarith [mul_le_mul_of_nonneg_left hr (Nat.cast_nonneg (α := ℝ) m)] |
| |
| lemma entropy_loss_envelope {X Y : Type*} [MeasurableSpace X] [MeasurableSpace Y] |
| (μ ν : Measure X) [IsProbabilityMeasure μ] [IsProbabilityMeasure ν] |
| (hkl : klDiv μ ν ≠ ∞) {ψ : X → Y} (hψ : Measurable ψ) |
| {g : Y → ℝ} (hg : Measurable g) |
| (he : Integrable (fun y => Real.exp (g y)) (ν.map ψ)) |
| (hmean : (∫ y, Real.exp (g y) ∂ν.map ψ) ≤ 1) |
| {F : X → ℝ} (hF : Integrable F μ) |
| (hbound : ∀ᵐ x ∂μ, llr μ ν x - g (ψ x) ≤ F x) : |
| (klDiv μ ν).toReal ≤ (klDiv (μ.map ψ) (ν.map ψ)).toReal + ∫ x, F x ∂μ := by |
| have herr : Integrable (fun x => max (llr μ ν x - g (ψ x)) 0) μ := by |
| apply hF.norm.mono' ((measurable_llr μ ν).sub (hg.comp hψ) |>.max measurable_const |>.aestronglyMeasurable) |
| filter_upwards [hbound] with x hx |
| rw [Real.norm_eq_abs,abs_of_nonneg (le_max_right _ 0),Real.norm_eq_abs] |
| exact max_le (hx.trans (le_abs_self _)) (abs_nonneg _) |
| obtain ⟨_,hi,h⟩ := entropy_loss_test μ ν hkl hψ hg he hmean herr |
| exact h.trans (add_le_add le_rfl (integral_mono_ae hi hF hbound)) |
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| end PosteriorReplicas |
| end |
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| end OAI |
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