openAI-math-dataset / lean /OAI /Probability /PosteriorReplicas /Information /FiniteConditionalEntropy.lean
| import OAI.Probability.PosteriorReplicas.Information.FiniteEntropyBounds | |
| namespace OAI | |
| open MeasureTheory ProbabilityTheory InformationTheory Set Filter | |
| open scoped ENNReal NNReal | |
| noncomputable section | |
| namespace PosteriorReplicas | |
| lemma neg_mul_log_product {p q : ℝ} (hp : 0 ≤ p) (hq : 0 ≤ q) : | |
| -(p*q)*Real.log (p*q) = q*(-p*Real.log p)+p*(-q*Real.log q) := by | |
| rcases hp.eq_or_lt with hp0 | hp0 | |
| · simp [← hp0] | |
| rcases hq.eq_or_lt with hq0 | hq0 | |
| · simp [← hq0] | |
| rw [Real.log_mul (ne_of_gt hp0) (ne_of_gt hq0)] | |
| ring | |
| section | |
| variable {V W : Type*} [Fintype V] [Fintype W] [MeasurableSpace V] [MeasurableSpace W] | |
| [MeasurableSingletonClass V] [MeasurableSingletonClass W] | |
| (μ : Measure V) [IsProbabilityMeasure μ] (κ : Kernel V W) [IsMarkovKernel κ] | |
| omit [Fintype W] [IsProbabilityMeasure μ] in | |
| lemma compProd_real_singleton (v : V) (w : W) : | |
| (μ ⊗ₘ κ).real {(v,w)} = μ.real {v} * (κ v).real {w} := by | |
| rw [measureReal_def,← singleton_prod_singleton, | |
| Measure.compProd_apply_prod (measurableSet_singleton v) (measurableSet_singleton w), | |
| lintegral_singleton,ENNReal.toReal_mul] | |
| exact mul_comm _ _ | |
| omit [IsProbabilityMeasure μ] in | |
| lemma finiteEntropy_compProd : | |
| finiteEntropy (μ ⊗ₘ κ) = finiteEntropy μ + ∑ v, μ.real {v} * finiteEntropy (κ v) := by | |
| unfold finiteEntropy | |
| rw [Fintype.sum_prod_type] | |
| simp_rw [compProd_real_singleton,neg_mul_log_product measureReal_nonneg measureReal_nonneg] | |
| rw [← Finset.sum_add_distrib] | |
| apply Finset.sum_congr rfl | |
| intro v _ | |
| rw [Finset.sum_add_distrib,← Finset.sum_mul,← Finset.mul_sum, | |
| sum_probability_atoms (κ v),one_mul] | |
| omit [IsProbabilityMeasure μ] in | |
| lemma finiteEntropy_compProd_conditional : | |
| finiteEntropy (μ ⊗ₘ κ) - finiteEntropy μ = ∑ v, μ.real {v} * finiteEntropy (κ v) := by | |
| rw [finiteEntropy_compProd] | |
| ring | |
| end | |
| end PosteriorReplicas | |
| end | |
| end OAI | |