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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 | 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
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