| import Mathlib |
| import OAI.Combinatorics.IndependentSets.Repetition.Sampling |
| import OAI.Combinatorics.IndependentSets.Repetition.Conditionals |
|
|
| namespace OAI |
|
|
| namespace IndependentSetsGames.Foundations.Games.FiniteDistribution |
|
|
| open scoped BigOperators |
| noncomputable section |
|
|
| variable {Ω : Type*} [Fintype Ω] |
| |
| def totalVariation (μ ν : FiniteDistribution Ω) : ℝ := |
| (∑ x, |μ.weight x - ν.weight x|) / 2 |
|
|
| theorem totalVariation_nonnegative (μ ν : FiniteDistribution Ω) : |
| 0 ≤ μ.totalVariation ν := |
| div_nonneg (Finset.sum_nonneg fun _ _ => abs_nonneg _) (by norm_num) |
|
|
| theorem totalVariation_comm (μ ν : FiniteDistribution Ω) : |
| μ.totalVariation ν = ν.totalVariation μ := by |
| unfold totalVariation |
| congr 1 |
| apply Finset.sum_congr rfl |
| intro x _ |
| exact abs_sub_comm _ _ |
|
|
| theorem probability_sub_le_totalVariation (μ ν : FiniteDistribution Ω) |
| (event : Ω → Bool) : |
| μ.probability event - ν.probability event ≤ μ.totalVariation ν := by |
| have pointwise (x : Ω) : |
| 2 * ((if event x then μ.weight x else 0) - |
| (if event x then ν.weight x else 0)) ≤ |
| |μ.weight x - ν.weight x| + (μ.weight x - ν.weight x) := by |
| have h₁ := le_abs_self (μ.weight x - ν.weight x) |
| have h₂ := neg_le_abs (μ.weight x - ν.weight x) |
| cases event x <;> simp only [Bool.false_eq_true, ↓reduceIte] <;> linarith |
| have summed := Finset.sum_le_sum (s := Finset.univ) fun x _ => pointwise x |
| simp only [← Finset.mul_sum, Finset.sum_add_distrib, Finset.sum_sub_distrib, |
| μ.normalized, ν.normalized, sub_self, add_zero] at summed |
| unfold probability totalVariation |
| linarith |
|
|
| theorem probability_abs_sub_le_totalVariation (μ ν : FiniteDistribution Ω) |
| (event : Ω → Bool) : |
| |μ.probability event - ν.probability event| ≤ μ.totalVariation ν := by |
| apply abs_le.mpr |
| constructor |
| · have h := ν.probability_sub_le_totalVariation μ event |
| rw [totalVariation_comm] at h |
| linarith |
| · exact μ.probability_sub_le_totalVariation ν event |
|
|
| theorem probability_le_add_totalVariation (μ ν : FiniteDistribution Ω) |
| (event : Ω → Bool) : |
| μ.probability event ≤ ν.probability event + μ.totalVariation ν := by |
| have h := μ.probability_sub_le_totalVariation ν event |
| linarith |
|
|
| end |
| end IndependentSetsGames.Foundations.Games.FiniteDistribution |
|
|
| namespace IndependentSetsGames.Foundations.Information |
|
|
| open scoped BigOperators |
|
|
| variable {α β : Type*} [Fintype α] [Fintype β] |
|
|
| theorem gameLaw_isProbability (μ : Games.FiniteDistribution α) : IsProbability μ.weight := |
| ⟨μ.nonnegative, μ.normalized⟩ |
|
|
| def toGameLaw (p : α → ℝ) (hp : IsProbability p) : Games.FiniteDistribution α where |
| weight := p |
| nonnegative := hp.1 |
| normalized := hp.2 |
|
|
| def gameLawEquiv : Games.FiniteDistribution α ≃ {p : α → ℝ // IsProbability p} where |
| toFun μ := ⟨μ.weight, gameLaw_isProbability μ⟩ |
| invFun p := toGameLaw p.1 p.2 |
| left_inv μ := by cases μ; rfl |
| right_inv p := by cases p; rfl |
| |
| theorem totalVariation_gameLaw (μ ν : Games.FiniteDistribution α) : |
| totalVariation μ.weight ν.weight = μ.totalVariation ν := rfl |
|
|
| noncomputable def gameConditionalKernel (μ : Games.FiniteDistribution (α × β)) |
| (fallback : Games.FiniteDistribution β) (a : α) : Games.FiniteDistribution β := |
| toGameLaw (conditionalKernel μ.weight fallback.weight a) |
| (conditionalKernel_isProbability μ.weight fallback.weight |
| (gameLaw_isProbability μ) (gameLaw_isProbability fallback) a) |
|
|
| theorem gameConditionalKernel_recombine (μ : Games.FiniteDistribution (α × β)) |
| (fallback : Games.FiniteDistribution β) (a : α) (b : β) : |
| firstMarginal μ.weight a * (gameConditionalKernel μ fallback a).weight b = |
| μ.weight (a, b) := |
| marginal_mul_conditionalKernel μ.weight fallback.weight (gameLaw_isProbability μ) a b |
|
|
| noncomputable def gameConditionalProduct (μ : Games.FiniteDistribution (α × β)) |
| (fallback : Games.FiniteDistribution β) (newInput : Games.FiniteDistribution α) : |
| Games.FiniteDistribution (α × β) := |
| toGameLaw (fun ab => newInput.weight ab.1 * conditionalKernel μ.weight fallback.weight ab.1 ab.2) |
| (alternativeInput_conditionalProduct_isProbability μ.weight fallback.weight newInput.weight |
| (gameLaw_isProbability μ) (gameLaw_isProbability fallback) (gameLaw_isProbability newInput)) |
|
|
| theorem gameConditionalProduct_weight (μ : Games.FiniteDistribution (α × β)) |
| (fallback : Games.FiniteDistribution β) (newInput : Games.FiniteDistribution α) |
| (a : α) (b : β) : |
| (gameConditionalProduct μ fallback newInput).weight (a, b) = |
| newInput.weight a * (gameConditionalKernel μ fallback a).weight b := rfl |
|
|
| theorem gameCondition_weight (μ : Games.FiniteDistribution α) (event : α → Bool) |
| (positive : 0 < μ.probability event) : |
| (μ.condition event positive).weight = |
| posterior μ.weight (fun a => if event a then 1 else 0) (μ.probability event) := by |
| funext a |
| cases he : event a <;> simp [Games.FiniteDistribution.condition, posterior, he] |
| |
| theorem gameCondition_relativeEntropy_le (μ : Games.FiniteDistribution α) (event : α → Bool) |
| (positive : 0 < μ.probability event) : |
| relativeEntropy (μ.condition event positive).weight μ.weight ≤ |
| Real.log (1 / μ.probability event) := by |
| rw [gameCondition_weight] |
| apply posterior_relativeEntropy_le μ.weight (fun a => if event a then 1 else 0) |
| (gameLaw_isProbability μ) |
| · intro a |
| cases event a <;> norm_num |
| · intro a |
| cases event a <;> norm_num |
| · exact positive |
| · unfold Games.FiniteDistribution.probability |
| apply Finset.sum_congr rfl |
| intro a _ |
| cases event a <;> simp |
| |
| end IndependentSetsGames.Foundations.Information |
| |
| end OAI |
| |
