openAI-math-dataset / lean /OAI /Combinatorics /BalancedRyser /Selection /SelectionHistoryMarginal.lean
| import OAI.Combinatorics.BalancedRyser.Selection.SelectionHistoryInvariant | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped MeasureTheory ProbabilityTheory BigOperators | |
| namespace Balanced.Selection.Process | |
| attribute [local instance] Classical.propDecidable | |
| variable {I V : Type*} [Fintype I] [Fintype V] | |
| theorem integrable_stateObservable (p : Parameters) (q : ℕ) (G : Input I V) | |
| (t : ℕ) (f : Option (State I V) → ℝ) : | |
| Integrable (fun ω => f (ω t).1) (snapshotPathLaw p q G) := by | |
| have hi : Integrable (fun s : Snapshot I V => f s.1) | |
| ((snapshotPathLaw p q G).map (fun ω => ω t)) := .of_finite | |
| exact hi.comp_measurable (measurable_pi_apply t) | |
| theorem sum_snapshotWeight_stateObservable (p : Parameters) (q t : ℕ) | |
| (G : Input I V) (before : Snapshot I V) (f : Option (State I V) → ℝ) : | |
| (∑ out, snapshotWeight p q t G before out * f out.1) = | |
| ∑ out, transitionWeight p q t G before.1 out * f out := by | |
| simp only [Snapshot, Fintype.sum_prod_type] | |
| simp_rw [← Finset.sum_mul, sum_snapshotWeight_state] | |
| theorem integral_stateObservable_eq_historyWeight (p : Parameters) (q : ℕ) | |
| (G : Input I V) (t : ℕ) (f : Option (State I V) → ℝ) : | |
| (∫ ω, f (ω t).1 ∂snapshotPathLaw p q G) = | |
| ∑ out, historyWeight p q G t out * f out := by | |
| induction t generalizing f with | |
| | zero => | |
| have hi := FiniteHistory.ae_initial_state | |
| (fun t => snapshotPMF p q t G) (initialSnapshot (I := I) (V := V)) | |
| have heq : (fun ω => f (ω 0).1) =ᵐ[snapshotPathLaw p q G] | |
| fun _ => f (some initialState) := by | |
| filter_upwards [hi] with ω hω | |
| simp only [hω, initialSnapshot] | |
| rw [integral_congr_ae heq] | |
| simp [historyWeight] | |
| | succ t ih => | |
| have hce := condExp_snapshot_step p q G t (fun _ out => f out.1) | |
| have hce' : (snapshotPathLaw p q G)[ | |
| (fun ω => f (ω (t + 1)).1) | snapshotFiltration t] | |
| =ᵐ[snapshotPathLaw p q G] | |
| fun ω => ∑ out, transitionWeight p q t G (ω t).1 out * f out := by | |
| change (snapshotPathLaw p q G)[ | |
| (fun ω => f (ω (t + 1)).1) | snapshotFiltration t] =ᵐ[snapshotPathLaw p q G] _ at hce | |
| simpa only [sum_snapshotWeight_stateObservable] using hce | |
| calc | |
| (∫ ω, f (ω (t + 1)).1 ∂snapshotPathLaw p q G) = | |
| ∫ ω, ((snapshotPathLaw p q G)[ | |
| (fun ω => f (ω (t + 1)).1) | snapshotFiltration t]) ω | |
| ∂snapshotPathLaw p q G := | |
| (integral_condExp (snapshotFiltration.le t)).symm | |
| _ = ∫ ω, (∑ out, transitionWeight p q t G (ω t).1 out * f out) | |
| ∂snapshotPathLaw p q G := integral_congr_ae hce' | |
| _ = ∑ out, ∫ ω, transitionWeight p q t G (ω t).1 out * f out | |
| ∂snapshotPathLaw p q G := | |
| integral_finsetSum _ (fun out _ => | |
| integrable_stateObservable p q G t (fun s => transitionWeight p q t G s out * f out)) | |
| _ = ∑ out, ∑ before, historyWeight p q G t before * | |
| (transitionWeight p q t G before out * f out) := by | |
| apply Finset.sum_congr rfl | |
| intro out _ | |
| exact ih (fun before => transitionWeight p q t G before out * f out) | |
| _ = ∑ out, historyWeight p q G (t + 1) out * f out := by | |
| simp only [historyWeight, Finset.sum_mul, mul_assoc] | |
| theorem state_event_real_eq_historyWeight (p : Parameters) (q : ℕ) | |
| (G : Input I V) (t : ℕ) (A : Set (Option (State I V))) : | |
| (snapshotPathLaw p q G).real {ω | (ω t).1 ∈ A} = | |
| ∑ out, if out ∈ A then historyWeight p q G t out else 0 := by | |
| have hm : MeasurableSet {ω : FiniteHistory.Path (Snapshot I V) | (ω t).1 ∈ A} := | |
| (show MeasurableSet {s : Snapshot I V | s.1 ∈ A} from .of_discrete).preimage | |
| (measurable_pi_apply t) | |
| rw [← integral_indicator_one hm] | |
| have h := integral_stateObservable_eq_historyWeight p q G t | |
| (fun out => if out ∈ A then 1 else 0) | |
| simpa only [Set.indicator_apply, Set.mem_ofPred_eq, Pi.one_apply, mul_ite, | |
| mul_one, mul_zero] using h | |
| theorem map_state_snapshotPathLaw | |
| [MeasurableSpace (Option (State I V))] | |
| [MeasurableSingletonClass (Option (State I V))] | |
| (p : Parameters) (q : ℕ) (G : Input I V) (t : ℕ) : | |
| (snapshotPathLaw p q G).map (fun ω => (ω t).1) = | |
| (historyPMF p q G t).toMeasure := by | |
| apply Measure.ext_of_singleton | |
| intro out | |
| have hm : Measurable (fun ω : FiniteHistory.Path (Snapshot I V) => (ω t).1) := | |
| (measurable_of_countable (fun s : Snapshot I V => s.1)).comp (measurable_pi_apply t) | |
| apply (ENNReal.toReal_eq_toReal_iff' (measure_ne_top _ _) (measure_ne_top _ _)).1 | |
| rw [Measure.map_apply hm (measurableSet_singleton out), | |
| PMF.toMeasure_apply_singleton _ _ (measurableSet_singleton out), historyPMF_apply_real] | |
| change (snapshotPathLaw p q G).real {ω | (ω t).1 ∈ ({out} : Set (Option (State I V)))} = _ | |
| rw [state_event_real_eq_historyWeight] | |
| simp | |
| end Balanced.Selection.Process | |
| end | |
| end OAI | |