| import OAI.Probability.Localization.Model | |
| namespace OAI | |
| open MeasureTheory ProbabilityTheory Set | |
| open scoped BigOperators ENNReal | |
| namespace RepeatedLocalization | |
| lemma continuous_chart (n : ℕ) : Continuous (@chart n) := by | |
| apply continuous_pi | |
| intro a | |
| refine Fin.lastCases ?_ (fun j => ?_) a | |
| · simp only [chart, Fin.lastCases_last] | |
| fun_prop | |
| · simp only [chart, Fin.lastCases_castSucc] | |
| fun_prop | |
| lemma measurable_chart (n : ℕ) : Measurable (@chart n) := | |
| (continuous_chart n).measurable | |
| lemma continuous_exactLabel (n : ℕ) : | |
| Continuous (fun Xz : GaussianBlock n × Coordinate n => exactLabel Xz.1 Xz.2) := by | |
| apply continuous_pi | |
| intro row | |
| unfold exactLabel | |
| apply continuous_finsetSum | |
| intro coordinate _ | |
| exact ((continuous_apply coordinate).comp ((continuous_apply row).comp continuous_fst)).mul | |
| ((continuous_apply coordinate).comp ((continuous_chart n).comp continuous_snd)) | |
| lemma measurable_initialCube (n : ℕ) : MeasurableSet (initialCube n) := by | |
| apply MeasurableSet.univ_pi | |
| intro _ | |
| exact measurableSet_Ico | |
| lemma volume_initialCube (n : ℕ) : | |
| volume (initialCube n) = ENNReal.ofReal ((1 / Real.sqrt n) ^ n) := by | |
| unfold initialCube | |
| rw [Real.volume_pi_Ico] | |
| have hw : 1 / (2 * Real.sqrt n) - -1 / (2 * Real.sqrt n) = 1 / Real.sqrt n := by | |
| ring | |
| simp only [hw, Finset.prod_const, Finset.card_univ, Fintype.card_fin] | |
| exact (ENNReal.ofReal_pow (by positivity) n).symm | |
| lemma volume_initialCube_pos {n : ℕ} (hn : 0 < n) : | |
| 0 < volume (initialCube n) := by | |
| rw [volume_initialCube] | |
| exact ENNReal.ofReal_pos.mpr (pow_pos (by positivity) n) | |
| lemma volume_initialCube_ne_top (n : ℕ) : volume (initialCube n) ≠ ∞ := by | |
| rw [volume_initialCube] | |
| exact ENNReal.ofReal_ne_top | |
| lemma cubePrior_isProbabilityMeasure {n : ℕ} (hn : 0 < n) : | |
| IsProbabilityMeasure (cubePrior n) := by | |
| exact cond_isProbabilityMeasure_of_finite | |
| (ne_of_gt (volume_initialCube_pos hn)) (volume_initialCube_ne_top n) | |
| instance gaussianBlockLaw_isProbabilityMeasure (n : ℕ) : | |
| IsProbabilityMeasure (gaussianBlockLaw n) := by | |
| unfold gaussianBlockLaw | |
| infer_instance | |
| lemma BlockRule.prob_le_one {n N : ℕ} (rule : BlockRule n N) | |
| (data : GaussianBlock n × Label n) (w : Fin N) : rule.prob data w ≤ 1 := by | |
| rw [← rule.sum_one data] | |
| exact Finset.single_le_sum (fun i _ => rule.nonneg data i) (Finset.mem_univ w) | |
| lemma measurable_actualRule {n N : ℕ} (rule : BlockRule n N) (w : Fin N) : | |
| Measurable (fun Xz : GaussianBlock n × Coordinate n => | |
| rule.prob (Xz.1, exactLabel Xz.1 Xz.2) w) := | |
| (rule.measurable_prob w).comp | |
| (measurable_fst.prodMk (continuous_exactLabel n).measurable) | |
| lemma integrable_actualRule {n N : ℕ} (rule : BlockRule n N) | |
| (z : Coordinate n) (w : Fin N) : | |
| Integrable (fun X => rule.prob (X, exactLabel X z) w) (gaussianBlockLaw n) := by | |
| apply (integrable_const (1 : ℝ)).mono' | |
| · exact ((measurable_actualRule rule w).comp | |
| (measurable_id.prodMk measurable_const)).aestronglyMeasurable | |
| · exact Filter.Eventually.of_forall (fun X => by | |
| rw [Real.norm_eq_abs, abs_of_nonneg (rule.nonneg _ w)] | |
| exact rule.prob_le_one _ w) | |
| lemma measurable_blockProbability {n N : ℕ} (rule : BlockRule n N) (w : Fin N) : | |
| Measurable (fun z => blockProbability rule z w) := by | |
| exact (measurable_actualRule rule w).stronglyMeasurable.integral_prod_left'.measurable | |
| lemma blockProbability_nonneg {n N : ℕ} (rule : BlockRule n N) | |
| (z : Coordinate n) (w : Fin N) : 0 ≤ blockProbability rule z w := by | |
| exact integral_nonneg (fun X => rule.nonneg _ w) | |
| lemma blockProbability_sum {n N : ℕ} (rule : BlockRule n N) (z : Coordinate n) : | |
| ∑ w, blockProbability rule z w = 1 := by | |
| unfold blockProbability | |
| rw [← integral_finsetSum _ (fun w _ => integrable_actualRule rule z w)] | |
| simp [rule.sum_one] | |
| lemma blockProbability_le_one {n N : ℕ} (rule : BlockRule n N) | |
| (z : Coordinate n) (w : Fin N) : blockProbability rule z w ≤ 1 := by | |
| rw [← blockProbability_sum rule z] | |
| exact Finset.single_le_sum (fun j _ => blockProbability_nonneg rule z j) | |
| (Finset.mem_univ w) | |
| lemma measurable_historyLikelihood {n N t : ℕ} (rules : Rules n N t) (w : History N t) : | |
| Measurable (fun z => historyLikelihood rules z w) := by | |
| exact Finset.measurable_prod _ (fun i _ => measurable_blockProbability (rules i (past w i)) (w i)) | |
| lemma historyLikelihood_nonneg {n N t : ℕ} (rules : Rules n N t) | |
| (z : Coordinate n) (w : History N t) : 0 ≤ historyLikelihood rules z w := by | |
| exact Finset.prod_nonneg (fun i _ => blockProbability_nonneg _ _ _) | |
| lemma historyLikelihood_le_one {n N t : ℕ} (rules : Rules n N t) | |
| (z : Coordinate n) (w : History N t) : historyLikelihood rules z w ≤ 1 := by | |
| exact Finset.prod_le_one₀ (fun _ _ => blockProbability_nonneg _ _ _) | |
| (fun _ _ => blockProbability_le_one _ _ _) | |
| end RepeatedLocalization | |
| end OAI | |