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dd14c26 | 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 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 | 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
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