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import OAI.Probability.FreeIsing.Gaussian.Attenuation
import OAI.Probability.FreeIsing.Trees.RootCovariance
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped NNReal ENNReal
namespace Problem367.GaussianTreeCovariance
def covariance {d : ℕ} (beta : ℝ) (E : Finset (TreeVertex d))
(r v : TreeVertex d) (h : TreeVertex d → ℝ) : ℝ :=
LeafGibbs.covariance
(fun eta => GibbsLeaf.energy beta h E (extendFiniteSpin E eta))
(fun eta => spinSign (extendFiniteSpin E eta r))
(fun eta => spinSign (extendFiniteSpin E eta v))
lemma measurable_covariance {d : ℕ} (beta : ℝ) (E : Finset (TreeVertex d))
(r v : TreeVertex d) : Measurable (covariance beta E r v) := by
classical
unfold covariance LeafGibbs.covariance LeafGibbs.expectation GibbsLeaf.energy
fun_prop
lemma covariance_congr {d : ℕ} (beta : ℝ) (E : Finset (TreeVertex d))
(r v : TreeVertex d) {h k : TreeVertex d → ℝ}
(heq : ∀ x ∈ E, h x = k x) : covariance beta E r v h = covariance beta E r v k := by
have he : (fun eta => GibbsLeaf.energy beta h E (extendFiniteSpin E eta)) =
(fun eta => GibbsLeaf.energy beta k E (extendFiniteSpin E eta)) := by
funext eta
unfold GibbsLeaf.energy
congr 1
exact Finset.sum_congr rfl (fun x hx => by rw [heq x hx])
unfold covariance
rw [he]
lemma covariance_bounds {d : ℕ} {beta : ℝ} (hb : 0 ≤ beta)
{E : Finset (TreeVertex d)} {r v : TreeVertex d}
(hE : ConnectedFinset E) (hr : r ∈ E) (hv : v ∈ E)
(hmin : ∀ x ∈ E, r.1.length ≤ x.1.length) (h : TreeVertex d → ℝ) :
0 ≤ covariance beta E r v h ∧
covariance beta E r v h ≤ Real.tanh beta ^ (v.1.length - r.1.length) := by
simpa only [covariance, extendFiniteSpin, dite_eq_left hr, dite_eq_left hv] using
tree_gibbs_root_covariance_bounds hb hE hr hmin h v hv
lemma covariance_erase {d : ℕ} {E : Finset (TreeVertex d)}
{w p r v : TreeVertex d} (beta : ℝ) (h : TreeVertex d → ℝ)
(hw : w ∈ E) (hp : p ∈ E.erase w) (hr : r ∈ E.erase w) (hv : v ∈ E.erase w)
(hedges : inducedEdges E = insert (p, w) (inducedEdges (E.erase w))) :
covariance beta E r v h = covariance beta (E.erase w) r v
(Function.update h p (h p + cavityMessage (Real.tanh beta) (h w))) := by
simpa [covariance, extendFiniteSpin, (Finset.mem_erase.mp hr).2,
(Finset.mem_erase.mp hv).2, hr, hv, finiteSpinEraseEquiv] using
tree_gibbs_covariance_erase beta h hw hp hedges
(fun eta => spinSign (eta ⟨r, hr⟩)) (fun eta => spinSign (eta ⟨v, hv⟩))
lemma covariance_leaf {d : ℕ} {E : Finset (TreeVertex d)}
{w p r : TreeVertex d} (beta : ℝ) (h : TreeVertex d → ℝ)
(hw : w ∈ E) (hp : p ∈ E.erase w) (hr : r ∈ E.erase w)
(hedges : inducedEdges E = insert (p, w) (inducedEdges (E.erase w))) :
covariance beta E r w h =
(Real.tanh beta * responseAttenuation (Real.tanh beta) (h w)) *
covariance beta (E.erase w) r p
(Function.update h p (h p + cavityMessage (Real.tanh beta) (h w))) := by
have hleaf := tree_gibbs_covariance_leaf beta h hw hp hedges
(fun eta => spinSign (eta ⟨r, hr⟩))
have he : covariance beta E r w h =
(Real.tanh beta * responseAttenuation (Real.tanh beta) (h w)) *
covariance beta E r p h := by
simpa [covariance, extendFiniteSpin, (Finset.mem_erase.mp hr).2,
(Finset.mem_erase.mp hp).2, hw, finiteSpinEraseEquiv] using hleaf
rw [he, covariance_erase beta h hw hp hr hp hedges]
lemma measurable_covariance_sq {d : ℕ} (beta : ℝ) (E : Finset (TreeVertex d))
(r v : TreeVertex d) (h : TreeVertex d → ℝ) :
Measurable (fun x : TreeVertex d → ℝ => ENNReal.ofReal
(covariance beta E r v (fun u => h u + x u) ^ 2)) := by
exact (((measurable_covariance beta E r v).comp
(Measurable.of_eval (fun u => measurable_const.add (measurable_pi_apply u)))).pow_const 2).ennreal_ofReal
lemma covariance_updated_fields {d : ℕ} {E : Finset (TreeVertex d)}
{w p : TreeVertex d} (beta : ℝ) (r v : TreeVertex d)
(hp : p ∈ E.erase w) (h x : TreeVertex d → ℝ) (z : ℝ)
(y : E.erase w → ℝ) :
covariance beta (E.erase w) r v
(Function.update (fun u => h u + Function.updateFinset (Function.update x w z) (E.erase w) y u)
p (h p + Function.updateFinset (Function.update x w z) (E.erase w) y p +
cavityMessage (Real.tanh beta)
(h w + Function.updateFinset (Function.update x w z) (E.erase w) y w))) =
covariance beta (E.erase w) r v
(fun u => Function.update h p (h p + cavityMessage (Real.tanh beta) (h w + z)) u +
Function.updateFinset x (E.erase w) y u) := by
classical
apply covariance_congr
intro u hu
by_cases hup : u = p
· subst u
simp only [Function.update_self, Function.updateFinset, dite_eq_left hp,
Finset.notMem_erase, dite_false, Function.update_self]
ring
· simp only [Function.update_of_ne hup, Function.updateFinset, dite_eq_left hu]
lemma lmarginal_leaf {d : ℕ} {E : Finset (TreeVertex d)}
{w p r : TreeVertex d} (beta : ℝ) (h x : TreeVertex d → ℝ) (z : ℝ)
(hw : w ∈ E) (hp : p ∈ E.erase w) (hr : r ∈ E.erase w)
(hedges : inducedEdges E = insert (p, w) (inducedEdges (E.erase w)))
(μ : TreeVertex d → Measure ℝ) :
lmarginal μ (E.erase w)
(fun y => ENNReal.ofReal (covariance beta E r w (fun u => h u + y u) ^ 2))
(Function.update x w z) =
ENNReal.ofReal (Real.tanh beta ^ 2 * responseAttenuation (Real.tanh beta) (h w + z) ^ 2) *
lmarginal μ (E.erase w)
(fun y => ENNReal.ofReal (covariance beta (E.erase w) r p
(fun u => Function.update h p (h p + cavityMessage (Real.tanh beta) (h w + z)) u + y u) ^ 2)) x := by
classical
unfold lmarginal
rw [← lintegral_const_mul' _ _ (ENNReal.ofReal_ne_top)]
apply lintegral_congr
intro y
dsimp only
rw [covariance_leaf beta _ hw hp hr hedges, covariance_updated_fields beta r p hp h x z y]
simp only [Function.updateFinset, Finset.notMem_erase, dite_false, Function.update_self,
mul_pow, ENNReal.ofReal_mul (mul_nonneg (sq_nonneg _) (sq_nonneg _))]
lemma lmarginal_nonleaf {d : ℕ} {E : Finset (TreeVertex d)}
{w p r v : TreeVertex d} (beta : ℝ) (h x : TreeVertex d → ℝ) (z : ℝ)
(hw : w ∈ E) (hp : p ∈ E.erase w) (hr : r ∈ E.erase w) (hv : v ∈ E.erase w)
(hedges : inducedEdges E = insert (p, w) (inducedEdges (E.erase w)))
(μ : TreeVertex d → Measure ℝ) :
lmarginal μ (E.erase w)
(fun y => ENNReal.ofReal (covariance beta E r v (fun u => h u + y u) ^ 2))
(Function.update x w z) =
lmarginal μ (E.erase w)
(fun y => ENNReal.ofReal (covariance beta (E.erase w) r v
(fun u => Function.update h p (h p + cavityMessage (Real.tanh beta) (h w + z)) u + y u) ^ 2)) x := by
classical
unfold lmarginal
apply lintegral_congr
intro y
dsimp only
rw [covariance_erase beta _ hw hp hr hv hedges, covariance_updated_fields beta r v hp h x z y]
theorem lmarginal_covariance_sq_le
{d : ℕ} {beta : ℝ} (hb : 0 ≤ beta)
(μ : TreeVertex d → Measure ℝ) [∀ u, IsProbabilityMeasure (μ u)]
(q : ℝ≥0∞)
(hq : ∀ (u : TreeVertex d) (a : ℝ),
(∫⁻ z, ENNReal.ofReal (responseAttenuation (Real.tanh beta) (a + z) ^ 2) ∂μ u) ≤ q)
{E : Finset (TreeVertex d)} {r : TreeVertex d}
(hE : ConnectedFinset E) (hr : r ∈ E)
(hmin : ∀ u ∈ E, r.1.length ≤ u.1.length)
(h x : TreeVertex d → ℝ) (v : TreeVertex d) (hv : v ∈ E) :
lmarginal μ E (fun y => ENNReal.ofReal
(covariance beta E r v (fun u => h u + y u) ^ 2)) x ≤
(ENNReal.ofReal (Real.tanh beta ^ 2) * q) ^ (v.1.length - r.1.length) := by
classical
revert hE hr hmin h x v
refine Finset.strongInductionOn E ?_
intro E ih hE hr hmin h x v hv
by_cases hvr : v = r
· subst v
simp only [Nat.sub_self, pow_zero]
unfold lmarginal
calc
_ ≤ ∫⁻ _ : E → ℝ, (1 : ℝ≥0∞) ∂Measure.pi (fun u : E => μ u) := by
apply lintegral_mono
intro y
apply ENNReal.ofReal_le_one.mpr
have hc := covariance_bounds hb hE hr hr hmin
(fun u => h u + Function.updateFinset x E y u)
simp only [Nat.sub_self, pow_zero] at hc
nlinarith
_ = 1 := by simp
have hsub : ({r} : Finset (TreeVertex d)) ⊆ E := Finset.singleton_subset_iff.mpr hr
have hne : ({r} : Finset (TreeVertex d)) ≠ E := by
intro heq
exact hvr (Finset.mem_singleton.mp (heq.symm ▸ hv))
obtain ⟨w, p, hwE, _, hpE, hpw, hEr, hsubr, hedges⟩ :=
(connectedFinset_singleton r).exists_prunable_vertex_of_shared_min
hE hsub (Finset.mem_singleton_self r) hmin hne
have hrEr : r ∈ E.erase w := hsubr (Finset.mem_singleton_self r)
have hpEr : p ∈ E.erase w := Finset.mem_erase.mpr ⟨hpw, hpE⟩
have hErE : E.erase w ⊆ E := Finset.erase_subset _ _
have hmin' : ∀ u ∈ E.erase w, r.1.length ≤ u.1.length :=
fun u hu => hmin u (hErE hu)
rw [lmarginal_erase _ (measurable_covariance_sq beta E r v h) hwE x]
by_cases hvw : v = w
· subst v
have hedge : (p, w) ∈ inducedEdges E := by
rw [hedges]
exact Finset.mem_insert_self _ _
have hdata := Finset.mem_filter.mp hedge
have hadj : treeAdj p w := hdata.2.1
have hlower : p.1.length < w.1.length := hdata.2.2
have hlen := treeAdj_length hadj
have hdist : w.1.length - r.1.length = (p.1.length - r.1.length) + 1 := by
have := hmin p hpE
omega
let B := ENNReal.ofReal (Real.tanh beta ^ 2) * q
have hm : Measurable (fun z : ℝ => ENNReal.ofReal
(responseAttenuation (Real.tanh beta) (h w + z) ^ 2)) := by
simp only [responseAttenuation, Real.tanh_eq_sinh_div_cosh]
fun_prop
calc
_ ≤ ∫⁻ z, ENNReal.ofReal (Real.tanh beta ^ 2 *
responseAttenuation (Real.tanh beta) (h w + z) ^ 2) *
B ^ (p.1.length - r.1.length) ∂μ w := by
apply lintegral_mono
intro z
dsimp only
rw [lmarginal_leaf beta h x z hwE hpEr hrEr hedges]
exact mul_le_mul_right (ih (E.erase w) (Finset.erase_ssubset hwE)
hEr hrEr hmin' _ x p hpEr) _
_ = ENNReal.ofReal (Real.tanh beta ^ 2) *
(∫⁻ z, ENNReal.ofReal (responseAttenuation (Real.tanh beta) (h w + z) ^ 2) ∂μ w) *
B ^ (p.1.length - r.1.length) := by
simp_rw [ENNReal.ofReal_mul (sq_nonneg (Real.tanh beta))]
have hcm : Measurable (fun z : ℝ => ENNReal.ofReal (Real.tanh beta ^ 2) *
ENNReal.ofReal (responseAttenuation (Real.tanh beta) (h w + z) ^ 2)) :=
measurable_const.mul hm
rw [lintegral_mul_const _ hcm, lintegral_const_mul _ hm]
_ ≤ ENNReal.ofReal (Real.tanh beta ^ 2) * q * B ^ (p.1.length - r.1.length) := by
gcongr
exact hq w (h w)
_ = _ := by
rw [hdist]
change B * B ^ (p.1.length - r.1.length) = B ^ ((p.1.length - r.1.length) + 1)
exact (pow_succ' B _).symm
· have hvEr : v ∈ E.erase w := Finset.mem_erase.mpr ⟨hvw, hv⟩
calc
_ ≤ ∫⁻ _ : ℝ, (ENNReal.ofReal (Real.tanh beta ^ 2) * q) ^
(v.1.length - r.1.length) ∂μ w := by
apply lintegral_mono
intro z
dsimp only
rw [lmarginal_nonleaf beta h x z hwE hpEr hrEr hvEr hedges]
exact ih (E.erase w) (Finset.erase_ssubset hwE) hEr hrEr hmin' _ x v hvEr
_ = _ := by simp
def gaussianContraction (beta : ℝ) (δ : ℝ≥0) : ℝ :=
1 - ((1 - Real.tanh beta ^ 2) * Real.tanh (gaussianSmallRadius δ) ^ 2) / 2
lemma gaussianContraction_pos (beta : ℝ) (δ : ℝ≥0) :
0 < gaussianContraction beta δ := by
have ht := Real.tanh_sq_lt_one (gaussianSmallRadius δ)
have hp := mul_nonneg (sq_nonneg (Real.tanh beta))
(sq_nonneg (Real.tanh (gaussianSmallRadius δ)))
unfold gaussianContraction
nlinarith
lemma gaussianContraction_lt_one (beta : ℝ) {δ : ℝ≥0} (hδ : 0 < δ) :
gaussianContraction beta δ < 1 := by
have ht : 0 < Real.tanh (gaussianSmallRadius δ) := by
rw [Real.tanh_eq_sinh_div_cosh]
exact div_pos (Real.sinh_pos_iff.mpr (gaussianSmallRadius_pos hδ)) (Real.cosh_pos _)
have hg : 0 < (1 - Real.tanh beta ^ 2) * Real.tanh (gaussianSmallRadius δ) ^ 2 :=
mul_pos (sub_pos.mpr (Real.tanh_sq_lt_one beta)) (sq_pos_of_pos ht)
unfold gaussianContraction
linarith
lemma lintegral_gaussian_attenuation_le (beta m a : ℝ) {δ t : ℝ≥0}
(hδ : 0 < δ) (ht : δ ≤ t) :
(∫⁻ z, ENNReal.ofReal (responseAttenuation (Real.tanh beta) (a + z) ^ 2)
∂gaussianReal m t) ≤ ENNReal.ofReal (gaussianContraction beta δ) := by
have hm : Measurable (fun z : ℝ => responseAttenuation (Real.tanh beta) (a + z) ^ 2) := by
simp only [responseAttenuation, Real.tanh_eq_sinh_div_cosh]
fun_prop
have hint : Integrable (fun z : ℝ => responseAttenuation (Real.tanh beta) (a + z) ^ 2)
(gaussianReal m t) := by
apply (integrable_const (1 : ℝ)).mono' hm.aestronglyMeasurable
exact Filter.Eventually.of_forall fun z => by
rw [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg _)]
exact GaussianAttenuation.attenuation_sq_le_one (Real.tanh_sq_lt_one beta).le _
rw [← ofReal_integral_eq_lintegral_ofReal hint (Filter.Eventually.of_forall fun _ => sq_nonneg _)]
apply ENNReal.ofReal_le_ofReal
have hlaw : HasLaw (fun z : ℝ => a + z) (gaussianReal (m + a) t) (gaussianReal m t) := by
refine ⟨(measurable_const.add measurable_id).aemeasurable, ?_⟩
simpa only [add_comm a] using (gaussianReal_map_add_const (μ := m) (v := t) a)
exact GaussianAttenuation.integral_attenuation_sq_le_of_hasLaw hlaw
(Real.tanh_sq_lt_one beta) hδ ht
theorem lintegral_gaussian_covariance_sq_le
{d : ℕ} {beta : ℝ} (hb : 0 ≤ beta)
(m : TreeVertex d → ℝ) (t : TreeVertex d → ℝ≥0)
{δ : ℝ≥0} (hδ : 0 < δ) (ht : ∀ u, δ ≤ t u)
{E : Finset (TreeVertex d)} {r : TreeVertex d}
(hE : ConnectedFinset E) (hr : r ∈ E)
(hmin : ∀ u ∈ E, r.1.length ≤ u.1.length)
(h x : TreeVertex d → ℝ) (v : TreeVertex d) (hv : v ∈ E) :
(∫⁻ y : E → ℝ, ENNReal.ofReal (covariance beta E r v
(fun u => h u + Function.updateFinset x E y u) ^ 2)
∂Measure.pi (fun u : E => gaussianReal (m u) (t u))) ≤
ENNReal.ofReal ((Real.tanh beta ^ 2 * gaussianContraction beta δ) ^
(v.1.length - r.1.length)) := by
have he := lmarginal_covariance_sq_le hb (fun u => gaussianReal (m u) (t u))
(ENNReal.ofReal (gaussianContraction beta δ))
(fun u a => lintegral_gaussian_attenuation_le beta (m u) a hδ (ht u))
hE hr hmin h x v hv
simpa only [lmarginal, ← ENNReal.ofReal_mul (sq_nonneg (Real.tanh beta)),
← ENNReal.ofReal_pow (mul_nonneg (sq_nonneg _) (gaussianContraction_pos beta δ).le)] using he
lemma integrable_covariance_sq {d : ℕ} {beta : ℝ} (hb : 0 ≤ beta)
{Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsFiniteMeasure P]
{X : Ω → TreeVertex d → ℝ} (hX : Measurable X)
{E : Finset (TreeVertex d)} {r : TreeVertex d}
(hE : ConnectedFinset E) (hr : r ∈ E)
(hmin : ∀ u ∈ E, r.1.length ≤ u.1.length)
(h : TreeVertex d → ℝ) (v : TreeVertex d) (hv : v ∈ E) :
Integrable (fun ω => covariance beta E r v (fun u => h u + X ω u) ^ 2) P := by
have hm : Measurable (fun ω => covariance beta E r v (fun u => h u + X ω u) ^ 2) :=
((measurable_covariance beta E r v).comp
(Measurable.of_eval (fun u => measurable_const.add
((measurable_pi_apply u).comp hX)))).pow_const 2
apply (integrable_const (1 : ℝ)).mono' hm.aestronglyMeasurable
refine Filter.Eventually.of_forall fun ω => ?_
rw [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg _)]
have hc := covariance_bounds hb hE hr hv hmin (fun u => h u + X ω u)
have ht : 0 ≤ Real.tanh beta := by
rw [Real.tanh_eq_sinh_div_cosh]
exact div_nonneg (Real.sinh_nonneg_iff.mpr hb) (Real.cosh_pos beta).le
have hp : Real.tanh beta ^ (v.1.length - r.1.length) ≤ 1 :=
pow_le_one₀ ht (Real.tanh_lt_one beta).le
nlinarith
theorem integral_gaussian_covariance_sq_le
{d : ℕ} {beta : ℝ} (hb : 0 ≤ beta)
(m : TreeVertex d → ℝ) (t : TreeVertex d → ℝ≥0)
{δ : ℝ≥0} (hδ : 0 < δ) (ht : ∀ u, δ ≤ t u)
{E : Finset (TreeVertex d)} {r : TreeVertex d}
(hE : ConnectedFinset E) (hr : r ∈ E)
(hmin : ∀ u ∈ E, r.1.length ≤ u.1.length)
(h x : TreeVertex d → ℝ) (v : TreeVertex d) (hv : v ∈ E) :
(∫ y : E → ℝ, covariance beta E r v
(fun u => h u + Function.updateFinset x E y u) ^ 2
∂Measure.pi (fun u : E => gaussianReal (m u) (t u))) ≤
(Real.tanh beta ^ 2 * gaussianContraction beta δ) ^
(v.1.length - r.1.length) := by
apply (ENNReal.ofReal_le_ofReal_iff
(pow_nonneg (mul_nonneg (sq_nonneg _) (gaussianContraction_pos beta δ).le) _)).mp
have hint := integrable_covariance_sq hb
(P := Measure.pi (fun u : E => gaussianReal (m u) (t u)))
(X := fun y => Function.updateFinset x E y) (by fun_prop)
hE hr hmin h v hv
rw [ofReal_integral_eq_lintegral_ofReal hint (Filter.Eventually.of_forall fun _ => sq_nonneg _)]
exact lintegral_gaussian_covariance_sq_le hb m t hδ ht hE hr hmin h x v hv
end Problem367.GaussianTreeCovariance
end
end OAI
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