File size: 3,096 Bytes
f61e547 | 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 | import OAI.Probability.FreeIsing.Posterior.FieldCalculus
namespace OAI
noncomputable section
open scoped BigOperators
namespace Problem367.FinitePosterior
variable {S : Type*} [Fintype S] [Nonempty S]
def sourceResponse (energy signal observable : S → ℝ) (t : ℝ) : ℝ :=
tiltCovariance energy signal observable t /
(1 - mean energy signal signal t ^ 2)
theorem hasDerivAt_sourceResponse (energy signal observable : S → ℝ) (t : ℝ)
(hsignal : ∀ s, signal s = -1 ∨ signal s = 1)
(hneg : ∃ s, signal s = -1) (hpos : ∃ s, signal s = 1) :
HasDerivAt (sourceResponse energy signal observable) 0 t := by
have hden := ne_of_gt (spin_mean_variance_pos energy signal signal t hsignal hneg hpos)
have hd := (hasDerivAt_tiltCovariance energy signal observable t hsignal).div
(((hasDerivAt_spin_mean energy signal t hsignal).pow 2).const_sub 1) hden
change HasDerivAt (sourceResponse energy signal observable) _ t at hd
apply hd.congr_deriv
simp only [Pi.pow_apply, Nat.cast_ofNat, Nat.reduceSub, pow_one]
field_simp [hden]
ring
theorem sourceResponse_eq (energy signal observable : S → ℝ) (t u : ℝ)
(hsignal : ∀ s, signal s = -1 ∨ signal s = 1)
(hneg : ∃ s, signal s = -1) (hpos : ∃ s, signal s = 1) :
sourceResponse energy signal observable t = sourceResponse energy signal observable u := by
exact is_const_of_deriv_eq_zero
(fun z => (hasDerivAt_sourceResponse energy signal observable z hsignal hneg hpos).differentiableAt)
(fun z => (hasDerivAt_sourceResponse energy signal observable z hsignal hneg hpos).deriv) t u
section Fields
variable {V : Type*} [Fintype V] [DecidableEq V]
omit [Nonempty S] in
theorem fieldMean_update_eq_mean (energy : S → ℝ) (spin : S → V → ℝ)
(observable : S → ℝ) (x : V → ℝ) (r : V) (t : ℝ) :
fieldMean energy spin observable (Function.update x r t) =
mean (fieldEnergy energy spin (Function.update x r 0))
(fun s => spin s r) observable t := by
have hx : Function.update x r t =
(fun v => Function.update x r 0 v + t * (if v = r then 1 else 0)) := by
funext v
by_cases hv : v = r <;> simp [hv]
rw [hx, fieldMean_along_line]
simp only [ite_mul, one_mul, zero_mul, Finset.sum_ite_eq', Finset.mem_univ, ite_true]
theorem normalizedFieldCovariance_update_source (energy : S → ℝ) (spin : S → V → ℝ)
(x : V → ℝ) (r v : V) (t u : ℝ)
(hsignal : ∀ s, spin s r = -1 ∨ spin s r = 1)
(hneg : ∃ s, spin s r = -1) (hpos : ∃ s, spin s r = 1) :
normalizedFieldCovariance energy spin (fun s => spin s r) (Function.update x r t) v =
normalizedFieldCovariance energy spin (fun s => spin s r) (Function.update x r u) v := by
have h := sourceResponse_eq (fieldEnergy energy spin (Function.update x r 0))
(fun s => spin s r) (fun s => spin s v) t u hsignal hneg hpos
unfold normalizedFieldCovariance fieldCovariance
simp only [fieldMean_update_eq_mean]
simpa only [sourceResponse, tiltCovariance, mul_comm] using h
end Fields
end Problem367.FinitePosterior
end
end OAI
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