| import Mathlib |
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| namespace OAI |
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| noncomputable section |
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| open scoped BigOperators |
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| namespace Problem367.FinitePosterior |
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| variable {ι : Type*} [Fintype ι] |
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| def weight (energy signal : ι → ℝ) (t : ℝ) (s : ι) : ℝ := |
| Real.exp (energy s + t * signal s) |
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| def partition (energy signal : ι → ℝ) (t : ℝ) : ℝ := |
| ∑ s, weight energy signal t s |
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| def mean (energy signal observable : ι → ℝ) (t : ℝ) : ℝ := |
| (∑ s, observable s * weight energy signal t s) / partition energy signal t |
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| def mass (energy signal : ι → ℝ) (t : ℝ) (s : ι) : ℝ := |
| weight energy signal t s / partition energy signal t |
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| omit [Fintype ι] in |
| lemma weight_pos (energy signal : ι → ℝ) (t : ℝ) (s : ι) : |
| 0 < weight energy signal t s := Real.exp_pos _ |
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| lemma partition_pos [Nonempty ι] (energy signal : ι → ℝ) (t : ℝ) : |
| 0 < partition energy signal t := by |
| exact Finset.sum_pos (fun s _ => weight_pos energy signal t s) Finset.univ_nonempty |
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| lemma partition_ne_zero [Nonempty ι] (energy signal : ι → ℝ) (t : ℝ) : |
| partition energy signal t ≠ 0 := ne_of_gt (partition_pos energy signal t) |
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| lemma mass_pos [Nonempty ι] (energy signal : ι → ℝ) (t : ℝ) (s : ι) : |
| 0 < mass energy signal t s := |
| div_pos (weight_pos energy signal t s) (partition_pos energy signal t) |
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| lemma sum_mass [Nonempty ι] (energy signal : ι → ℝ) (t : ℝ) : |
| ∑ s, mass energy signal t s = 1 := by |
| simp only [mass, ← Finset.sum_div] |
| change partition energy signal t / partition energy signal t = 1 |
| exact div_self (partition_ne_zero energy signal t) |
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| lemma mean_eq_sum_mass (energy signal observable : ι → ℝ) (t : ℝ) : |
| mean energy signal observable t = ∑ s, observable s * mass energy signal t s := by |
| simp only [mean, mass, ← mul_div_assoc, ← Finset.sum_div] |
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| lemma mean_const [Nonempty ι] (energy signal : ι → ℝ) (t c : ℝ) : |
| mean energy signal (fun _ => c) t = c := by |
| rw [mean_eq_sum_mass, ← Finset.mul_sum, sum_mass, mul_one] |
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| omit [Fintype ι] in |
| lemma hasDerivAt_weight (energy signal : ι → ℝ) (t : ℝ) (s : ι) : |
| HasDerivAt (fun x => weight energy signal x s) |
| (signal s * weight energy signal t s) t := by |
| simpa [weight, mul_comm] using |
| (((hasDerivAt_id t).mul_const (signal s)).const_add (energy s)).exp |
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| lemma hasDerivAt_partition (energy signal : ι → ℝ) (t : ℝ) : |
| HasDerivAt (partition energy signal) |
| (∑ s, signal s * weight energy signal t s) t := by |
| exact HasDerivAt.fun_sum (fun s _ => hasDerivAt_weight energy signal t s) |
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| theorem hasDerivAt_mean [Nonempty ι] (energy signal observable : ι → ℝ) (t : ℝ) : |
| HasDerivAt (mean energy signal observable) |
| (mean energy signal (fun s => observable s * signal s) t - |
| mean energy signal observable t * mean energy signal signal t) t := by |
| have hn : HasDerivAt (fun x => ∑ s, observable s * weight energy signal x s) |
| (∑ s, (observable s * signal s) * weight energy signal t s) t := by |
| simpa only [mul_assoc] using HasDerivAt.fun_sum |
| (fun s (_ : s ∈ Finset.univ) => |
| (hasDerivAt_weight energy signal t s).const_mul (observable s)) |
| convert! hn.div (hasDerivAt_partition energy signal t) |
| (partition_ne_zero energy signal t) using 1 |
| dsimp [mean] |
| field_simp |
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| lemma hasDerivAt_log_partition [Nonempty ι] (energy signal : ι → ℝ) (t : ℝ) : |
| HasDerivAt (fun x => Real.log (partition energy signal x)) |
| (mean energy signal signal t) t := by |
| simpa only [mean] using (hasDerivAt_partition energy signal t).log |
| (partition_ne_zero energy signal t) |
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| lemma hasDerivAt_mass [Nonempty ι] (energy signal : ι → ℝ) (t : ℝ) (s : ι) : |
| HasDerivAt (fun x => mass energy signal x s) |
| ((signal s - mean energy signal signal t) * mass energy signal t s) t := by |
| convert! (hasDerivAt_weight energy signal t s).div |
| (hasDerivAt_partition energy signal t) (partition_ne_zero energy signal t) using 1 |
| dsimp [mass, mean] |
| field_simp [partition_ne_zero energy signal t] |
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| lemma variance_eq_sum [Nonempty ι] (energy signal observable : ι → ℝ) (t : ℝ) : |
| mean energy signal (fun s => (observable s) ^ 2) t - |
| (mean energy signal observable t) ^ 2 = |
| ∑ s, (observable s - mean energy signal observable t) ^ 2 * |
| mass energy signal t s := by |
| have heq (s : ι) : |
| (observable s - mean energy signal observable t) ^ 2 * mass energy signal t s = |
| (observable s) ^ 2 * mass energy signal t s - |
| (2 * mean energy signal observable t) * (observable s * mass energy signal t s) + |
| (mean energy signal observable t) ^ 2 * mass energy signal t s := by ring |
| simp_rw [heq] |
| rw [Finset.sum_add_distrib, Finset.sum_sub_distrib, |
| ← Finset.mul_sum, ← Finset.mul_sum, sum_mass, |
| ← mean_eq_sum_mass, ← mean_eq_sum_mass] |
| ring |
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| lemma variance_nonneg [Nonempty ι] (energy signal observable : ι → ℝ) (t : ℝ) : |
| 0 ≤ mean energy signal (fun s => (observable s) ^ 2) t - |
| (mean energy signal observable t) ^ 2 := by |
| rw [variance_eq_sum] |
| exact Finset.sum_nonneg (fun s _ => mul_nonneg (sq_nonneg _) (mass_pos energy signal t s).le) |
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| lemma variance_pos [Nonempty ι] (energy signal observable : ι → ℝ) (t : ℝ) |
| (h : ∃ i j, observable i ≠ observable j) : |
| 0 < mean energy signal (fun s => (observable s) ^ 2) t - |
| (mean energy signal observable t) ^ 2 := by |
| rw [variance_eq_sum] |
| apply Finset.sum_pos' |
| · intro s _ |
| exact mul_nonneg (sq_nonneg _) (mass_pos energy signal t s).le |
| · obtain ⟨i, j, hij⟩ := h |
| have hex : ∃ s, observable s ≠ mean energy signal observable t := by |
| by_contra! hn |
| exact hij ((hn i).trans (hn j).symm) |
| obtain ⟨s, hs⟩ := hex |
| exact ⟨s, Finset.mem_univ s, |
| mul_pos (sq_pos_of_ne_zero (sub_ne_zero.mpr hs)) (mass_pos energy signal t s)⟩ |
| |
| theorem hasDerivAt_spin_mean [Nonempty ι] (energy spin : ι → ℝ) (t : ℝ) |
| (hspin : ∀ s, spin s = -1 ∨ spin s = 1) : |
| HasDerivAt (mean energy spin spin) |
| (1 - (mean energy spin spin t) ^ 2) t := by |
| have heq : (fun s => spin s * spin s) = (fun _ : ι => (1 : ℝ)) := by |
| funext s |
| rcases hspin s with hs | hs <;> simp [hs] |
| have hd := hasDerivAt_mean energy spin spin t |
| rw [heq, mean_const] at hd |
| simpa only [← sq] using hd |
| |
| theorem mean_monotone [Nonempty ι] (energy signal : ι → ℝ) : |
| Monotone (mean energy signal signal) := by |
| apply monotone_of_deriv_nonneg |
| · intro t |
| exact (hasDerivAt_mean energy signal signal t).differentiableAt |
| · intro t |
| rw [(hasDerivAt_mean energy signal signal t).deriv] |
| simpa only [← sq] using variance_nonneg energy signal signal t |
| |
| lemma mean_lt [Nonempty ι] (energy signal observable : ι → ℝ) (t c : ℝ) |
| (hle : ∀ s, observable s ≤ c) (hlt : ∃ s, observable s < c) : |
| mean energy signal observable t < c := by |
| apply (div_lt_iff₀ (partition_pos energy signal t)).2 |
| rw [partition, Finset.mul_sum] |
| apply Finset.sum_lt_sum |
| · intro s _ |
| exact mul_le_mul_of_nonneg_right (hle s) (weight_pos energy signal t s).le |
| · obtain ⟨s, hs⟩ := hlt |
| exact ⟨s, Finset.mem_univ s, mul_lt_mul_of_pos_right hs (weight_pos energy signal t s)⟩ |
| |
| lemma lt_mean [Nonempty ι] (energy signal observable : ι → ℝ) (t c : ℝ) |
| (hle : ∀ s, c ≤ observable s) (hlt : ∃ s, c < observable s) : |
| c < mean energy signal observable t := by |
| apply (lt_div_iff₀ (partition_pos energy signal t)).2 |
| rw [partition, Finset.mul_sum] |
| apply Finset.sum_lt_sum |
| · intro s _ |
| exact mul_le_mul_of_nonneg_right (hle s) (weight_pos energy signal t s).le |
| · obtain ⟨s, hs⟩ := hlt |
| exact ⟨s, Finset.mem_univ s, mul_lt_mul_of_pos_right hs (weight_pos energy signal t s)⟩ |
| |
| theorem spin_mean_mem_Ioo [Nonempty ι] (energy signal spin : ι → ℝ) (t : ℝ) |
| (hspin : ∀ s, spin s = -1 ∨ spin s = 1) |
| (hneg : ∃ s, spin s = -1) (hpos : ∃ s, spin s = 1) : |
| mean energy signal spin t ∈ Set.Ioo (-1 : ℝ) 1 := by |
| constructor |
| · apply lt_mean energy signal spin t (-1) |
| · intro s |
| rcases hspin s with hs | hs <;> simp [hs] |
| · obtain ⟨s, hs⟩ := hpos |
| exact ⟨s, by simp [hs]⟩ |
| · apply mean_lt energy signal spin t 1 |
| · intro s |
| rcases hspin s with hs | hs <;> simp [hs] |
| · obtain ⟨s, hs⟩ := hneg |
| exact ⟨s, by simp [hs]⟩ |
| |
| end Problem367.FinitePosterior |
| |
| end |
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| end OAI |
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