层3 E2: estimator.py——式(4) 几何均值先验 chunk_prior + 式(5) 贝叶斯目标 bayesian_target,17 单测
对应 docs/04 §4 E2。detach 双防线(chunk_prior 内为本质防线、 bayesian_target 末尾为防御性第二道,分别对应参考 trainer:2196/2205); log 域抗下溢;clamp 下限守定理 4.1(b);k_sem∈[0,N] 口径校验; 方差收缩 toy 模拟对拍 validate_chunk_mc_estimator.py 精神 (MSE_bayes < MSE_freq,含劣先验 ±0.1 稳健性)。 Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
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"""MC 估计与 Dirichlet 贝叶斯平滑——论文 §3.2.2 式(4)(5)。
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把 similarity.py 产出的软计数 k_sem(外部 teacher 信号)与学生自身的
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chunk 置信度 π̄(内部先验)融合成有界目标 π̂ ∈ (0, 1],供层 5 的 chunk
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损失当乘子:loss_c = −π̂ · mean(log p)。定理 4.1 三性质由此获得:
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(a) π̂ 有界 → 无白盒式(2) 的梯度爆炸;(b) π̂ > 0 → k_sem=0 也不塌缩;
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(c) 先验收缩 → 方差小于频率估计 k/N。
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纯逻辑模块(CLAUDE.md §2):只依赖 torch,toy 张量本地 CPU 可测。
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"""
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import torch
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def chunk_prior(log_probs: torch.Tensor) -> torch.Tensor:
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"""式(4):π̄ = exp((1/C)·Σ_t log p_t)——学生对整个 chunk 的几何均值置信度。
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C 个 token 概率的几何均值,充当式(5) 的贝叶斯先验:teacher 采样(k_sem)
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是主信号,π̄ 只是"学生自己觉得这段有多稳"的地板,防 k_sem=0 时目标归零。
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参数:
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log_probs: 学生对 chunk 内各 token 的对数概率,shape (C,),值 ≤ 0。
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(调用方从 log_softmax 后 gather 标签位置所得,层 5 负责。)
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返回:
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π̄,标量张量 shape ()、值 ∈ [1e-8, 1],**不带梯度**。
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实现细节:
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- log 域先均值再 exp:直接连乘 C=50 个小概率会下溢
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(50 个 0.01 → 1e-100,超出 fp32 下限 ~1e-38),log 域安全。
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- detach 命门(参考实现 distillation_trainer.py:2196 同):π̄ 是学生
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自身概率的函数,若保留梯度,优化器会发现"压低自己的 chunk 概率
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→ π̄→0 → π̂ 变小 → 损失权重变小"这条逃逸路径——恰在 k_sem=0
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(teacher 否定)的 chunk 上最有利可图,这些 chunk 最先塌缩。
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π̄ 只能当常数先验,不能当优化变量。锁死断言见
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tests/test_estimator_detach.py。
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- clamp 下限 1e-8:极端负的均值 exp 后可能下溢为 0,而定理 4.1(b)
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的反塌缩要求 π̄ 严格为正。差异标注:参考实现 clamp(1e-8, 1.0),
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上限实为冗余——log p ≤ 0 ⇒ mean ≤ 0 ⇒ exp ≤ 1,此处省去。
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"""
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if log_probs.numel() == 0:
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raise ValueError("log_probs 为空:chunk 至少要含 1 个 token")
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log_pi_bar = log_probs.detach().mean() # (C,) -> ()
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return log_pi_bar.exp().clamp(min=1e-8)
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def bayesian_target(
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k_sem: float,
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pi_bar: torch.Tensor,
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n_rollouts: int,
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alpha: float,
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) -> torch.Tensor:
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"""式(5):π̂ = (k_sem + α·π̄) / (N + α)——chunk 接受概率的贝叶斯估计。
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等价凸组合视角(论文式10):
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π̂ = N/(N+α) · (k_sem/N) + α/(N+α) · π̄
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即"teacher 频率估计"与"学生先验"的加权平均;默认 N=10、α=1 时权重
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约 91% : 9%,teacher 主导,先验只兜底。
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参数:
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k_sem: 式(3) 的软匹配计数,∈ [0, N](aggregate_similarity 产出)。
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pi_bar: 式(4) 的先验 π̄,标量张量(chunk_prior 产出)。
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n_rollouts: teacher rollout 数 N。**必须等于算 k_sem 时的
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len(teacher_rollouts)**——分子分母口径不一致会系统性偏移 π̂。
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alpha: 先验强度 α ≥ 0。α=0 退化为频率估计 k/N(层 6 的
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no_bayesian 消融,参考 config.py:315),失去定理 4.1(b) 保护。
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返回:
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π̂,标量张量 shape ()、值 ∈ [1e-8, 1],**不带梯度**。
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实现细节:
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- 差异标注:参考实现(distillation_trainer.py:2205)在此对 π̂ 整体
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detach;我们的 π̄ 在 chunk_prior 内已 detach,此处的 detach 是
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第二道防线——防止将来有人把带梯度的张量传进 pi_bar。
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- clamp(1e-8, 1.0):下限防 α=0 且 k_sem=0 时 π̂=0(乘子归零则该
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chunk 完全失去监督);上限防 pi_bar 越界传入时 π̂ 溢出概率语义。
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"""
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if n_rollouts < 1:
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raise ValueError(f"n_rollouts 必须 ≥ 1,得到 {n_rollouts}")
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if alpha < 0:
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raise ValueError(f"alpha 必须 ≥ 0,得到 {alpha}")
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if not 0.0 <= k_sem <= n_rollouts:
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raise ValueError(
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f"k_sem={k_sem} 越界 [0, {n_rollouts}]:检查是否与"
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f" len(teacher_rollouts) 口径一致"
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)
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# 式(5): π̂ = (k_sem + α·π̄) / (N + α)
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pi_hat = (k_sem + alpha * pi_bar) / (n_rollouts + alpha) # () -> ()
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return pi_hat.clamp(1e-8, 1.0).detach()
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"""estimator.py 单测——docs/04 §5.2:公式对拍 + 定理 4.1 性质 + 方差收缩。
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对拍精神源自参考实现 validate_chunk_mc_estimator.py(比 MSE_freq vs
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MSE_bayes),但全用 toy 数据本地 CPU 跑,不连真 teacher。
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detach 命门的两个世界断言在 tests/test_estimator_detach.py(E3)。
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"""
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import math
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import pytest
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import torch
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from ars_opd.estimator import bayesian_target, chunk_prior
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# ------------------------------------------------------------- chunk_prior
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def test_prior_is_geometric_mean():
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# 式(4) 手算:p = [0.9, 0.1] → π̄ = exp((log .9 + log .1)/2) = √0.09 = 0.3
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log_probs = torch.log(torch.tensor([0.9, 0.1]))
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assert math.isclose(chunk_prior(log_probs).item(), 0.3, rel_tol=1e-6)
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def test_prior_uniform_probs():
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# 全同概率的几何均值 = 该概率本身
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log_probs = torch.full((50,), math.log(0.5))
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assert math.isclose(chunk_prior(log_probs).item(), 0.5, rel_tol=1e-6)
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def test_prior_shape_and_range():
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pi_bar = chunk_prior(torch.log(torch.rand(50).clamp(1e-6, 1.0)))
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assert pi_bar.shape == () # (C,) -> 标量
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assert 0.0 < pi_bar.item() <= 1.0
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def test_prior_log_domain_survives_underflow():
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# 50 个 p=0.01 直接连乘 = 1e-100(fp32 下溢为 0);log 域算出 0.01
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log_probs = torch.full((50,), math.log(0.01))
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assert math.isclose(chunk_prior(log_probs).item(), 0.01, rel_tol=1e-4)
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def test_prior_clamp_floor():
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# 极端负 log 均值 → exp 下溢,clamp 兜到 1e-8 保持严格为正(定理 4.1b 前提)
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log_probs = torch.full((5,), -1e9)
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assert chunk_prior(log_probs).item() == pytest.approx(1e-8)
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def test_prior_is_detached():
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# detach 命门:π̄ 不带梯度(逃逸机制的完整断言在 test_estimator_detach.py)
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log_probs = torch.log(torch.tensor([0.5, 0.5], requires_grad=True))
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pi_bar = chunk_prior(log_probs)
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assert not pi_bar.requires_grad
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def test_prior_empty_raises():
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with pytest.raises(ValueError, match="为空"):
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chunk_prior(torch.tensor([]))
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# --------------------------------------------------------- bayesian_target
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def test_target_formula_hand_computed():
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# 式(5) 手算:k=8, π̄=0.5, N=10, α=1 → π̂ = (8 + 0.5)/11 = 0.77272…
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pi_hat = bayesian_target(8.0, torch.tensor(0.5), n_rollouts=10, alpha=1.0)
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assert math.isclose(pi_hat.item(), 8.5 / 11, rel_tol=1e-6)
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def test_target_convex_combination_identity():
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# 式(10) 恒等:π̂ = N/(N+α)·(k/N) + α/(N+α)·π̄,任取参数逐点核对
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k, pi_bar, n, alpha = 3.7, torch.tensor(0.42), 10, 1.5
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direct = bayesian_target(k, pi_bar, n, alpha).item()
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convex = (n / (n + alpha)) * (k / n) + (alpha / (n + alpha)) * pi_bar.item()
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assert math.isclose(direct, convex, rel_tol=1e-6)
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def test_target_anti_collapse_at_k_zero():
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# 定理 4.1(b):k=0(teacher 全否定)时 π̂ = α·π̄/(N+α) > 0,监督不归零
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pi_hat = bayesian_target(0.0, torch.tensor(0.3), n_rollouts=10, alpha=1.0)
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assert math.isclose(pi_hat.item(), 0.3 / 11, rel_tol=1e-6)
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assert pi_hat.item() > 0
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def test_target_full_score_shrinks_below_one():
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# 贝叶斯收缩:k=N 满分时 π̂ = (N+α·π̄)/(N+α) < 1(只要 π̄<1)——
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# 先验把估计从两端往中间拉,这正是方差收缩的来源
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pi_hat = bayesian_target(10.0, torch.tensor(0.5), n_rollouts=10, alpha=1.0)
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assert math.isclose(pi_hat.item(), 10.5 / 11, rel_tol=1e-6)
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assert pi_hat.item() < 1.0
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def test_target_bounded_in_unit_interval():
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# 定理 4.1(a):任意合法参数下 π̂ ∈ (0, 1]
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for k in [0.0, 2.5, 10.0]:
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for p in [1e-8, 0.5, 1.0]:
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v = bayesian_target(k, torch.tensor(p), 10, 1.0).item()
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assert 0.0 < v <= 1.0
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def test_target_alpha_zero_is_frequency_estimate():
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# α=0 退化为 k/N(no_bayesian 消融);k=0 时被 clamp 兜到 1e-8 而非 0
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assert math.isclose(
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bayesian_target(7.0, torch.tensor(0.5), 10, 0.0).item(), 0.7, rel_tol=1e-6
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)
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assert bayesian_target(0.0, torch.tensor(0.5), 10, 0.0).item() == pytest.approx(
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1e-8
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)
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def test_target_is_detached_even_with_grad_input():
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# 第二道防线:pi_bar 带梯度传入,π̂ 仍必须 detach
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pi_bar = torch.tensor(0.5, requires_grad=True)
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pi_hat = bayesian_target(5.0, pi_bar, 10, 1.0)
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assert not pi_hat.requires_grad
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def test_target_validation_raises():
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pi_bar = torch.tensor(0.5)
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with pytest.raises(ValueError, match="n_rollouts"):
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bayesian_target(0.0, pi_bar, 0, 1.0)
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with pytest.raises(ValueError, match="alpha"):
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bayesian_target(0.0, pi_bar, 10, -0.1)
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with pytest.raises(ValueError, match="越界"):
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bayesian_target(11.0, pi_bar, 10, 1.0) # k_sem > N:口径不一致
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with pytest.raises(ValueError, match="越界"):
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bayesian_target(-0.5, pi_bar, 10, 1.0)
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# --------------------------------------------- 方差收缩(定理 4.1c,toy 模拟)
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def test_variance_shrinkage_beats_frequency_estimate():
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"""toy 模拟对拍 validate_chunk_mc_estimator.py 的 MSE_freq vs MSE_bayes。
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设真值 μ:每次试验采 N=10 个相似度 sim_i(均值 μ 的噪声),
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频率估计 = mean(sim) = k/N,贝叶斯估计 = (k + α·π̄)/(N+α)。
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先验 π̄ = μ(理想先验)时,收缩纯降方差、零偏差代价,MSE 必更小。
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"""
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torch.manual_seed(0)
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mu, n, alpha = 0.7, 10, 1.0
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trials = 2000
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# (trials, N) 的相似度样本:均值 μ、截断到 [0,1]
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sims = (mu + 0.25 * torch.randn(trials, n)).clamp(0.0, 1.0)
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k = sims.sum(dim=1) # (trials,) 每次试验的 k_sem
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freq = k / n
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bayes = (k + alpha * mu) / (n + alpha)
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mse_freq = ((freq - mu) ** 2).mean().item()
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mse_bayes = ((bayes - mu) ** 2).mean().item()
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assert mse_bayes < mse_freq
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def test_variance_shrinkage_robust_to_imperfect_prior():
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# 先验偏离真值(π̄ = μ±0.1)仍应赢:α=1、N=10 时先验权重仅 1/11,
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# 引入的偏差平方远小于省下的方差(定理 4.1c 在论文设定下的稳健性)
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torch.manual_seed(1)
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mu, n, alpha, trials = 0.6, 10, 1.0, 2000
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sims = (mu + 0.25 * torch.randn(trials, n)).clamp(0.0, 1.0)
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k = sims.sum(dim=1)
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mse_freq = ((k / n - mu) ** 2).mean().item()
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for prior in [mu - 0.1, mu + 0.1]:
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bayes = (k + alpha * prior) / (n + alpha)
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assert ((bayes - mu) ** 2).mean().item() < mse_freq
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