不确定性估计Uncertainty Estimation

前面几章训练出来的网络都只回答一个数：这次抓取能成功的概率是 0.93，这个像素的深度是 1.52 m，这个动作之后关节会转到这里。机器人拿到这个数之后还得做决定：要不要执行这次抓取？这个深度值在卡尔曼滤波（4.8 节）里该给多大权重？是自己继续，还是停下来请人帮忙？这些决定需要的不只是“最可能的答案”，还有这个答案有多可能错。卡尔曼滤波里每个测量都必须带协方差，否则增益没法算；学出来的模型如果没有不确定性，就没法放进同一个框架里。

“不知道”有两种，处理方法完全不同。第一种来自世界本身：同一个距离，深度相机每次读数都在抖；同样的抓取姿态，物体有时滑、有时不滑。这叫偶然不确定性（aleatoric，源自拉丁语“骰子”），再多数据也消不掉，模型能做的只是把它的大小如实报告出来。第二种来自模型自己：数据没覆盖到的距离、没见过的物体，模型只能瞎猜。这叫认知不确定性（epistemic，源自希腊语“知识”），去那里采数据就能把它降下来。

机器人在乎这个区分，因为两种不确定性要求不同的动作。认知不确定性大：去采数据（主动学习），换个保守动作，或者请人帮忙。偶然不确定性大：采再多同样的数据也没用，该做的是换传感器、换策略，或在规划里留出余量。16.2 节已经看到，网络在训练范围外会自信地给出错误答案，集成的分歧只涨了一点。本节系统地讲怎样估计、检验和使用这两种不确定性。

The networks of the previous chapters each answer with a single number: this grasp succeeds with probability 0.93, this pixel is 1.52 m deep, after this action the joint will be here. The robot then still has to decide: execute the grasp or not? How much weight does that depth get in a Kalman filter (Lesson 4.8)? Carry on alone, or stop and ask a person? These decisions need more than "the most likely answer"; they need how likely that answer is to be wrong. In a Kalman filter every measurement must come with a covariance or the gain cannot be computed, and a learned model without uncertainty simply does not fit into that framework.

"Not knowing" comes in two kinds that are handled in completely different ways. The first comes from the world itself: at the same distance a depth camera's reading jitters from frame to frame; with the same grasp pose an object sometimes slips and sometimes does not. This is aleatoric uncertainty (from the Latin for "dice"). No amount of data removes it; the best a model can do is report its size honestly. The second comes from the model: at distances the data never covered, for objects it has never seen, the model can only guess. This is epistemic uncertainty (from the Greek for "knowledge"), and collecting data there brings it down.

Robots care about the distinction because the two call for different actions. Large epistemic uncertainty: collect data (active learning), fall back to a cautious action, or ask for help. Large aleatoric uncertainty: more of the same data will not help; change the sensor or the strategy, or leave a margin in the plan. Lesson 16.2 showed a network confidently wrong outside its training range while the disagreement of its ensemble grew only a little. This lesson is about how to estimate, check and use both kinds of uncertainty.

本节反复用到一个例子：一台廉价深度相机的读数有系统误差，误差大小随距离变化（距离 z 处的误差是一条带波动的曲线，单位 mm），而且读数噪声随距离增大。我们把标定板放在不同距离上，测出“读数减真实距离”，想学出这条误差曲线，用来修正以后的读数。

先用最简单、可以精确求解的模型：贝叶斯线性回归。把误差写成 25 个高斯小鼓包的加权和 \(y=\boldsymbol\phi(z)^\top\mathbf w+\varepsilon\)，权重有高斯先验，噪声 \(\varepsilon\sim\mathcal N(0,\sigma^2)\)。这样权重的后验也是高斯的（4.4 节的条件化，4.6 节的 MAP），新距离处的预测方差可以写成两项之和：

One example runs through this lesson. A cheap depth camera has a systematic error that depends on distance (at distance z the error follows a wavy curve, in mm), and its reading noise grows with distance. We place a calibration board at various distances, measure "reading minus true distance", and want to learn the error curve so as to correct future readings.

Start with the simplest model that can be solved exactly: Bayesian linear regression. Write the error as a weighted sum of 25 Gaussian bumps, \(y=\boldsymbol\phi(z)^\top\mathbf w+\varepsilon\), with a Gaussian prior on the weights and noise \(\varepsilon\sim\mathcal N(0,\sigma^2)\). Then the posterior over the weights is Gaussian too (conditioning from Lesson 4.4, MAP from Lesson 4.6), and the predictive variance at a new distance splits into two terms:

第一项是偶然部分，即传感器噪声，与数据多少无关；第二项是认知部分：权重后验协方差 \(\Sigma_{\mathbf w}\) 投影到这个距离上的结果，只在数据覆盖的地方变小。拖动下图的测量次数和噪声，或者把一段距离留空不测，看两项各自怎么变。把鼠标放到公式的两项上，图里对应的带子和曲线会加粗。

The first term is the aleatoric part, the sensor noise, and does not depend on how much data you have; the second is the epistemic part, the posterior covariance of the weights \(\Sigma_{\mathbf w}\) projected onto this distance, and it shrinks only where data exist. Move the number of measurements and the noise in the figure below, or leave a range of distances unmeasured, and watch each term. Hover either term of the formula to thicken the matching band and curve.

再看两端：数据只到 2.4 m，在 2.8 m 处（N = 200）认知标准差回升到 8.75 mm，往 3 m 走继续增大，模型在“告诉你它不知道”。这是贝叶斯模型最有用的性质。下一节换成神经网络：它们没有这样干净的后验，我们得用别的办法近似出同样的两项。

Look at the ends too: the data stop at 2.4 m, and at 2.8 m (N = 200) the epistemic std is back up to 8.75 mm and keeps growing towards 3 m; the model is telling you it does not know. This is the most useful property of a Bayesian model. The next section moves to neural networks, which have no such clean posterior, so we need other ways to approximate the same two terms.

偶然部分：异方差回归。让网络输出两个数：均值 \(\mu_\theta(z)\) 和标准差 \(\sigma_\theta(z)\)（第二个输出经过 softplus 保证为正），训练时最小化高斯负对数似然（NLL），而不是 MSE：

The aleatoric part: heteroscedastic regression. Let the network output two numbers, a mean \(\mu_\theta(z)\) and a standard deviation \(\sigma_\theta(z)\) (the second passes through a softplus so it is positive), and train it by minimizing the Gaussian negative log-likelihood (NLL) instead of the MSE:

第二项是被 \(1/\sigma^2\) 加权的平方误差，第一项惩罚“把 σ 报得很大来逃避”。两者平衡时 \(\sigma_\theta^2(z)\) 等于该处残差平方的期望：网络自己学出“这里噪声有多大”（异方差，heteroscedastic，指噪声随输入变化；同方差则处处一样）。这就是 16.2 节“损失 = 负对数似然”的直接推广：MSE 对应固定 σ 的高斯，这里把 σ 也交给网络。

认知部分：深度集成与 MC dropout。训练 \(K\) 个结构相同、初值和小批量顺序不同的网络（深度集成，Lakshminarayanan 等 2017）。在数据多的地方，所有成员都被数据“按住”，预测几乎相同；在数据外，每个成员按自己的初值随意外推，彼此分歧就是认知不确定性的估计。MC dropout（Gal 与 Ghahramani 2016）只训练一个带 dropout 的网络，测试时不关掉 dropout，用 20 个随机掩码各预测一次，把它们当成 20 个成员。两种情况下都按全方差公式合并，第一项是偶然部分，第二项是认知部分：

The second term is the squared error weighted by \(1/\sigma^2\); the first penalizes "escaping by reporting a huge σ". They balance when \(\sigma_\theta^2(z)\) equals the expected squared residual there, so the network learns for itself how noisy each place is (heteroscedastic: the noise varies with the input; homoscedastic: the same everywhere). This is the direct extension of "losses are negative log-likelihoods" from Lesson 16.2: the MSE is a Gaussian with fixed σ, and here σ is handed to the network too.

The epistemic part: deep ensembles and MC dropout. Train \(K\) networks with the same architecture but different initializations and minibatch orders (a deep ensemble, Lakshminarayanan et al. 2017). Where data are plentiful every member is pinned down by them and they predict nearly the same thing; outside the data each member extrapolates according to its own initialization, and their disagreement is the estimate of epistemic uncertainty. MC dropout (Gal and Ghahramani 2016) trains a single network with dropout and does not switch dropout off at test time: 20 random masks give 20 predictions, treated as 20 members. Either way, combine them with the law of total variance; the first term is the aleatoric part, the second the epistemic part:

分类器输出的概率只有在校准的时候才能当概率用：在所有它说“成功概率 0.8”的抓取里，应当恰好约 80% 成功。检验办法是可靠性图：把测试样本按预测概率分成 10 个区间，每个区间里比较平均预测概率和实际成功率；点都在对角线上就是校准的。偏离的大小用期望校准误差概括：

A classifier's output probabilities can be used as probabilities only if they are calibrated: of all the grasps it calls "success probability 0.8", about 80% should actually succeed. The check is a reliability diagram: sort the test samples into 10 bins by predicted probability and compare, in each bin, the mean predicted probability with the observed success rate; points on the diagonal mean calibrated. The size of the deviation is summarized by the expected calibration error:

\(\overline p_b\) 和 \(\overline y_b\) 是第 \(b\) 个区间里的平均预测概率和实际成功率，\(n_b\) 是区间里的样本数。右边是最简单的修正，温度缩放（Guo 等 2017）：网络照旧，只把 logit \(z\) 除以一个标量 \(T\)。\(T>1\) 把概率往 0.5 拉，\(T<1\) 往两端推。\(T\) 在一个留出的验证集上通过最小化 NLL 来拟合，只有一个参数，几乎不会过拟合，而且不改变任何样本的预测类别。

下图是一个抓取成功率分类器：输入是夹爪相对物体中心的横向偏差和接近角误差（都归一化到 [−1, 1]），真实的成功概率是一个平滑函数（中心附近约 95%，往外下降），标签按这个概率随机抽取，所以就算知道真实概率，平均准确率也只有约 82.5%（在本页的 3000 次测试抓取上恰好是 83.5%）：剩下的是偶然不确定性。300 次训练抓取，一个 [2, 16, 16, 1] 的 ReLU 网络，全批量 Adam，学习率 0.01。看训练越久，网络怎样变得越来越过度自信。

\(\overline p_b\) and \(\overline y_b\) are the mean predicted probability and the observed success rate in bin \(b\), and \(n_b\) is the number of samples in it. On the right is the simplest repair, temperature scaling (Guo et al. 2017): keep the network, just divide the logit \(z\) by a scalar \(T\). \(T>1\) pulls probabilities towards 0.5, \(T<1\) pushes them to the ends. \(T\) is fitted by minimizing the NLL on a held-out validation set; with one parameter it hardly overfits, and it changes no sample's predicted class.

The figure below is a grasp-success classifier. Its inputs are the gripper's lateral offset from the object centre and its approach-angle error (both normalized to [−1, 1]). The true success probability is a smooth function (about 95% near the centre, falling off outwards) and labels are drawn at random with that probability, so even knowing the true probability gives only about 82.5% accuracy on average (83.5% on this page's 3000 test grasps): the rest is aleatoric uncertainty. 300 training grasps, a [2, 16, 16, 1] ReLU network, full-batch Adam, learning rate 0.01. Watch the network grow more and more overconfident the longer it trains.

注意准确率在训练中先升后降（第 100 步 82.6%，第 1500 步 76.9%），NLL 从 0.40 涨到 0.95，这就是 16.2 节的过拟合，只是这里它主要表现为过度自信。温度缩放修好了置信度，修不好准确率，也修不好地图上那些弯弯扭扭的错误边界：它是一种“事后校准”，不是更好的模型。而且它只在与验证集同分布的数据上有效，分布一偏移，校准就会重新变坏（Ovadia 等 2019 的大规模实验）。

Note that the accuracy first rises and then falls during training (82.6% at step 100, 76.9% at step 1500) and the NLL climbs from 0.40 to 0.95: this is the overfitting of Lesson 16.2, showing up here mainly as overconfidence. Temperature scaling repairs the confidence but not the accuracy, nor the wiggly wrong boundary on the map: it is post-hoc calibration, not a better model. And it holds only on data distributed like the validation set; under distribution shift calibration degrades again (the large-scale study of Ovadia et al. 2019).

前面所有的区间都依赖模型假设（高斯噪声、网络学对了 σ）。共形预测换了个思路：不管模型多差，拿一组模型没见过的校准数据 \((z_i,y_i)_{i=1}^n\)，算每个点的“不合格分数”，例如绝对残差 \(s_i=|y_i-\hat y(z_i)|\)，取其中第 \(k=\lceil (n+1)(1-\alpha)\rceil\) 小的分数作为阈值 \(\hl{qhat}{\hat q}\)，预测区间就是 \(\hat y(z)\pm\hat q\)。只要新数据和校准数据可交换（例如独立同分布），就有

Every interval so far relied on model assumptions (Gaussian noise, a correctly learned σ). Conformal prediction takes another route: however bad the model, take a set of calibration data \((z_i,y_i)_{i=1}^n\) the model has not seen, compute a "nonconformity score" for each, for example the absolute residual \(s_i=|y_i-\hat y(z_i)|\), and take the \(k\)-th smallest score, \(k=\lceil (n+1)(1-\alpha)\rceil\), as the threshold \(\hl{qhat}{\hat q}\); the prediction interval is \(\hat y(z)\pm\hat q\). As long as the new data and the calibration data are exchangeable (for example i.i.d.),

（上界要求分数没有并列。）理由只是排序：新点的分数和 \(n\) 个校准分数可交换，它落在 \(n+1\) 个位置中任意一个的可能性相同，落在前 \(k\) 个以内的概率至少是 \(k/(n+1)\ge1-\alpha\)。没有用到任何关于模型或噪声分布的假设。下图里的“模型”故意选了一个错的：贝叶斯线性回归假设处处同样的噪声，估计出 \(\sigma_{\mathrm m}=3.39\) mm，可真实噪声从 0.6 mm 涨到 7.4 mm。

(The upper bound requires no ties among the scores.) The reason is nothing but ranking: the new point's score and the \(n\) calibration scores are exchangeable, so it is equally likely to land in any of the \(n+1\) positions, and the probability that it is among the first \(k\) is at least \(k/(n+1)\ge1-\alpha\). No assumption about the model or the noise distribution is used. The "model" in the figure below is deliberately wrong: a Bayesian linear regression that assumes the same noise everywhere and estimates \(\sigma_{\mathrm m}=3.39\) mm, whereas the true noise grows from 0.6 mm to 7.4 mm.

三点观察。(1) 保证是对“校准数据和测试点一起随机”而言的平均：某一组校准数据可能偏低或偏高（右下的直方图：n = 50 时 5%–95% 分位约为 81.5%–96.5%，n = 500 时收窄到 85.5%–94.5%），平均值才稳稳落在 1 − α 上。(2) 保证是边缘的（对所有距离平均），不是条件的（对每个距离）：绝对分数在噪声小的地方太宽、噪声大的地方太窄，总体却刚好 90%。好的分数函数（这里除以 u(z)，或者用分位数回归）让区间随难度伸缩，条件覆盖率更接近目标，但仍没有逐点保证。(3) α 很小、n 又少时，\(k=\lceil(n+1)(1-\alpha)\rceil\) 可能超过 \(n\)，此时 q̂ = ∞：数据太少，诚实的答案就是“什么都可能”。

Three observations. (1) The guarantee is an average over the random draw of calibration data and test point together: one particular calibration set may come out low or high (the lower-right histogram: the 5–95% range is about 81.5–96.5% at n = 50, narrowing to 85.5–94.5% at n = 500), and only the average sits firmly at 1 − α. (2) The guarantee is marginal (averaged over all distances), not conditional (at each distance): the absolute score is too wide where the noise is small and too narrow where it is large, yet exactly 90% overall. A good score function (here, dividing by u(z); or quantile regression) makes the interval stretch with the difficulty and brings the conditional coverage closer to the target, but still with no pointwise guarantee. (3) When α is small and n is small, \(k=\lceil(n+1)(1-\alpha)\rceil\) can exceed \(n\) and then q̂ = ∞: with that little data the honest answer is "anything is possible".

贝叶斯的说法是：参数 \(\theta\) 本身不确定，有后验 \(p(\theta\mid\mathcal D)\)；对新输入 \(x\) 的预测要对它积分：

In Bayesian terms the parameters \(\theta\) are themselves uncertain, with posterior \(p(\theta\mid\mathcal D)\), and a prediction at a new input \(x\) integrates over it:

右式是全方差公式（4.3 节的期望与方差）：总方差 = 噪声方差的平均（偶然）+ 均值的方差（认知），期望和方差都对后验取。集成和 MC dropout 都是用 \(K\) 个样本 \(\theta_k\) 近似这个后验，于是得到 FIG 2 用的公式。数据越多，后验越集中，第二项趋于 0；第一项则趋于真实的噪声方差。

The right-hand side is the law of total variance (expectation and variance from Lesson 4.3): total variance = the mean of the noise variance (aleatoric) + the variance of the mean (epistemic), both taken over the posterior. Ensembles and MC dropout approximate the posterior with \(K\) samples \(\theta_k\), which gives the formula used in FIG 2. With more data the posterior concentrates and the second term goes to 0, while the first tends to the true noise variance.

特征矩阵 \(\Phi\)（第 \(i\) 行是 \(\boldsymbol\phi(z_i)^\top\)），先验 \(\mathbf w\sim\mathcal N(\mathbf 0,\Sigma_0)\)，噪声方差 \(\sigma_i^2\) 可以逐点不同，记 \(\Lambda=\operatorname{diag}(1/\sigma_i^2)\)。后验是高斯的：

Feature matrix \(\Phi\) (row \(i\) is \(\boldsymbol\phi(z_i)^\top\)), prior \(\mathbf w\sim\mathcal N(\mathbf 0,\Sigma_0)\), noise variances \(\sigma_i^2\) allowed to differ per point, \(\Lambda=\operatorname{diag}(1/\sigma_i^2)\). The posterior is Gaussian:

\(\mathbf m\) 就是岭回归（4.6 节的 MAP）。\(\Sigma_{\mathbf w}^{-1}\) 是“信息”：每个测量把 \(\boldsymbol\phi(z_i)\boldsymbol\phi(z_i)^\top/\sigma_i^2\) 加进去，所以在一个地方测 \(N\) 次，那里的认知方差大约按 \(\sigma^2/N\) 下降（FIG 1 的 \(1/\sqrt N\)）；噪声大的测量贡献的信息少（FIG 6 里远处的板“不值钱”）。注意 \(\sigma^2_{\mathrm{epi}}\) 不依赖 \(\mathbf y\)：认知不确定性只取决于在哪里测过、测得多准，而不是测到了什么。本页用 25 个宽 0.2 m 的高斯鼓包加一个常数项，鼓包权重的先验标准差为 6 mm，常数项为 10 mm；数值上用 Cholesky 分解，\(\sigma^2_{\mathrm{epi}}=\|L^{-1}\boldsymbol\phi\|^2\)。

\(\mathbf m\) is ridge regression (the MAP of Lesson 4.6). \(\Sigma_{\mathbf w}^{-1}\) is the "information": each measurement adds \(\boldsymbol\phi(z_i)\boldsymbol\phi(z_i)^\top/\sigma_i^2\), so measuring \(N\) times at one place makes the epistemic variance there fall roughly like \(\sigma^2/N\) (the \(1/\sqrt N\) of FIG 1), and a noisy measurement contributes little information (the "cheap" far boards of FIG 6). Note that \(\sigma^2_{\mathrm{epi}}\) does not depend on \(\mathbf y\): epistemic uncertainty depends only on where you measured and how precisely, not on what you measured. This page uses 25 Gaussian bumps of width 0.2 m plus a constant, with prior standard deviations of 6 mm on the bump weights and 10 mm on the constant; numerically it uses a Cholesky factor, \(\sigma^2_{\mathrm{epi}}=\|L^{-1}\boldsymbol\phi\|^2\).

单个样本的损失 \(\ell=\log\sigma+\frac{r^2}{2\sigma^2}\)，\(r=y-\mu\)。对两个输出求导：

Per-sample loss \(\ell=\log\sigma+\frac{r^2}{2\sigma^2}\) with \(r=y-\mu\). Differentiating with respect to both outputs:

第一式说明：均值的梯度被 \(1/\sigma^2\) 加权，噪声大的点“拉力”小，这是逐点加权最小二乘，与卡尔曼滤波按协方差加权是同一个道理。它也是 NLL 的一个坑：网络可以把某些难拟合的区域的 σ 报大，从而几乎不再学那里的均值（Seitzer 等 2022 提出的 β-NLL 用 \(\sigma^{2\beta}\) 重新加权来缓解）。第二式说明 σ² 学的是残差平方的条件期望：如果均值本身拟合得不好，系统误差也会被当成“噪声”（FIG 2 里近处 σ 偏大）。本页的 \(\sigma_{\min}\) 取 0.2 mm，防止除以零。

The first expression says the gradient on the mean is weighted by \(1/\sigma^2\): noisy points pull less. This is pointwise weighted least squares, the same idea as weighting by covariance in a Kalman filter. It is also a trap of the NLL: the network can report a large σ for regions that are hard to fit and then hardly learn the mean there (β-NLL, proposed by Seitzer et al. 2022, reweights by \(\sigma^{2\beta}\) to mitigate this). The second says σ² learns the conditional expectation of the squared residual: if the mean itself fits poorly, systematic error is booked as "noise" too (the inflated near-range σ of FIG 2). On this page \(\sigma_{\min}\) is 0.2 mm, to avoid dividing by zero.

分类器的集成给出 \(K\) 个概率 \(p_k\)，平均 \(\bar p\)。预测熵（4.11 节）同样分成两部分：

An ensemble of classifiers gives \(K\) probabilities \(p_k\) with mean \(\bar p\). The predictive entropy (Lesson 4.11) splits in the same way:

左边是总不确定性，右边第一项（各成员熵的平均）是偶然部分，第二项互信息是认知部分。互信息 \(I\) 是“如果知道了真参数，能对 \(y\) 多知道多少”：成员都说 0.5 时总熵最大，但互信息为 0，这是纯偶然（这个抓取本来就看运气）；成员一个说 0.05、一个说 0.95 时，平均也是 0.5，互信息却很大，这是认知的（模型没见过）。主动学习的 BALD（Houlsby 等 2011）就是挑互信息最大的样本去标注。

The left side is the total uncertainty; on the right the first term (the members' mean entropy) is the aleatoric part and the second, the mutual information, the epistemic part. The mutual information \(I\) is "how much more we would know about \(y\) if we knew the true parameters". If all members say 0.5 the total entropy is maximal but the mutual information is 0: pure aleatoric (this grasp is simply a matter of luck). If one says 0.05 and another 0.95 the mean is still 0.5, but the mutual information is large: epistemic (the model has not seen this). BALD (Houlsby et al. 2011) in active learning labels exactly the samples with the largest mutual information.

校准的定义：\(\mathbb P(Y=1\mid\hat p(X)=p)=p\) 对所有 \(p\) 成立。ECE 用分箱近似它（分箱数会影响数值，结果要连同分箱方式一起报告）。温度缩放求 \(T^\ast=\arg\min_T\sum_{i\in\mathcal D_{\mathrm{val}}}\ell_{\mathrm{BCE}}\big(z_i/T,\,y_i\big)\)，这是一维凸问题（对 \(1/T\) 是凸的），本页用黄金分割搜索。因为 sigmoid 单调，\(z/T\) 与 \(z\) 同号，所以预测类别和准确率都不变；多分类时 softmax\((\mathbf z/T)\) 的最大分量位置也不变。NLL 和 Brier 分数这类严格正常评分规则只在预测分布等于真实分布时取到最优，所以既奖励准确也奖励校准，这也是深度集成用 NLL 训练的理由。

The definition of calibration: \(\mathbb P(Y=1\mid\hat p(X)=p)=p\) for every \(p\). ECE approximates it with bins (the number of bins changes the value, so report it with the binning). Temperature scaling solves \(T^\ast=\arg\min_T\sum_{i\in\mathcal D_{\mathrm{val}}}\ell_{\mathrm{BCE}}\big(z_i/T,\,y_i\big)\), a one-dimensional convex problem (convex in \(1/T\)), solved here by golden-section search. Because the sigmoid is monotone, \(z/T\) has the same sign as \(z\), so the predicted class and the accuracy do not change; with many classes the position of the largest entry of softmax\((\mathbf z/T)\) does not change either. Strictly proper scoring rules such as the NLL and the Brier score are optimal only when the predicted distribution equals the true one, so they reward both accuracy and calibration, which is also why deep ensembles are trained with the NLL.

设分数 \(s_1,\dots,s_n,s_{n+1}\)（最后一个是测试点）可交换，并假设没有并列。则 \(s_{n+1}\) 在这 \(n+1\) 个数中的排名在 \(\{1,\dots,n+1\}\) 上均匀分布。记 \(s_{(k)}\) 为校准分数中第 \(k\) 小的：

Let the scores \(s_1,\dots,s_n,s_{n+1}\) (the last one belongs to the test point) be exchangeable, with no ties. Then the rank of \(s_{n+1}\) among the \(n+1\) numbers is uniform on \(\{1,\dots,n+1\}\). Write \(s_{(k)}\) for the \(k\)-th smallest calibration score:

集合 \(C(x)=\{y:s(x,y)\le\hl{qhat}{\hat q}\}\) 就是预测区间（或分类时的预测集合）。分数函数可以任选：绝对残差、除以某个尺度 u(x) 的归一化残差、分位数回归的上下界之差（CQR）、分类时 \(1-\hat p_y\)。选得好，集合更小、更贴合每个输入的难度，但覆盖率保证与选择无关。代价是要留出一份不参与训练的校准数据（分割共形），并且保证只在可交换时成立：时间序列、闭环控制、分布偏移都会破坏它，需要加权或在线的变体。

The set \(C(x)=\{y:s(x,y)\le\hl{qhat}{\hat q}\}\) is the prediction interval (or, for classification, the prediction set). Any score function will do: the absolute residual, the residual normalized by a scale u(x), the spread of quantile-regression bounds (CQR), or \(1-\hat p_y\) for classification. A good choice gives smaller sets that track each input's difficulty, but the coverage guarantee does not depend on the choice. The price is a held-out calibration set that is not used for training (split conformal), and the guarantee holds only under exchangeability: time series, closed-loop control and distribution shift all break it and need weighted or online variants.

在候选位置 \(z\) 中选下一次测量。FIG 6 比较三种规则：随机；\(\hl{acq}{\arg\max_z\,\sigma^2_{\mathrm{epi}}(z)+\sigma^2(z)}\)（总方差）；\(\hl{acq}{\arg\max_z\,\sigma^2_{\mathrm{epi}}(z)}\)（认知方差）。对高斯模型，一次测量带来的信息增益是 \(\tfrac12\log\big(1+\sigma^2_{\mathrm{epi}}(z)/\sigma^2(z)\big)\)，它同样只关心认知部分，并且会因噪声大而打折扣。总方差规则把偶然方差算了进去，会一次次去噪声最大的地方，而那里的噪声正是采多少次都消不掉的。

Choose the next measurement among candidate locations \(z\). FIG 6 compares three rules: random; \(\hl{acq}{\arg\max_z\,\sigma^2_{\mathrm{epi}}(z)+\sigma^2(z)}\) (total variance); \(\hl{acq}{\arg\max_z\,\sigma^2_{\mathrm{epi}}(z)}\) (epistemic variance). For a Gaussian model the information a measurement brings is \(\tfrac12\log\big(1+\sigma^2_{\mathrm{epi}}(z)/\sigma^2(z)\big)\); it too cares only about the epistemic part, discounted where the noise is large. The total-variance rule counts the aleatoric variance as well and keeps going to wherever the noise is largest, which is exactly the noise no amount of measuring removes.

一个最直接的用法：机器人在不确定时停下来求助（或者换一个保守动作）。问题是用哪个数当“不确定”。下图训练了一个抓取分类器的 5 成员集成，训练抓取都在单位圆内（输入是两个归一化的抓取参数）。把琥珀圈（一个新物体的抓取参数）拖到各处，比较三种分数：单个网络的置信度 \(\max(p,1-p)\)（Hendrycks 与 Gimpel 2017 的基线），集成的互信息 \(\hl{mi}{I}\)（认知），以及到最近训练样本的距离（一种最朴素的密度估计）。

The most direct use: when uncertain, the robot stops and asks for help (or switches to a cautious action). The question is which number counts as "uncertain". The figure below trains a 5-member ensemble of grasp classifiers whose training grasps all lie inside the unit disk (the inputs are two normalized grasp parameters). Drag the amber ring (the grasp parameters of a new object) around and compare three scores: a single network's confidence \(\max(p,1-p)\) (the baseline of Hendrycks and Gimpel 2017), the ensemble's mutual information \(\hl{mi}{I}\) (epistemic), and the distance to the nearest training sample (the plainest density estimate).

这张图值得多看一会儿。单个网络的置信度在圆外几乎处处接近 100%：ReLU 网络离数据越远，logit 越像一个线性函数，绝对值越来越大（Hein 等 2019 证明了这在 ReLU 网络里几乎不可避免）。集成的互信息只在“边界延长线”附近变大：那里成员们对边界往哪边弯意见不一；在别的方向上，5 个网络都学到了“越往右越成功、越往左越失败”，一起自信地外推，互信息几乎为 0。圆内边界附近预测熵很大（最大可到 \(\log 2\approx0.69\) nats），但互信息很小：这是偶然的，这类抓取本来就是一半一半，请人帮忙也没用。所以实际系统常把几种信号组合起来：集成或 dropout 的分歧、特征空间里的距离或密度，再加上任务层面的检查（比如抓取后检测是否真的抓住了）。

This figure rewards a longer look. The single network's confidence is close to 100% almost everywhere outside the disk: the farther a ReLU network gets from the data, the more its logit behaves like a linear function whose magnitude keeps growing (Hein et al. 2019 showed this is almost unavoidable for ReLU networks). The ensemble's mutual information grows only near the "extension of the boundary", where the members disagree about which way the boundary bends; in other directions all 5 networks learned "further right succeeds, further left fails" and extrapolate confidently together, with mutual information close to 0. Near the boundary inside the disk the predictive entropy is large (up to \(\log 2\approx0.69\) nats) but the mutual information is small: that is aleatoric, such grasps really are fifty-fifty, and asking a person will not help. So real systems often combine several signals: ensemble or dropout disagreement, distance or density in feature space, plus checks at the task level (such as verifying after a grasp that the object is really held).

回到深度相机。每次摆一块标定板都要人工时间，我们想用尽量少的测量把整个 0–3 m 的误差曲线学好。现在假设噪声模型 σ(z) 已经知道（例如用 FIG 2 学出来），模型是 FIG 1 的贝叶斯线性回归，但按每个距离的噪声加权。一开始只有 0.6、1.0、1.4、1.8 m 四个测量，之后每一步按某个规则从 0–3 m（步长 2 cm）里选下一个距离。

Back to the depth camera. Every board placement costs human time, and we want to learn the whole 0–3 m error curve with as few measurements as possible. Assume the noise model σ(z) is now known (learned as in FIG 2, say) and the model is FIG 1's Bayesian linear regression, weighted by the noise at each distance. It starts with only four measurements at 0.6, 1.0, 1.4 and 1.8 m; at each step a rule picks the next distance from 0–3 m (in 2 cm steps).

总方差规则的失败很有代表性：3 m 处的噪声（11.3 mm）比任何地方都大，总方差永远最大，规则一遍遍把板放在那里，可每次测量带来的信息很少，别处的认知不确定性也一直降不下来。认知方差规则先把没测过的两端（0 m 附近和 2–3 m）补上，然后在远处多放几块，因为远处噪声大，每块板“值”得少，要多测几次才能把那里的认知方差压到和别处一样。随机规则不算差，但浪费在已经测得很准的地方。单次运行的 RMSE 受测量噪声影响会起伏；对 40 个随机种子平均，24 次测量时认知规则 2.15 mm，随机 2.53 mm，总方差 3.14 mm，排序不变。

The failure of the total-variance rule is typical: the noise at 3 m (11.3 mm) is larger than anywhere else, so the total variance is always largest there and the rule keeps putting the board there; each measurement brings little information and the epistemic uncertainty elsewhere never comes down. The epistemic rule first fills the unmeasured ends (near 0 m and 2–3 m), then places a few more boards far away, because there the noise is large, each board is "worth" less, and more measurements are needed to push the epistemic variance down to the level elsewhere. Random is not bad but wastes boards where the curve is already well measured. The RMSE of a single run fluctuates with the measurement noise; averaged over 40 random seeds, at 24 measurements the epistemic rule gets 2.15 mm, random 2.53 mm and total variance 3.14 mm, the same ranking.