Reliable Decisions with Threshold Calibration
Authors: Roshni Sahoo
Shengjia Zhao
Alyssa Chen
Stefano Ermon
Venue: NeurIPS 2021 Poster (2021)
Keywords: uncertainty quantification
calibration

Abstract: Decision makers rely on probabilistic forecasts to predict the loss of different decision rules before deployment. When the forecasted probabilities match the true frequencies, predicted losses will be accurate. Although perfect forecasts are typically impossible, probabilities can be calibrated to match the true frequencies on average. However, we find that this \textit{average} notion of calibration, which is typically used in practice, does not necessarily guarantee accurate decision loss prediction. Specifically in the regression setting, the loss of threshold decisions, which are decisions based on whether the forecasted outcome falls above or below a cutoff, might not be predicted accurately. We propose a stronger notion of calibration called threshold calibration, which is exactly the condition required to ensure that decision loss is predicted accurately for threshold decisions. We provide an efficient algorithm which takes an uncalibrated forecaster as input and provably outputs a threshold-calibrated forecaster. Our procedure allows downstream decision makers to confidently estimate the loss of any threshold decision under any threshold loss function. Empirically, threshold calibration improves decision loss prediction without compromising on the quality of the decisions in two real-world settings: hospital scheduling decisions and resource allocation decisions.

Reliable Decisions with Threshold Calibration
Roshni Sahoo Stanford University rsahoo@stanford edu
Shengjia Zhao Stanford University sj zhao@stanf ord. edu
Alyssa Chen
Stefano Ermon Stanford University ermon@stanford. edu
UTSW Medical Center alyssa. chen@utsw... edu
Abstract
Decision makers rely on probabilistic forecasts to predict the loss of different decision rules before deployment. When the forecasted probabilities match the true frequencies, predicted losses will be accurate. Although perfect forecasts are typically impossible, probabilities can calibrated to match the true frequencies average. However, we find that this average notion of calibration, which is typically used practice, does not necessarily guarantee accurate decision loss prediction. Specifically in the regression setting, the loss of threshold decisions, which are decisions based on whether the forecasted outcome falls above or below cutoff, might not be predicted accurately. We propose stronger notion of calibration called threshold calibration, which is exactly the condition required to ensure that decision loss is predicted accurately for threshold decisions. We provide an efficient algorithm which takes an uncalibrated forecaster as input and provably outputs threshold-calibraee decision makers to confidently estimate the loss of any threshold decision under any threshold loss function. Empirically, threshold calibration improves decision loss prediction without compromising on the quality of the decisions in two real-world settings: hospital scheduling decisions and resource allocation decisions.
forecaster. Our procedure allows downstream
Introduction
Decision makers need understand the consequences of their decisions prior making them. When decisions are based on predictions from a machine learning model, the decision loss the loss incurred under decision rule based on the predictions summarizes the consequences of these decisions. As an example, suppose machine learning practitioner develops model to predict patient length-of-stay in the hospital [17, 3). A hospital decides whether they have capacity admit new patients based the model's predictions of current patients' length-of-stay (e.g. for each current patient who is predicted to have a length-of-stay that is less than days, the hospital schedules new patient). Incorrect decisions due to the model's is predictions cause the hospital to accrue costs from under-utilizing resources or overbooking procedures. The decision loss an aggregation of the costs incurred from incorrect decisions. To determine whether decision rule is safe use, the hospital would like to have an accurate estimate the decision loss under different choices of and different costs associated with errors. This type of decision-maie maaking scenario occurs in many high-stakes settings such as designing interventions for adverse weather events [33, 9] and resource allocation decisions using economic estimates [15, 32).
Probabilistic predictions (probabilistic forecasts) can be used to estimate decision loss prior to deployment. In this work, we consider the regression setup, where forecast is represented by a cumulative distribution function over the possible outcomes. the forecasted probabilities of
35th Conference on Neural Information Processing Systems (NcurIPS 2021).

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Average-Calirre Forecaster
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Figure/ 1: We evaluate wwerage-calibaate and threshold-calibraee patient length-of-star forecasters across range of threshold decision rules. Left: The average-calibrate forecaster underestimates the true decision loss for some decision rules and overestimates it on others, resulting in nonzero reliability gap. Middle: The reliability gap minimized under the threshold-calibrated forecaster but not under the average-calibrated forecaster. Right: The threshold-calibrated forecaster accurately predicts the true decision loss across a range of decision rules.
incorrect decisions match the true frequencies of these events, then the average decision loss can be accurately predicted from the forecasts. However, forecasted probabilities of incorrect decisions do not typically match the true ones, yielding inaccurate decision loss predictions. We refer to the absolute difference between the average loss predicted by forecaster and the true average loss as the reliability gap.
Many previous works in calibration and uncertainty quantification motivated by the assumption that calibrated uncertainty estimates will yield safer or more reliable downstream decisions [31, 2, 22, 24, 25]. However, find that the standard notion of calibration, average calibration, does not guarantee zero reliability gap for even simple class of decision rules: threshold decision rules (left, Figure 1). In threshold decision, decision maker takes of possible actions depending on whether an outcome falls above or below a cutoff (e.g. the hospital schedules a new patient if a current patient's length-of-stay
less than 3 days, otherwise the hospital does not schedule new patient). Stronger calibration properties, such as distribution calibration [31], are theoretically guaranteed to yield zero reliability gap but are difficult to achieve in practice. In particular, flexible distribution families can better approximate the true distribution than simple ones and yield lower decision loss, but applying distribution calibration to such forecasters can increase the decision loss and enlarge the reliability gap compared to the uncalibrated forecaster. Thus, existing calibration definitions are either insufficient or impractical for minimizing the reliability gap under threshold decision rules.
To address these shortcomings. we propose a new notion of calibration called threshold calibration. Threshold calibration strikes balance between average and distribution calibration; it is exactly the condition required to guarantee zero reliability gap under threshold decisions and is practical to enforce (Figure 1, Right). First, we establish that threshold calibration the necessary and sufficient condition to guarantee zero reliability gap for any threshold decision under any threshold loss. Second, we design an efficient algorithm that takes an uncalibrated forecaster as input and provably outputs a wreshold-calibrann forecaster. Third, we show that empirically, threshold calibration is practical solution; in two real-world settings and a suite of benchmark regression tasks, we find that threshold calibration minimizes the reliability gap across decision makers with different threshold loss functions while achieving similar or improved decision loss compared to the baselines.
2 Preliminaries
2.1 Notation and Forecasting Setup
We consider the regression setup with feature space X' and a label space y C R. The input is random variable X E X' and the label is a random variable Y E y. We use capital letters to denote random variables X, Y and lower case letters denote their values v, y-
Let F()) be the space of cumulative distribution functions (CDFs) over V. forecaster h: X F()) is function that maps input from the feature space to CDF on Y. In other

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words, given fixed input x' E X. the forecaster outputs the predicted CDF h[z] E F()). Ideally, the forecaster aims to predict the CDF of Y given X.
To further clarify the notation, for fixed input-label pair (z, v) E x V. M/a] is a CDF over the predicted label values and (v) [0,1] is the value of the CDF M(ar] at the point y. We note that h[X] is random variable that takes values in F()) and h([](() is a random variable that takes values in [0, 1].
Let h [X] be the true conditional CDF of given X. We use ~ to denote the distribution random variable. We have that Y ~ ['" We introduce a new random variable y to represent a label distributed according to the h[X], the forecasted conditional distribution, so Y ~ h[X].
2.2 Decision-Making
Let A be a countable action space. A decision rule 5 & - A is any map from an input x (e.g. current patient's attributes) an action a (e.g. admit a new patient). We assume that a decision maker has a loss function É X x y x A + R, describing the loss incurred when choosing an action input-label pair (Ir, y). Because the labels y are unobserved, the decision maker often wants to minimize their expected loss assuming that the labels are distributed according to the forecasted distribution. As a result, they use the Bayes decision rule with respect to h.
Definition 1 (Bayes Decision Rule). Given a space of decision rules A, the Bayes decision rule with respect to the forecaster h is the decision rule in A that minimizes the expected loss under the forecasted distribution
%% = arg inf Ex ( BEA
2.3 Threshold Decisions
We focus on the setting where the decision maker aims to minimize a threshold loss function. The action space A consists of two actions so A {0, 1}. A threshold loss function é is defined as follows
((, = E
Cyy I yo, a = i) + Σ 1E/0,1}
Co,IIly > yo. a =
iE(U,1}
where C.j E R. The Ca. 's denote decision costs, costs associated with different outcome-action pairs, and yo is decision threshold. Let be the space of threshold loss functions, which are all losses of this form with any Cij E IR. and yo E IR.
Given threshold loss function É, the decision maker can use the Bayes decision rule 8 in Definition 1 to select which action to take. We show that the resulting decision rules always take the form of
5% (x) = (vo) 2 a) 0% (x) = T(h (())) S o:)
for some parameters E [0. 1] and yo y that depends on the loss function (proved in Appendix B). We call such decision rules threshold decision rules because intuitively, they choose the action based on whether h (1x))))))))))))))))))))))))))( is greater (or less) than a threshold a. We denote the space of such decision rules as An. Since the decision maker's loss function is a threshold loss function, the decision maker restrict the space of decision rules they consider threshold decision rules the forecasted CDFs.
3 Reliable Decision-Making with Threshold Calibration
3.1 Problem Sctup
Forecasts are often produced by one group, such as machine learning practitioners or scientists, and consumed by another, such policy makers or private agents [14]. Motivated by this paradigm, model these two entities separately:
1. A forecaster h takes inputs x X' and produces CDFs h(x) over the possible outcomes in y. The provider of h does not know the specific downstream tasks for which h is used.
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2. A decision maker has a dataset of unlabeled inputs D = {z, 1i- binary action space A {0, 1), and threshold loss function & E C of interest. The decision maker must take an action a; E A for each unlabeled input zę. The decision maker uses the forecaster h to select {a,};= because (1) the decision maker does not have enough labeled data to build their own model locally or (2) building the model requires domain expert.
Multiple decision makers may rely on the same forecaster but have different loss functions. Further, decision maker's loss function change if their decision costs decision threshold change. In this setting, we identify the conditions on h that the provider can enforce to ensure reliable decision-making under threshold decisions.
3.2 Reliability Gap
Decision makers often need to accurately estimate the average decision loss incurred under decision rule prior to deployment. To quantify the accuracy of these decision loss predictions, we define the reliability gap.
Definition 2 (Reliability Gap). Given a forecaster h, we define the the reliability gap 7 E) of particular decision rule 8 under loss function & as
766 e) -E ((X,,),))
((x()) -
The first term in the equation is the average decision loss predicted by the forecaster. Under the forecasted distribution, the labels are distributed according to h[X]. As result, the first term does not depend on the true labels and can be computed by the decision maker using the unlabeled data prior to deployment. The second term is the true average decision loss. Under the true conditional distribution, the labels are distributed according to h h=[X]. So, the second term can be thought of as the loss that is incurred at test-time. One caveat is that the reliability gap quantifies the reliability of average decision loss prediction and obtaining zero reliability gap does not imply any instance-based guarantees for individual decisions.
When the forecaster perfectly matches the true distribution (i.e. h h²), we have (8, = 0 for any decision rule 5 and any loss function f. However, in practice, we cannot assume that the forecaster predicts the true distribution. In addition, we would like the forecaster to applicable for different downstream decision makers. As result, we study the necessary and sufficient conditions on the forecaster that guarantee zero reliability gap for any threshold decision on the forecasted CDFs and any threshold loss function.
3.3 Threshold Calibration
We define the property of threshold calibration and show that it necessary and sufficient to ensure zero reliability gap under any threshold decision on the forecasted CDFs and any threshold loss function. The lemma and theorem in this section are proven Appendix
We define the property of threshold calibration below.
Definition (Threshold Calibration). A forecaster satisfies threshold calibration if
Pr[b[[](Y) s h[x](1)) S a] Vyo E VV,a E [0, 10, [0.
(1)
A threshold-calibrated forecaster is cerage-calibrated on subsets of the predicted CDFs that sat- isfy h[X](1)) I a. We make the following observation about conditioning on the complementary predicted CDFs.
Lemma 1. Given forecaster h that satisfies Definition 3, then we have that Vyo E V,a E [o, 1], Vc [0, 1], Pr[h([](Y)) S b[[][())) =
In threshold decision task, decision maker will take action given inputs with predicted CDFs satisfying (1)) S a (and take a complementary action given inputs with predicted CDFs
satisfying (36) > a). Intuitively, threshold calibration ensures that the forecaster satisfies
average calibration on the subsets of predicted CDFs where the decision maker chooses a = 0 and =

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Threshold calibration is specific type of group calibration [28], where calibration across the collection of groups (X x [1X] S oee is desired. Since threshold calibration requires achieving calibration on intersecting groups, it is also related to the notion of multicalibration [18]. In Section 4, we give an efficient algorithm for achieving threshold calibration that is inspired by previous work on multicalibration.
Using Definition 3 and Lemma I, we define the threshold calibration error (TCE) measure deviation from threshold calibration at a threshold E y and quantile a E [0, 1].
Definition (Threshold Calibration Error).
TCE(h, ao = [ (P(MXI()) - a) -cl de
L Pr[MX](()) S (X) al de.
Threshold calibration is desirable property due to its connection to achieving zero reliability gap.
Theorem 1. Let be the space of threshold loss functions. Given a forecaster h, let D, be the space of threshold decision rules the forecasted CDFs of h. A forecasrrr h satisfies threshold calibration if and only (6, E) = 0 45 E An, 4E L.
We obtain this result by observing that the expected decision loss under the true distribution can be decomposed into two terms. The first term corresponds to the cost incurred from "false positive" errors and the second term corresponds to the cost incurred from "false negative" errors. Under threshold calibration, the forecaster's ' predicted eгrоr rates match the true eπror rates. Since the decision loss (with any choice of costs) is linear combination of these error rates, the expected decision loss predicted by the forecaster matches the expected decision loss under the true distribution. Thus, under threshold-calibratee forecaster, we achieve zero reliability gap under any threshold decision on the forecasted CDFs and any threshold loss function.
3.4 Comparison to Existing Calibration Definitions
We compare threshold calibration to other methods for calibrating probabilistic forecasts. Average calibration is the standard definition of calibration for regression [23, 12].
Definition 5 (Average Calibration). A forecaster h satisfies average calibration if
Pr[M[X](() s = Vc [0, 1].
In other words, forecaster averago-calibrated if the true label is below the e-th quantile of the forecasted CDF (([x]] exactly percent of the time.
In contrast, distribution calibration is much stronger definition of calibration [31]. Intuitively, distribution calibration requires a forecaster to be calibrated for every distribution in the forecaster's model family.
Definition 6 (Distribution Calibration). A forecaster h satisfies distribution calibration if
Pr[M[x](Y) S c M(x] g] = Vg E F()),,
where F is space of CDFs corresponding to the forecaster'ss 's model family.
We outline the relationship between average, threshold, and distribution calibration in the following proposition.
Proposition 1. If a forecaster satisfies distribution calibration, then it satisfies threshold calibration. a forecaster satisfies threshold calibration, then it satisfies average calibration.
We note that the converses of the statements in Proposition 1 are not necessarily true. A threshold- calibrated forecaster does not necessarily satisfy distribution calibration. An average-calibrated forecaster does not necessarily satisfy threshold calibration or distribution calibration (see Appendix C). This implies that an average-calibrated forecaster does not satisfy the necessary condition of Theorem I, meaning that the reliability gap under threshold decisions may not be zero. So, decision
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makers who rely on forecaster that only satisfies average calibration (but not threshold calibration) are not guaranteed accurately estimate their decision loss under threshold decisions.
From Proposition 1, we have that a distribution-calibaaeed forecaster satisfies the necessary condition of Theorem 1. However, distribution calibration be challenging to achieve in practice because the same CDF is rarely predicted more than one time on the training samples, making it difficult to guarantee calibration without compromising the sharpness of the forecasts. Sharpness corresponds to the width of the prediction intervals generated from the forecasts, and sharp forecasts yield short prediction intervals. Although distribution calibration theoretically guaranteed yield zero reliability gap. we observe that achieving distribution calibration is challenging when the model family complex (Section 5).
Finally, we emphasize the threshold calibration exactly the condition needed to guarantee the reliability gap is zero in Theorem 1.
4 Achieving Threshold Calibration
We design a recalibration algorithm that takes an uncalibrated forecaster as input and provably outputs careshold-calibrated forecaster. Our algorithm is iterative procedure that terminates when the maximum TCE is less than a user specified threshold e. Our key result is that the algorithm must terminate after (L))²²) iterations.
"seudo-code for the algorithm is shown in Algorithm 1. Intuitively, at each iteration of the algorithm, find the y and a' where the TCE in Definition 4 is maximized. This partitions the input X' into two parts: those where h (16) < a' and those where [=x](16) > a'. For each partition, we use a standard recalibration algorithm (Isotonic regression [23]) achieve average calibration. Intuitively, after the recalibration step, the forecaster should satisfy average calibration for each partition, and hence the TCE in Definition 4 must be (close to) for vó and a.. We repeat this procedure until the TCE is less than e for every possible and a.
Algorithm 1: Threshold Recalibration
Input: Forecaster h X' - F()",, maximum error 0
2 Output: threshold-calibratee forecaster
3 Set h +
4 for t 1, until maximum threshold calibration error sup
TCE(h'-1, ,30,, a) S do
Find the yo and a that maximize threshold calibration error.
yú, a² + argsup, (ng-n)eyx(o.1 TCB(h'-1, yo, a)
Partition input features X' into Xn + far E X' h² [x][6]] S Saa} and Xi = X Xo.
Use Isotonic regression to learn recalibration maps ob. of F()) + F())) on X, and X; respectively.
Apply the recalibration map to obtain new prediction functions. (h' [x]]) if Xa (1+-(()))) otherwise
end
10 return AT where the final iteration count.
The following theorem shows that our iterative threshold calibration procedure converges in small number iterations. The intuition of the proof that after each iteration, the L2 distance between the prediction functions h and the true CDF ht must decrease by least e². Therefore, the algorithm must terminate before the 1.2 distance decreases below 0 (which is impossible). A full proof is provided in Appendix B.
Theorem 2. Algorithm converges after at most O(1/e) iterations and outputs forecaster with threshold calibration error at most e.
For simplicity, we do not consider finite sample approximation of the TCE in line 5 of Algorithm 1. Line 5 can interpreted in two ways: line 5 estimates the TCE on the true distribution (which we can only do with infinite samples), the empirical distribution (i.c. the uniform distribution

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the recalibration data). Under the former interpretation, Theorem 2 holds assuming that line 5 can estimate the true TCE (which is the ideal scenario with infinite data). Under the latter interpretation. Theorem 2 holds for the empirical distribution, i.e. it guarantees that Algorithm will output forecaster with threshold calibration error at most on the empirical distribution rather than the true distribution. We will instead use experiments to show that Algorithm can generalize to the true distribution. Note that under both interpretations, Algorithm will converge after at most ((1/²²) iterations. For completeness, we describe the finite sample version of the algorithm in Appendix A.
5 Experiments
In the following experiments, we demonstrate that threshold calibration minimize the reliability gap (1) across a range of decision costs, (2) across a range of decision thresholds, and (3) in simple and complex model families. Across all datasets and forecaster model families that we consider, we find that threshold calibration outperforms the baselines in reducing the size of the reliability gap while attaining similar improved decision loss compared to the baselines.
5.1 Datasets
We consider datasets that relate to real-world decision decision-making tasks and standard benchmarks. In the main paper, we show results on the UCI Protein and the MIMIC-III datasets. All remaining results can be found in Appendix A.
MIMIC-IIL Patient length-of-stay predictions are used for hospital scheduling and resource man- agement [17]. We consider a patient length-of-stay forecaster trained on patient admission laboratory values from the MIMIC-III dataset [20]. In our decision task, the hospital decides to schedule a new patient for an elective procedure if a current patient is predicted to have a short length of stay.
Demographic and Health Survey (DHS). Local wealth measurements allocation decisions. We use the DHS data from Sheehan et al. [30] to predict asset wealth from satellite images as done in Yeh et al. [32] and Sheehan et al. [30]. Our experimental setup is motivated by the decision task defined in Yeh et al. [32], where aid is allocated to regions where the predicted asset wealth falls below particular threshold.
used to inform resource
UCI Regression Datasets. We evaluate on a suite of UCI regression datasets (Naval, Protein, Energy, Crime) [11]. They are common benchmarks in the uncertainty quantification literature [31, 2, 8, 23].
5.2 Experimental Setup and Baselines
Experimental Setup. We consider a forecaster that outputs Gaussian distributions and a forecaster that outputs Gaussian-Laplace mixture distributions. We train/validationntest split. The un- calibrated forecaster is a neural network trained on the training set with the validation set used for early stopping. For large datasets (Protein, Energy, Naval, MIMIC-II), the recalibration transform is trained on the validation set. For small datasets (Crime, DHS), the recalibration transform is trained on the training and validation set. On the test set, we evaluate our method and the baselines using decision decision-making making metrics (Section 5.3). Calibration metrics are also measured and results are provided in Appendix A.
Baselines. We compare the uncalibrated forecaster to the forecaster after enforcing average, threshold, or distribution calibration through posthoc recalibration procedure. Methods for achieving these properties are described in Appendix A.
5.3 Decision-Making Metrics
We simulate decision makers enumerated 1. who use a probabilistic forecaster h for their threshold decision tasks. We assume that there is no cost associated with true positives or true negatives, and the total cost of a false positive plus a false nega- tive is equal to 10 for all decision makers. As a result, decision maker i's task is deter- mined by a decision threshold y and decision cost ratio G.. Each decision maker has a
loss function (1x y. a) 10c,1l(a 1, 2 vd) + 10(1 c.)l((a 0, y < vo) and decision rule
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Figure 2: Under the Gaussian forecaster and across different decision thresholds, threshold calibration reduces the reliability gap on both datasets while average calibration does not reduce the reliability gap on the Protein dataset (Left, Middle Left), and all calibration methods yield improved or comparable decision loss compared to the uncalibrated forecaster (Middle Right, Right). Error bars represent 95% confidence intervals and are generated over random trials.
(x) ((][x] 2 a.. We consider decision makers with (36- c,) Vo x C. where Vo and C each consist of 50 uniformly spaced points that span the label space and [0.05, 0.95], respectively.
For each decision maker i, we compute the decision loss (the loss incurred by the Bayes decision rule 5% (X)) and the reliability gap (averaged over the possible threshold decision rules).
1 [C] Σ h18h ())
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Reliability Gap
Aggregate statisties can obtained by averaging over all M decision makers, all decision makers who share the same threshold yo. or all decision makers who share the same cost ratio c.
5.4 Results
Using the MIMIC-III and UCI Protein datasets, we study the effect of recalibration the reliability gap and the decision loss achieved by decision makers with different decision thresholds and cost ratios. Furthermore, we examine the effect of recalibration forecasters that output CDFs from simple (Gaussian) and complex (Gaussian-Laplace mixture) model families.
Threshold Calibration Minimizes Reliability Gap Across Decision Thresholds. We evaluate the effect of recalibrating the Gaussian forecaster on decision makers with different decision thresh- olds. On both datasets, threshold calibration yields the largest decrease in the reliability gap (left plots, Figure 2). Distribution calibration also decreases the reliability gap across decision thresholds, relative to the uncalibrated Gaussian forecaster. Average calibration does not consistently reduce the reliability gap: on the UCI Protein dataset, the reliability gap of the werage-callibaate forecaster enjoys slight decrease at some decision thresholds but is increased at others, relative to the un- calibrated Gaussian forecaster (middle left, Figure 2). Lastly, these calibration methods achieve similar decision loss the uncalibrated forecaster (right plots, Figure 2). These trends are consistent with the results obtained on the other datasets under the Gaussian forecaster. Threshold calibration outperforms baselines across different decision thresholds under the Laussian-Laplace forecaster, as well (Appendix A).
Threshold Calibration Minimizes Reliability Gap Across Decision Cost Ratios. Across deci- sion makers with different cost ratios, distribution and threshold calibration reduce the reliability gap relative to the uncalibrated forecaster, with threshold calibration yielding the largest decreases in the reliability gap (left plots, Figure 3). Mcanwhile, average calibration does not consistently reduce the reliability gap; the UCI Protein dataset, it achieves similar reliability gap the uncalibrated forecaster (middle left, Figure 3). As before, these calibration methods achieve similar decision loss to the uncalibrated forecaster (right plots, Figure 3). These trends are consistent with results obtained on the other datasets under the Gaussian forecaster. Threshold calibration outperforms baselines across different decision cost ratios under the Caussian-Laplace forecaster, as well (Figure 4).
Distribution Calibration Degrades Performance under Complex Model Families. Forecasters that can output CDFs from more flexible model families (c.g. -aussian-Laplace mixture distributions)
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Figure 3: Under the Gaussian forecaster and across different decision cost ratios, threshold calibration reduces the reliability gap on both datasets while average calibration does not reduce the reliability gap on the Protein dataset (Left, Middle Left), and calibration methods yield improved or comparable decision loss compared to the uncalibrated forecaster (Middle Right, Right). Error bars represent 95% confidence intervals and are generated over random trials.
MIMIC-III Patient Longth-af-Stay
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Figure 4: We consider the effect of recalibrating the Laussian-Laplace forecaster under a range of decision cost ratios. Threshold calibration reduces the reliability gap while distribution calibration can enlarge the reliability gap (Left, Middle Left), Average and threshold calibration achieve comparable or lower decision loss as the baseline forecaster, while distribution calibration increases the decision loss. Error bars represent 95% confidence intervals and generated over 6 random trials.
may be able to better capture the true conditional distribution of given x compared to Gaussian forecasters. As result, examine the effect of the recalibration procedures when the uncalibrated forecasts follow a more flexible distribution. The uncalibrated Gaussian-l .aplace forecaster (Figure 4) yields smaller reliability gap and smaller decision loss compared the uncalibrated Gaussian forecaster (Figure 3). Applying threshold calibration to the Gaussian-Laplace forecaster further reduces the reliability gap. However, under the Gaussian-Laplace forecaster, distribution calibration enlarges the size of the reliability gap and increases the decision loss. Although distribution calibration theoretically guaranteed minimize the reliability gap. is challenging achieve in finite samples without compromising the sharpness of the forecasts (in our case, decision loss). So, we find that decision loss and reliability gap increase. We hypothesize that the recalibration dataset may not contain many instances that yield similar distribution parameters, so the recalibration transform does not generalize well unseen data. We also observe these trends the UCI Crime, UCI Energy, and DHS datasets.
6 Related Work
Forecasting and Decision Making. The connection between forecasts and decision making was first studied in economies [1, 29]. The development of Bayesian decision analysis connected topics of forecasts and decision-based loss functions [10, 4]. Decision-makinn under uncertainty with probabilistic forecasts was then studied in econometrics
[7]. [19] also considers learning regression functions that minimize decision loss. While [19] focuses on transforming the predicted CDF to a point prediction, our method focuses transforming the predicted CDF into new CDF [19] also requires knowing the loss function to learn the transformation, while our method assumes that the loss function belongs to commonly used function family (threshold loss functions).
Calibration. Calibration definitions have been studied in the statisties literature [5, 6, 26]. For the regression setting, methods for ensuring that machine learning models satisfy average calibration

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have been studied in [23, 8]. In addition, methods for achieving stronger calibration notions have also been introduced such distribution calibration [3 and individual calibration [34]. Calibration and trustworthy predictions in the medical domain are also studied in [16]. [16] introduces the notion of D-calibration, which is related to our average calibration baseline, but is tailored to the survival analysis task. A perfectly average calibrated prediction function is also D-calibrated, and vice versa.
Multicalibration. Our definition of threshold calibration is most related to the line of work on multicalibration [18, 21J. Given a large collection & of potentially intersecting groups of the data, predictor is multicalibrated on it is simultaneously calibrated on every sufficiently large group in G [18]. Previous works give methods for achieving mean and moment multicalibration for predictor functions. Our iterative procedure for achieving threshold calibration inspired by methods for achieving multicalibration.
7 Limitations and Societal Impact
Our work demonstrates that certain types of calibration enable decision makers to estimate decision loss before deployment, which should not be confused with enabling decision makers to make optimal decisions. For example, forecaster that always outputs the marginal distribution of Y is threshold-calibrated but likely incurs high decision loss. Furthermore, posthoc recalibration is limited by the quality of the baseline model. If the baseline model outputs the marginal distribution of Y. then it is already preesold-ccaibratee but likely is not useful for decision making. Applying our threshold calibration method will not offer any benefit in this case.
Also, our work assumes that predictions of do not affect the true label Y. However, when predictions used to make decisions, they can often influence the outcome they aim to predict [27]. Our work does not account for these performative effects, so the decision loss may not be accurately estimated in these settings. Future work could focus developing calibration procedures that enable forecasters to be robust to such distribution shifts. In addition, we specifically focus on binary-action threshold decisions. Future work may generalize our results the setting where decision makers have loss functions involving multiple thresholds and multiple actions.
There is potential for negative societal impact if threshold calibration is incompatible with fairness criteria. Nevertheless, we note that the perfect predictor (that predicts the true conditional probability) satisfies calibration definition. Consequently, if the perfect predictor satisfies some faimesss notion (such as group calibration), then our calibration definition is also compatible with that fairness notion. Note that the perfect predictor does not satisfy faimess notion called demographic parity, hence calibration definition is not compatible with demographic parity either.
Conclusion
We show that a threshold-calibrated forecaster theoretically guarantees accurate decision loss estima- tion under threshold decision losses and threshold decision rules. We provide an iterative procedure for achieving threshold calibration and show that in practice it minimizes the reliability gap relative to baselines without compromising the forecaster's s decision loss. Such estimates permit decision makers to reason about the consequences of their decisions prior deployment.
Acknowledgements
RS supported in part by a NSF GRFP under grant number DGE-1656518. SZ is supported in part by a JP Morgan fellowship and a Qualcomm innovation fellowship. SE is supported in part by NSF #1522054, #1733686). ONR (N000141912145) AFOSR (FA95501910024) ARO (W911NF-21-1-0123 and Sloan Fellowship. We are grateful for Rishi Bommasani, Kristy Choi, Matthew Jörke, Judy Shen, Rui Shu, Fan-Yur Sun, Rohan Taori, Ke Alexander Wang, Rose Wang, and Henry Zhu for insightful discussions.
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Checklist
The checklist follows the references. Please read the checklist guidelines carefully for information on how to answer these questions. For each question, change the default [TODO] [Yes], [No], or [N/A]. You are strongly encouraged to include justification to your answer, either by referencing the appropriate section of your paper or providing brief inline description. For example:
Did you include the license to the code and datasets? [Yes] See Appendix A.
Did you include the license to the code and datasets? [No] The code and the data are proprietary.
Did you include the license the code and datasets? [N/]]
Please do not modify the questions and only use the provided macros for your answers. Note that the Checklist section does not count towards the page limit. your paper, please delete this instructions block and only keep the Checklist section heading above along with the questions/answers below.
1. For all authors...
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(a) Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? [Yes]
(b) Did you describe the limitations of your work? [Yes] See Section
(c) Did you discuss any potential negative societal impacts of your work? [Yes] See Section 7.
(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
2. If you are including theoretical results...
(a) Did you state the full set of assumptions all theoretical results? [Yes]
(b) Did you include complete proofs of all theoretical results? [Yes] See Appendix B.
3. If you ran experiments...
(a) Did you include the code, data, and instructions needed to reproduce the main experi- mental results (either in the supplemental material or as URL)? [Yes]
(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] See Appendix
(c) Did you report error bars (e.g., with respect to the random seed after running experi- ments multiple times)? [Yes] See Section and Appendix A
(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] See Appendix A.
4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
(a) If your work uses existing assets, did you cite the creators? [Yes] See Appendix A.
(b) Did you mention the license of the assets? [Yes] See Appendix A.
(c) Did you include any new assets either in the supplemental material or as URL? [Yes] See Appendix A.
(d) Did you discuss whether and how consent was obtained from people whose data you't 're using/curating? [N/A] We did not directly obtain data from anyone but used publicly available datasets.
(e) Did you discuss whether the data you are using/curating contains personally identifiable information offensive content? [N/A] The creators the MIMIC-III dataset. which we use. deanonymized the data to remove any personally identifiable information. To the best of our knowledge, there no other potential source of personally identifiable or offensive content in the data we use.
5. If you used crowdsourcing or conducted research with human subjects...
(a) Did you include the full text of instructions given to participants and screenshots, applicable? [N/A] We did not do human subject research.
(b) Did you describe any potential participant risks, with links Institutional Review Board (IRB) approvals, if applicable? [N/A] We did not do human subject research.
(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A] We did do human subject research.
14


Reviews:
[{"review_id": "hbKZLMcS-cN", "reviewer": "Reviewer_Y7ES", "summary": "As pointed out by the paper, threshold calibration is related with practical hospital scheduling problem, in which hospital needs to make the decision to acquire new patients based on predicted length-of-stay to the existing patients. The paper proposed an efficient and practical threshold calibration method, which calibrated the non-calibrated results based on CDF prediction, and is shown to have better calibration results as compared to average calibration and distribution calibration.  Additionally, the paper proposed an algorithm to do threshold calibration, and empirical results on real data sets shows reduced reliability gap and true decision loss. ", "questions": "", "limitations": "", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "[1] The paper formulates the problems into regression problems, predicts the likelihood of stay less or equal than y_0 and makes decision based on the if the likelihood is above a threshold \\alpha, as mentioned in section 2.3. How are \\alpha and y_0 get selected, are they flexible to tune or pre-determined? Since the problem is mainly to predict the decision, what is the major reason to formulate the problem as a regression problem rather than a classification problem?\n\n[2] This paper proposes a min-max approach to find y_0, \\alpha to maximize TCE, followed by a greedy approach to fix the mis-calibration with the selected breakdown identified by y_0 and \\alpha, as described in algorithm 1. How to select y_0 and alpha to maximize TCE, is it via grid search? ", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "Not Applied", "review_text": "", "extra_scores": {"ethical_concerns": "No", "needs_ethics_review": "No", "ethics_review_area": ["I don’t know"], "time_spent_reviewing": "2"}, "extra_text": {}}, {"review_id": "YHoUD3Bobfp", "reviewer": "Reviewer_YN6e", "summary": "The paper proposes a new concept called threshold calibration of probabilistic regression models, and proposes a method with theoretical calibration guarantees. The method is evaluated on real datasets and is shown to outperform existing methods.\n", "questions": "", "limitations": "", "rating": 8, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "The paper fills an important gap between average calibration and distribution calibration. The ideas are simple and well justified, the intuitions have been explained well, a technically strong paper. I have only the following minor comments.\n\nFrom the definition of the reliability gap it is not obvious whether the gap could also become negative. I suggest to\n have an answer to this question either in the main paper or in the appendices.\n\nAll the brackets have the same size in all the formulas, I recommend using different sizes. In formulas like the one\nin Definition 4 it would be much easier to visually grasp where the subexpressions start and end. However, it is cert\nainly also a matter of taste.\n\nIn lines 182-188 on page 5 the claims (average calibration does not imply threshold calibration does not imply distribution calibration) are intuitively obvious, but perhaps particular examples would be useful (could also be presented in the appendix).\n\nAlgorithm 1 refers to the use of isotonic regression, but some additional details about the implementation should be mentioned. The isotonic function is uniquely determined on its training points, but it could be interpolated in various ways. One possible choice is the constant interpolation (e.g. each test point gets the same label as the closest training point that is less or equal to this test point), and another choice is linear interpolation.\n\nLine 261 states that $\\mathcal{Y}_0$ has 50 uniformly spaced points spanning the label space. It would be useful to mention that it is spanning the label space between its minimal and maximal value in the training (?) data set.\n\nThe sentence 'Let $\\mathcal{C}$ be the set of possible choices for $\\alpha$.' in page 7, line 262 could be rephrased because the previous sentence has basically already stated the same.\n\nThe notation $\\delta^i_{h,\\alpha_*}$ on line 263 and $\\delta^i_{h,\\alpha^*}$ on line 264 is confusing. If I understand correctly, then the asterisk should really be in the superscript of $\\delta$ in accordance to Definition 1 of the Bayes decision rule.\n\n==\nAfter discussion:\nI am happy to stay at my suggested grade 8.\n", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "The limitations and social impact have been covered adequately.", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "4"}, "extra_text": {}}, {"review_id": "GpsnMxqd_jN", "reviewer": "Reviewer_H5Nm", "summary": "This work applies the concept of calibrated regression into a decision making framework where the decision is confined to threshold decisions. The Bayes optimal decision w.r.t. the predicted distribution is derived, and the concept of threshold calibration is defined. Threshold calibration is a sufficient and necessary condition for the reliability gap to be 0, meaning the absolute difference between the predicted expected loss and the true expected loss is 0 for a given decision rule. Further, a simple algorithm is proposed which threshold recalibrates a distributional prediction.\n", "questions": "", "limitations": "", "rating": 8, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "This paper is very clearly written and the topics/problems discussed are well motivated. While there is a lot recent attention on achieving calibration for distributional predictions, exactly $\\textit{how}$ calibration can be used in a downstream task and to what degree it is desirable has not been discussed very much, and this paper provides a thorough analysis at least in the context of threshold decisions.\nThe newly defined concepts are also very relevant and theoretically sound.\nWhile the empirical results are not especially surprising given threshold recalibration is tailored to the problem setting and evaluation metrics used (reliability gap), it does provide a good baseline for future works in applications of calibration in decision making tasks.\nI believe the problem setting, concepts, derivations, and methods this work provides will be of value to the general UQ community.\n\nJust a couple of questions/points:\n1. In many of the definitions, while I understand what the authors mean, I believe the notation is slightly off. E.g. in average calibration (Definition 5, L170), the statement should not be for small $x$, but for the random variable $X$. Small $x$ is a realized value (as the authors also indicate in the Preliminaries), which implies conditioning each $x \\in \\mathcal{X}$. In this case, it would indicate another notion, \"individual calibration\" [33]. \nThe same point goes for other places too, e.g. Definition 6.\nI am aware the notation is correctly written in various places in the Appendix, e.g. L572.\n\n2. Was any hyperparameter tuning performed for the experiments? I cannot find any details in the Appendix. While distribution calibration is reported to degrade reliability gap (L293), can this simply be because there are not enough points in the recalibration set (as the authors also hypothesize)? I'm wondering if simply tuning the percentage of validation points used for recalibration (currently set to 10% of training) can alleviate this.\n\n3. Lastly, the code link in the Appendix doesn't work. \n", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "The limitations are well addressed in Section 7, with constructive statements about future directions for improvements. ", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "5"}, "extra_text": {}}, {"review_id": "DkLTpT-gjWN", "reviewer": "Reviewer_CBqV", "summary": "This paper studies calibrated prediction from a decision loss perspective. The main contribution is the proposal of a new notion of threshold calibration, an algorithm to learn predictors achieving this new notion of calibration, and well as experiments demonstrating the effectiveness of the proposed algorithm.", "questions": "", "limitations": "", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "This work studies an important problem of learning regression classifiers which are well calibrated for the purpose of downstream decision-making. While the setup is interesting, there are few points worth highlighting.\n- Definition 3 which introduces the notion of threshold calibration is quite opaque the way it is written. I would like to see some intuition into why this should be the correct notion (I can see from the proof of Theorem 1 how it arises, but am still unable to get an intuitive idea of this definition).\n- Proof sketch is missing (Theorem 1). While I understand that there is a page limit on submission, I think it is very important to highlight to both the reviewers and readers that the theorem is provably correct, either by providing some intuition or giving a sketch of the main idea.\n- One can ask this question of decision based calibration in a much more general framework (rather than just focussing on threshold functions for decisions). Can the authors perhaps point out how the notion of calibration change for general functions mapping cdfs to decisions.\n- On line 182, the authors mention that the converse of Proposition 1 is not true. Can the authors perhaps add in a proof of this to the paper, giving counter examples to the two statements.\n- Theorem 2: What happens to the prediction accuracy of the forecaster h^T as compared to the input forecaster h, when evaluated on the regression prediction task? What about sample complexity issues? How many samples are required to perform this recalibration -- this is quite important since I feel that the notion of threshold calibration as defined for all (x, y, c) tuple is indeed quite a strong one.\n- The experiment section of the paper is quite weak as well. I would like to see some some more simulation experiments which are directed towards studying the limitations of the proposed methods, some more loss functions l^i, a wider variety of decision rules, different choices of recalibration functions (based on quantile methods, optimizing some variant of the pinball loss) as well as additional baselines (optimizing directly for the decision loss, fine tuning the regression function for the decision loss).\n", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "yes, the authors address it", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "2"}, "extra_text": {}}]

Author rebuttal:

Meta-review: This paper looks at how to guarantee accurate decision loss prediction when the predictions are based on machine learning models. The authors propose the notion of "threshold calibration" which provides a stronger guarantee than the traditional average notion of calibration. This ensures that decision loss is predicted accurately when the threshold decision is used.

A threshold calibration is a promising approach compared to average calibration or distribution calibration. Reviewer H5Nm and YN6e concur that the results presented in this paper "fill an important gap between average calibration and distribution calibration". However, Reviewer CBqV raises a point that the empirical evaluation could be extended further, hence recommending a low score. Reviewer Y7ES thought the paper is borderline and did not provide a strong justification in favor of one way or another. Hence, it does not play an important role in my decision.

I find Reviewer CBqV's argument on the empirical evaluation relatively weak, especially for the venues like NeurIPS, and after reading the paper, I also find the studied problem to be quite timely. The paper is well-written as well as thought-provoking. The only suggestion I have for the authors is to move the related work section toward the beginning of the paper. This could help the non-expert readers better understand the literature.

Following the recommendation of Reviewer H5Nm and YN6e, I recommend this paper for publication at NeurIPS2021 as a poster.
Decision: Accept (Poster)