On Training-Conditional Conformal Prediction and Binomial Proportion Confidence Intervals
Authors: Rudi Coppola
Manuel Mazo Espinosa
Venue: Accepted by TMLR ()

Abstract: Estimating the expectation of a Bernoulli random variable based on $N$ independent trials is a classical problem in statistics, typically addressed using Binomial Proportion Confidence Intervals (BPCI). In the control systems community, many critical tasks—such as certifying the statistical safety of dynamical systems—can be formulated as BPCI problems.

Conformal Prediction (CP), a distribution-free technique for uncertainty quantification, has gained significant attention in recent years and has been applied to various control systems problems, particularly to address uncertainties in learned dynamics or controllers. A variant known as training-conditional CP was recently employed to tackle the problem of safety certification.

In this note, we highlight that the use of training-conditional CP in this context does not provide valid safety guarantees. We demonstrate why CP is unsuitable for BPCI problems and argue that traditional BPCI methods are better suited for statistical safety certification.

Published in Transactions on Machine Learning Research (02/2025)
On Training-Conddtionaal Conformal Prediction and Binomial Proportion Confidence Intervals
Rudi Coppola
r.coppolaSuuellt..t
Department of Mechanical Engineering Delft University of Technology
Manuel Mazo Jr.
m.mazo@tudelf..nl
Department of Mechanical Engineering Delft University of Technology
Reviewed on OpenReview: https: // openreview. net/ formm? id ddppkkgyytob
Abstract
Estimating the expectation of a Bernoulli random variable based on N independent trials is a classical problem statistics, typically addressed using Binomial Proportion Confidence Intervals (BPCI). In the control systems community, many critical tasks such as certifying the statistical safety of dynamical systems- can be formulated as BPCT problems.
Conformal Prediction (CP), a distribution-free technique for uncertainty quantification, has gained significant attention in recent years and has been applied various control systems problems, particularly to address uncertainties in learned dynamics or controllers. A variant known ihrining-comditionni CP was recently employed to tackle the problem of safety certification.
In this note, we highlight that the use of training-condditinnal CP in this context does not provide valid safety guarantees. We demonstrate why CP is unsuitable for BPCI problems and argue that traditional BPCI methods are better suited for statistical safety certification.
1 Introduction
Uncertainty quantification is critical aspect in fields where predictions influence safety and performance guarantecs, such as in control systems. Probabilistic guarantces, including those derived from the theory of Probably Approximately Correct (PAC) learning, play an important role providing bounds on the accuracy of predictions under limited training data.
Conformal Prediction (CP) is one of the approaches that has gained visibility due to its ability to provide valid prediction sets without requiring strong distributional assumptions. A distinctive characteristic. of CP is that, rather than providing point prediction of the variable of interest, it provides set predictions with a valid bound on the probability that the predicted set contains the true variable (Vovk et al., 2005). This technical note focuses on a specific formulation CP known as condiiiona CP (Vork, 2012). However, existing applications areas such as safety verification for dynamical systems have shown limitations in the CP safety
interpretation of these guarantees. In particular, recent works have applied training-condice verification problems in control systems (Chilakamarri et al., 2024; Lin & Bansal, 2024; Vincent et al., 2024). While promising, these applications have misinterpreted the implications of CP's set prediction framework, especially in cases where the underlying data can be modeled as Bernoulli random variables. This paper aims to rigorously analyze these limitations and provide an alternative framework for interpreting PAC- based guarantees in such contexts. In section 2 we recall existing methods for estimating the expectation of Bernoulli random variable. In section we introduce the formalism of training-conditional CP, followed by a detailed analysis of its PAC guarantees. In section 4 we present a special case of interest, where the nonconformity measure corresponds to indicator function, leading to Bernoulli-distribtted conformity

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scores. We demonstrate that the PAC guarantees derived from this setting unsuitable for estimating the expectation of Bernoulli random variable.
2 Binomial Proportion Confidence Intervals
Consider a setting where there are N + independent and identically distributed (i.i.d.) Bernoulli random variables (r.v.) R1 Rg,..., Rx, RN+1 with parameter b, ie. R4 ~ Bern, and PrBem, (R = 1) - b. Given a realization of the first I.V., the problem to estimate an interval of values for the probability that the N + 1-th variable will be equal to 1, or other words we want. to estimate the parameter b. This is a very well studied problem and it is known in the literature under the name of Binomial Proportion Confidence Intervals (BPCI), see Dean & Pagano (2015) for survey. We give below quick overview of the setting.
BinN,6 with N trials and probability of success b, defined by PrBinx,y (Y = v) = (P) -1-11-----0)) bb)" N- for y E 2 [o.N]: where 2 [oo.) denotes the integers 0, I.....N. NN. Let b 2o.N] + [0, 1] and b [o..v] - [0, 1] be two random variables serving interval estimators. The coverage probability of the interval estimator [6, b] for Y ~ Binn, is defined as
Define the new r.V. Y = RG. It is well known that Y has a binomial distribution
(b.b,6) = Pr <bs B(Y)).
(1)
In the expression above b is fixed and it's the true parameter of the binomial distribution describing Y. Note that 6 and 6 are a transformation of the same random variable Y. This expression can also be rewritten oquivalently as
b(b,b,8) = -EP Priing, (Y y).
PET
where -= {u E 10.N1 b(y) < bb b())) For a E 10 an interval estimator [ö, B) is a conservalively valid confidence interval if the coverage probability p(b, b B) is
(sometimes also called 'exact' or 'sccure') greater equal to 1 for all the values of b. An example of a conservatively valid interval estimator is given by the Kkopper-Pearsee method (Clopper & Pearson, 1934), see also Dean & Pagano (2015) for more estimators.
Before concluding this section, we rewrite equation 1 in an equivalent form that is more commonly found in the literature on Probably Approximately Correct (PAC) bounds. First, note that PrBern, (RN+1 = 1) b. Second, since Y is a r.v. obtained as transformation of the i.i.d. Bernoulli random variables R,,..., RN, the probability of any event M C p. Pr (Y E M) can be equivalently described by Pr ( (((r1, TN): ENSN F3 E M),, where Pry Preern, is the product probability mcasure induced by the N i.id. Bernoulli random variables. Hence, we rewrite the definition of a conservatively valid a confidence interval by revisiting equation 1:
Priiux, & (b(Y) <bb = Prefrrs
i
Σ IS
R
S
P = 1) =
Σ Rs iSN
2 21-a,
(2)
for all b E [0, 1). We will use this form of the coverage probability to draw comparison with the guarantees given by training-cont conditionl CP.
3 training-conditionaa Conformal Prediction
Conformal Prediction is a statistical tool that uses the available data sampled from identically and indepen- dently from an underlying distribution to output predictions for which an crror probability can be computed. The original formulation of CP can be informally explained as follows. Suppose that we want to solve a classification problem and we have method that given a feature x outputs a label g. Given a desired error probability e, conformal prediction uses the available data to generate a set of labels, typically containing Ú. containing the true label y corresponding to the feature æ with probability not smaller than 1-e e (Shafer & Vovk, 2008). It. is a method capable of augmenting (usually unreliable) point prediction to set prediction
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with probabilistic guarantees of correctness, ie. it construct a set predictor. The original formulation of CP has been successfully applied to both classification and regression problems, see Angelopoulos et al. (2023); Fontana et al. (2023) for a recent survey.
In this section we introduce instead the basic concepts of coniriiinnc CP (Vork, 2012). Training- conditional CP a variant of the original formulation of CP. While the quality of the guarantees differs from the original, the core idea remains the same, that is, constructing set predictions with some form of guarantecs: coaniing-condttionall CP produces PAC-style guarantees. In the following, we give self-contained overview of the theoretical details of training-conditioon
CP.
Let (Z, P)) be a probability space where Z, F and P denote a sample set, à o-algebra, and a probability measure respectively, and consider L + 1 i.id. random variables (r.v.) Zi, Zh Zi..., ZN and ZN+1 with == N + M. Let Z for = .. be the training set and Z; for i = I,....N be the calibration set, Note that ZN+1 not part of cither set. We use the lower case of a r.v. to denote a rcalization'. An Inductive Nonconformitty M-measure (INM) is measurable function A ZM Z + R. While no additional requirements are needed for A, intuitively effective INM will assign high real number to any element in Z that does not conform to a training set (in MM). An Inductive Nonconformal Predictor (INP) is a set predictor defined as
(" 2N, 21 ... x = {a E Z
p² E).
(3)
where e E [0, 1] is the signéficance level, the p-vauues are defined
p = 2 +1
(4)
N +1
and
R =
for i = =I,...N, R A(( 2),,,
(5)
are the nonconformity scores.
In the following, when it is clear from the context we omit the arguments of the INP and write r instead of (z 2N, zi...., Intuitively, = belongs to the INP Pe there are strictly more than [c(N + +1) -1] elements R2 in the calibration set with a higher (worse) or equal nonconformity score than R=. easy to see that e' < e" implies that - gr The INP is the set predictor mentioned in the discussion at the beginning of this section: similarly to the original formulation of CP, given some prediction method depending the training set, the INP uses the available calibration set to produce a set prediction guaranteed to contain the correct prediction. The elements included in the set prediction are all the E Z that conform well enough with the calibration set, according to the chosen INM. The following theorem specifies the PAC-style guarantecs for training-conditionn CP.
Theorem (Vovk (2012)) Choose e, E. [0, 1], for the training set Z 2/....g ...,, Z'A 2 let N be the size of the calibration set, and consider the event
Sg - ((a.....,N)) 2N) N P(ZN+1 I' 2N. 2 2,,,,, 2())) 2 1-111
(6)
in the o-algebra FN of the product probability space (NN, where I is defined according to equations 3-5. It holds that
(N 21
where 8 = BinN (J) (J) ES ( 1-00-E---- EYN is the cumulaatie binomial distribution with N trials and probability of success E, with = LE(N +1)-
'Our considerations hold also for the case where Z X x Y where X and Y represent a measurable feature space and label space respectively and each Z may be written = (z, u) where X is some feature and Y a label. For clarity omit the exact structure of Z.
A brief note the notation. In the original formulation of CP has double role: it is the significance level (appearing as the index to the INP P') and describes the coverage probability maalning-conditionn formulation the latter role covered by E, that E the coverage probability and remains the significance level, see Vork (2012).
c, see Shafer & Vovk (2008) for details. In the
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The quantities 1 8 and E are sometimes referred to the confidence and coverage probability (which is not the coverage probability mentioned in section 2). Theorem is to be understood in the following way. Given two values e and E, for the given training set, the event SE is the subset of Z containing all the tuples (21,...,yy such that the INP I contains a rcalization of ZN+1 with probability at least 1 - E, or, in other words, N returns a subset of Z of measure al least. 1 - E. By equation the measure of this set of tuples Sg is at least. 1 6, where depends on c, and N. This form of guarantees where double layer of nested probabilitiess distribution free result, that is, it holds for every P as long the samples used to construct the INP are i.i.d. and P-distributed. In particular, in this work we focus on Bernoull-diitributee r.v.s, and Theorem 1 holds for any value of the parameter 6 of a Bernoulli distribution. Observe that the confidence 1 - and the quantity 1 - mentioned in section 2 play similar role in that they described the outmost. layer of probability, compare for instance equations equation and equation Finally, we note that and E are chosen a priori; in other words, they cannot be defined as random variables depending on rcalization of the calibration set, as erroneously done in Lin & Bansal (2024).
present is called Probably Approximately Correct (PAC). Moreover, this is
4 A Special Case of Interest
In this section draw a parallel between the BPCI and training-conditionn CP and show the fundamental difference between the two approaches.
Let the INM be an indicator function for the set Q C Z, that is
if if if
A((z), x)
(8)
Typically Q depends on 2f,,.,, 2M For example, in binary classification problems, the training set may be used to train paramcterized function that assigns one of two labels to all Q, in Support Vector Machines. However, since Theorem assumes a given training set, omit this dependency here. A point z with a high nonconformity score is interpreted as poorly conforming to the training set. For this reason, in section 4.1 the set Q will represent the unsafe region of a dynamical system.
Given a fixed training set, the nonconformity scores of the calibration set follow an iid. Bernoulli distribution with parameter b, i.e. RG ~ Berng, where b = (Q)). Using a BPCT method it directly possible derive a conservatively valid confidence interval for the parameter b describing the probability of drawing sample in Q, as shown in section 2. Can praning-conditionall CP approach also provide conservatively valid confidence interval for b based on the calibration set?? The answer is no. We illustrate this with an example.
Example 1 Part 1.
Suppose that the calibration set has size 2, i.e. N 2. Up to reindering, there are three distinct outcomes. Case 1: With probability (1 - b)2 we have 21, 22 & Q, resulting in nonconformity scores Rí R 0. We construct the prediction set 1 following its definition equation 3.
For all z E Q, have that R = 1, meaning: has the highest (worst) nonconformitty score. Since Ifi < 2 R R = O the corresponding p-value is p² 3
For all a E Q we have that R = 0 resulting in and p² ===1.
The inclusion of z in the predicted set r depends on the significance level e.
Ife E 1) then any EQ Q is excluded from I since p² = S e, while all Q are included since pd =1 =1>6 Thus, De =QQ Q.
EfeE [0, 2) then any z E QUQ = has a sufficiently high p-value, meaning r = Z.
Case 2: With probability 20(1 - b) have (zr E @A E Q) (22 E EQAZ Q) hence Rí U RS (0. 1}.

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If R = 1 then px
If R 0 then p²
Thus:
I E [0, 2) then r = Z.
Ife ( 1) then r =Q Q.
Case 3: With probability 6 we have 21, Q and Rı R2 1.
If RF 1 then p²
Tf R = O then p² = 1 as well
Then for any significance level E [0, 1) i holds 1 ===0 Z.
In summary, for any fixed the INP is fully determined by the calibration set through equations 3-5; as a result, 1" can be thought equivalently as discrete random variable with support Q, Q and Z, see Figure 1.
Z
10
QxQ
QxQ
(2) 22)
0.6
Mass
0.4
Z
QxQ
Q Q
Q
0.0
1
Z
C 0.6
0.8
Figure On the left, representation of the product space Z Z Z, partitioned accordingly to the sets Q and Q, and hypothetical calibration set (21, 22) as in Case 1. On the right, summary of Case 1, 2 and 3. On the z--0-,2-axes are represented the values of c, the prediction (or support) of the INP, and the probability mass function respectively. For any given e, the INP I can be viewed as discrete random variable with support Q, Q and Z. the figure, for e = 0.8 and b = 0.3, the INP predicts Q with probability 0, with probability 62, and Q with probability 62.
Example 1 Part
Now, fir E e [0,1] and consider any 1). Theorem 1 implies that
P²(SE) 2 E
(9)
where
Sg = {(21, 22) E zZ P(7 (2 22, =(,--.,(m)) 21 -F.).

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Published in Transactions on Machine Learning Research (02/2025)
However, equation 9 does not provide confidence interval for the probability of drawing
sample in
Q, or conversely in Q. For E E N = Z with probobility b2 (from Case 3) and r = Q with probability 1 1- (from Case 1 and 2), see Figure 1. Theorem 1 is a distribution-free result and as such it holds for all values of b, leading to two cases b E and b > E:
1. bf < E (L.e 1-b21-E).
If 21,120 22 E EQ (Case 3) we have that P(ZN-1 E - E, hence Q x Q [ Sg. If at least one of and 22 belongs to Q (Case and 2) we have that P(Zx+1 E TT² F(Q)
1-b2 2 1 - E, hence Q x Q S Sp.
Thus, SE Zz Trivially, (2(S)) p2 (z²) = 12 E².
1f > E (L.e. 1 1-6-111 B)
If 21,22 E Q (Case 3), as before, P[ZN41 (T) P(Z) = 1 -E, and once again QxQCSa
. af least one of 2ι and 22 belongs to Q (Case 1 and 2) then P(ZN+1 E ET) P(Q) b 1-E, hence such 21 and 22 do not belong to Sg by definition.
. Thuus P²(SE) = (2 x Q) 622 E
Theorem 1 holds for both cases, since we have either p2 pz²) 1 12 E² or P2 x Q) = 62 2 E². Now, assume b > E and that the calibration set gives R1 = O and R2 = 1. What can we say about b?
For the given calibration set and significance level the INP predicts 1 = Q, hence it is tempting to say that (P(()() 2 1 E) = (2
1) E², or equivalently (² S E) 2 E²; recalling equation 2, we may conclude that [0, Ej is E² confidence interval for b. But this is clearly not true: since we assumed that b > F the interval [0, F)] will never contain the parameter b (note that none of the arguments of P2() depends on #2) in the preceding statement, unlike equation 2). We conclude from this example that this is not a viable polh to oblain PAC bound for b comparable to equation 2.
The example above leads us to the following remark and main message of this technical note.
Remark 1 Theorem 1 guarantees the correctness of the set predictor 1" Adopting the frequentist perspec- tive, it is statement on how often the set predictor I constructed from N samples attains the desired coverage level 11-E E for a new realization of ZN+1- In other words, since is unknown, if b >> E the INP attains the desired coverage level only when 1 = Z (which is a trivial prediction), and does not attain the desired coverage level when I = =Q. Essentially, the confidence level of E² is attained by making trivial predictions sufficiently often. If instead b S E, the INP always correct. Thus, Theorem does not estimate or provide information on the probahility of specific score class, which is the goal of BPCI methods. See the appendiæ for graphical representation.
To further clarify, consider the equivalent set predictor mapping the clements z predicted by I to their respective nonconformity score
T (E R,.,,,NN =(.,.(..., (E Ff......... (.
U
which amounts to P (0, 1) when - Z and r== {0} when r== Q. Let RN+1 = A(( *hs); ZN+1) be the score of the 1+t-th sample. Then, we can replace the event RN+1 E T with 7N+1 E [e in equation 6. In essence, both BPCI methods and training-conditionn while BPCI methods compute an interval containing the true value b describing the probability of the event that the N + 1-th score equals 1, Lee RN+1 = (with probability not less than - aa, training-condit ional CP computes a lower bound for the probability of the event that the N 1-th score contained in the predicted set of scores, L.e. RN+1 I (with probability not less than 1- 8).
CP provide PAC guarantees but differ in scope:
3Tf re predicts 2 implies that the nonconformity score of Zx+1 predicted to 0, whereas if it predicts Z then know is that the nonconformity score Z441 predicted be (0, 1) which is uninformative.
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(see their Sec. Proofs-D). However, Published in Transactions on Machine Learning Research (02/2025)
Remark 1 extends to any scenario where the nonconformity score takes values from finite set, effectively defining classification problem. 'Training-oonditiona CP provides framework for constructing set predictor that guarantees the desired coverage level with a minimum confidence. The predictor adapts to the calibration data: for 'good' calibration data, it produces tight sets (few classes), while for 'poor' calibration data, it putputs loose sets (many classes). On average, the probability that the calibration data is least 1- 6.
yields a predictor attaining the coverage level of
Depending the choice of e and E, we have shown that the 1-8 confidence level may be achieved simply by predicting the entire sample space (i.e., all classes) sufficiently often (see Figure 2). However, this approach does not provide meaningful information about the probability of specific class, which is the focus of equation 2 and, more generally, BPCI methods.
4.1 A Note on Safety Verification for Dynamical Systems
Recent studies have applied training-conditionaal CP, particularly Theorem 1, to provide PAC guarantees on the safety of control systems with neural network- based controllers (Chilakamarri et al., 2024; Lin Bansal, 2024), and more broadly, on the safety of autonomous systems (Vincent et al., 2024). this section we show that these works follow the reasoning outlined in section 4, and are therefore incorrect. Below, we follow the notation used in Lin & Bansal (2024), but the same applies to the other works.
Consider a dynamical system defined by == fxx) where E X S R", a fixed time horizon T E Rzo- Denote by (T) for r E [0, 7 the stale trajectory of the system al time T when initialized at x (for simplicity we assume that the solution to the differential equation exists and is unique) Let XA C X represent a set of undesirable states, and consider the cost function defined as
J(x) = min (T)(())), rE|O,T]
where d: X + R is function satisfying
d(x) S + x E XA. (x)>><< TEX XA.
for some threshold 7 E The function d measures the distance between a point in the domain and the unsafe set XA- An instructive example for the discussion is below is to choose y 0 and d X - {0, 1}, with d = > E XA and d 1 è E X Xa, but the same applies for any different choice. In this case, J assigns positive real number to a point E X if and only if the state trajectory from never intersects with XA- Let (X,F, P)) be a probability space. To quantify system safcty probabilistically, we soek to estimate P(fx J(x) > 0}), i.e. the probability of sampling an initial state that leads to a safe trajectory. In Lin & Bansal (2024) the authors define the nonconformity score as R3 J(x;) for i = 1, .... N, and are therefore interested estimating P((x : R > 0}). However, this is equivalent to defining a nonconformity
mcasure
A(x) l
(10)
x
and we have shown that this line of reasoning is not suitable for estimating the parameter b of Bernoulli r.V. given N i..dd realizations R; ~ Bern, of it.
Since Chilakamarri et al. (2024) relies on the framework of Lin & Bansal (2024), it suffers from the same issue. Additionally, in Vincent et al. (2024, Theorem 1), the authors re-derive Theorem 1, originally from Vovk (2012). They claim that training-conditionn. CP reduces to the Clopper-Pearson confidence interval when the underlying i.dd. random variables are have disproved this claim.
IIn the original paper the trajectory E depends learned controller and depends on training 7/,-.,, Zzyg For clarity omit this dependence here, since the training is given and is fixed.
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5 Conclusion
In this note we examined existing methodologies to use training-conditional CP for statistical safety verifi- cation, a problem that can be reduced to estimating the expectation of a Bernoulli random variable. While training-conditional CP remains a powerful tool for uncertainty quantification we have shown that it is not appropriate for BPCI problems. Specifically, we clarified the correct interpretation of confidence intervals and PAC-style guarantees for craiiing-condition CP. We do not rule out the possibility that a different formulation of CP could be applied to BPCI problems. This is left for future work.
References
Anastasios N Angelopoulos, Stephen Bates, et al. Conformal prediction: A gentle introduction. Foundations and Trends@ in Machine Learning, 16(4):494 591, 2023.
Vamsi Krishna Chilakamarri, Zeyuan Feng, and Somil Bansal. Reachability analysis for black-bor box dynamical systems. Xiv preprint arXiv:2410.0771 2024.
Charles J Clopper and Egon Pearson. The use of confidence or fiducial limits illustrated in the case of the binomial. Biometrika, 26(4):404-414) 1934.
Natalic Dean and Marcello Pagano. Evaluating confidence interval methods for binomial proportions in clustered surveys. Journal of Survey Statistics and Methodology, 3(4):484-500 503, 2015.
Matleo Fontana, Gianluca, Zeni, and Simone Vantini. Conformal prediction: unified review of theory and new challenges. Bernoulli, 29(1):1-23 23, 2023.
Albert Lin and Somil Bansal. Verification of neural reachable tubes via scenario optimization and conformal prodiction. In 6th Annual Learning for Dynamics & Control Conference, pp. 719-731777 119-731. PMLR, 2024.
Glenn Shafer and Vladimir Vovk. A tutorial on conformal prediction. Journal of Machine Learning Research, 9(3), 2008
Joseph A Vincent, Aaron O Feldman, and Mac Schwager. Guarantees robot system performance using stochastic simulation rollouts. IEEE Transactions Robolics, 2024.
Vladimir Vovk. Conditional validity of inductive conformal predictors. In Asian conference on machine learning, pp. 475-48 475-490. PMLR, 2012.
Vladimir Vork, Alexander Gammerman, and Glenn Shafer. Algorithmic learning in a random world, vol- ume 29. Springer, 2005.
A Appendix
We validate empirically equation 9 as follows and represent the results graphically in Figure 2.
We define a list of values for E by Eg = 0.01 + 0.01 * q for q = ..., 98. For every value Eg we consider an underlying Bernoulli distribution with parameter bigg F. F.g < Fg (right. figure) and an underlying Bernoulli distribution with parameter b2,0 E' + a1, % > E, (left figure) with a 0.005. For every value of 0,..., 98 examine the two situations b1.g I Eg and b2.g > Eq, as mentioned in Example 1 Part
2. The significance level is set to 2/3. We draw neat = 5 10 pairs of calibration points {2(1) (6) 1eat
:22 Jiel
For every pair of calibration points 2 22 we construct the resulting INP as Fí = (=( 222 ...., draw
(i)
(4) (i)
Riees = 5. 1 test points (1 Thest and compute the empirical frequency 9. [fj=1... 02 Er, as (iN+1 an approximation for P(ZN+1 (6);; finally we compute h = [ fum as an approximation to 22 (Sg) shown in the plots as the solid red line. The solid black line represents the curve given by E², which remains al always below the red line in both plots, expected. The area. shaded in blue represents the fraction
Bioss
Hoat
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contormal
conformal
0.8
0.5
0.4
0.1
0.0
0.
0.5
58
0.0
0.2
0.6
08
Figure On the left the curves resulting from bzgg Eg. on the right the curves resulting from di I Eg. for q 0,... 98.
of the 9.'s is for which the INP 1(s) is equal to Z, whereas the area shaded in red represents the fraction of the J for which the INP (()) is equal to Q and 9i is greater or equal than 1 - E. It is visible in the left plot that the only reason why the solid red line (approximating (² = bg.,) above E is that the INP is allowed to predict the entire set Z. In contrast, the right the solid red line approximates (?{SE,
((i) results in a prediction Ii) satisfying P(ZN+1 E (())) 2 Egg accordingly, for fixed g, the area shaded in red covers approximately - 1gg of the 'Probability' axis and the area shaded blue approximately ./...
since any pair of =1 =2
In summary, in both situations the theorem is confirmed empirically, since the red line is always above the black line. In the first case, where b2.4 Eg. the minimum confidence level of E is attained by predicting sufficiently often the entire sample space Z, precisely with frequency of bi as this is the only set prediction attaining the required coverage probability of 1 Eg Unfortunately, a prediction of the entire sample space is unnffrmative. In the second case, where b1.g I Eg. any predicted set between Q and Z attains the required coverage probability of Egg.


Reviews:
[{"review_id": "rff8mbFN5y", "reviewer": "Reviewer_R9Jz", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "# Strengths\n\nThe work is highly relevant to the TMLR community. It highlights an important issue in the misuse of training-conditional conformal prediction with concrete examples, serving as a crucial warning that could have significant impact on future research.\n\n# Weaknesses\n\nThe paper generally provides the necessary material to support its points, but it could benefit from clearer communication through more intuitive explanations and higher-level walk-throughs.\n\n- For instance, Example 1 is quite dense. It might be clearer by having a brief top-down explanation of how the example will demonstrate the point could help readers better anticipate what to expect. Also, you can perhaps start with something like, “We give an XXX answer with the following example,” rather than “We proceed with an example.”\n\n- The paper could also improve clarity by adding interpretations after definitions, explaining what high or low values of the terms (e.g., R^z, p^z) imply. For example, in Eq. (8), is it correct to interpret Q as the set of samples that do not conform? If so, clarifying this interpretation when introducing Q would help. Similarly, providing a verbal, intuitive explanation of the function d would make the discussion more accessible.\n\n- There are quite some similar notations about Binomial distributions and Bernoulli distributions which might bring some confusion. Also, $Bin_{N,E}(\\cdot)$ in Theorem 1 needs a definition.\n\n- The parameter $b$ mentioned just before Example 1, seems to appear abruptly. It would help to specify that b is the probability of $z\\in Q$ when the training set is fixed.\n\n# Questions\n\n1. In Theorem 1, do you need to fix $N$ before discussing the event? Also, this result also depends on the function $A$., so it might be clearer to explicitly include A in the statement.\n\n2. Could you elaborate more on what you mean by Theorem 1 being a distribution-free result? Does it simply mean that the result is independent of b? If so, it might be helpful to state this more explicitly for clarity.\n\n3. Regarding Eq. (9), I do not find the conclusion immediately straightforward. If Theorem 1 were applied directly, shouldn’t the result actually be a Binomial tail probability? The conclusion about $E^2$ seems to arise only from the argument that follows Eq. (9).\n\n4. I may have missed something, but there is still an aspect of Example 1 that is unclear to me. Specifically, when the training set is fixed, $Q$ and $b$, as defined through $A$, are mutually dependent and both dependent on the training set. Meanwhile, $p^z$ and $\\Gamma^\\epsilon$ also depend on the calibration set. How, then, can you discuss scenarios with different values of $b$ while seeming to hold the set $\\bar{Q}$ fixed? Could you clarify this part?", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper identifies a significant flaw that has been repeated in the literature. Specifically, several works have used training-conditional conformal prediction (CP), particularly in the form of Theorem 1, to provide PAC guarantees on the safety of dynamical systems. However, the theorem has been misinterpreted and applied incorrectly in scenarios where it is unsuitable, leading to problematic implications.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "See the weakness above and please answer the question.", "broader_impact_concerns": "No", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "h9flrNrdsE", "reviewer": "Reviewer_xgpv", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "# Strengths\n\n* The paper contributes to the field of conformal prediction and its applications to modern models and uncertainty estimation, a rapidly developing and important domain.\n\n* The authors provide a clear example demonstrating their findings, which highlights the shortcomings of the conformal prediction framework within the context of the Binomial Proportion Confidence Intervals (BPCI) problem. Through this example, they effectively illustrate why confidence interval estimation is not feasible in this framework, underscoring its limitations.\n\n# Weaknesses\n\n(The following parts are discussed in more detail in the next section, \"requested changes\")\n\n* Some parts of the presentation could be improved.  \n* The practical impact is unclear.  \n* Observing the described behavior in more challenging setups would be interesting.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "In this paper, the authors address an issue with using the conformal predictions framework to solve the Binomial Proportion Confidence Intervals (BPCI) problem, specifically deriving bounds or confidence intervals for the estimated Bernoulli probability $b$. Through a simple toy example, the authors demonstrate that estimating a confidence interval is not feasible within the conformal prediction framework. They argue that this approach could lead to incorrect predictions because, although conformal predictions are based on Theorem 1 (Vovk, 2012), the results of this theorem cannot be directly applied to confidence interval estimation. Furthermore, the authors discuss the practical implications of their findings in the context of the Safety Verification for Dynamical Systems task. They highlight that recent works applying conformal predictions to this task are, in fact, incorrect.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "1. **[strengthen]** Example 1 seems slightly overloaded and hard to follow. I recommend dividing its different cases more distinctly. A complementary table showcasing the results (e.g., what INP equals in different cases) would be helpful.\n\n2. **[strengthen]** Though the results of the toy example are appreciated and theoretically important, it is not entirely clear how this theoretical shortcoming of conformal prediction influences the real performance of systems. For instance, in [Vincent 2024], it was shown that the proposed intervals are useful—why is that the case here? A discussion would be much appreciated.\n\n3. **[strengthen]** Building on the previous point, the provided example is a counterexample showing why conformal prediction might be inefficient in the considered scenario. Are there any assumptions under which conformal predictions could provide tight enough confidence intervals? A discussion on this would be valuable.\n\n4. **[strengthen]** Could you elaborate on how the choice of INM influences the provided example? Is there a family of INMs that work for the BPCI problem, or is this a universal result?", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "W6IU0JMcLF", "reviewer": "Reviewer_N32P", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "The paper's strength is that it makes an important point via a relatively simple example. However, the paper is extremely concise, making it hard to follow for people who are not experts on the topic but might think about leveraging conformal prediction. Exactly that audience would highly benefit from such a paper.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The authors show by counter-example that the usage of conformal prediction for providing safety certificates in learning-based control is not valid.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "1) Some abbreviations that the authors maybe consider general knowledge are not introduced, or at least not at the first use. For accessibility of the paper, I think it would be important to introduce those (e.g., i.i.d., r.v.). Similarly, at the beginning of Section 3, the filtration of the probability space appears, but the word \"filtration\" does not appear; this variable is not explained. I think also the use of superscripts for probabilities should be explained.\n\n2) I think currently, in (4), the range of $i$ is not defined.\n\n3) It would help to provide some more intuition on how we want to use the INP and what role it will play later in in the paper. \n\n4) The example is nice and simple but not that simple to follow. Especially, the conclusion is not yet fully convincing. The bounds from Theorem 1 are probabilistic in nature. The confidence intervals hold with a certain probability. Thus, I may always construct a case where my confidence intervals are wrong. As long as this case has sufficiently low probability, this should not be a concern. I think in the end of the example, it should be made more precise what is the problem. Also in the discussion of the graphical example in the appendix, I don't think the point becomes sufficiently clear.\n\n5) There seems to be some issue with the usage of \\cite vs \\citep. I would also recommend using the \\secref commands provided in the TMLR papers - currently, it seems like the space between Sec. and the section number is too large.", "broader_impact_concerns": "No concerns.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: All the three reviewers recommend to accept this paper. After reading the paper, the reviews, and the author responses, I agree with their recommendations.

My only minor suggestion is that the authors should polish the writing of Section 4 a little bit to make it more readable.
Decision: Accept (poster)