An Unverifiable World9. Compressing the UnknownContents中文

Chapter 9: Compressing the Unknown

Thesis: Two moves take on uncertainty directly. On a slice you can inspect, deliver a guaranteed bound (a certificate). Spend your limited checks where they remove the most uncertainty (optimal screening).

Part II placed the moves in four concrete settings and let them play out there, tangled up with one another. From here on the angle changes. Each move is pulled out on its own, stripped of its field's jargon, and shown in pure form. Then it is laid across all the settings at once. This is the book's main contribution: the "table" that shows one move under its many vocabularies.

The eight moves pair off into four chapters, two moves to a chapter. The pairing is not for convenience. It is itself a claim: the two moves in each pair go after the same part of the problem, and Chapter 13 will pin down exactly where each one acts. This chapter's two moves, the certificate and optimal screening, go after the same target: uncertainty. They compress the unknown from opposite ends. At one end, you prove a guaranteed bound on a slice you are able to check. At the other, you spend your limited checks where they remove the most uncertainty.

And under the iron rule set down in Chapter 4, every move we propose and every side-by-side comparison of two fields has to face the same question again. Is the transfer from one field to another substantive (same mechanism, same failure mode, same trade-off), or just a pretty metaphor?

Certificate: Prove a Bound on a Slice

The first move in pure form: instead of verifying the whole, produce a local guarantee that is bounded and can be rechecked independently. What you deliver is not "all of it is right." It is "within this range, it is wrong by at most this much," together with a piece of evidence that anyone can check quickly.

Across fields, its forms are remarkably consistent.

In machine learning it is called a generalization bound. Valiant's 1984 PAC framework1 and the VC dimension of Vapnik and Chervonenkis (brought into that framework by Blumer and colleagues in 19894) give a guarantee of the form "with probability at least $1-\delta$, the true error does not exceed the empirical error plus a complexity penalty":

$$R(h)\ \le\ \hat R(h)+\sqrt{\frac{d\big(\ln(2n/d)+1\big)+\ln(4/\delta)}{n}}.$$

You cannot verify how the model will behave on all future data (that is the open world). But on one slice, the samples already seen, you can prove a bound that holds for unseen data at a stated level of confidence. Hoeffding's inequality17 is the probabilistic engine behind it, and PAC-Bayes (McAllester6) refines it.

In software it is called types and proofs. A type system does not prove that a program is "all correct." It proves only that one property holds (for instance, that the program will never treat an integer as a pointer). In exchange, the check becomes decidable and can be rechecked mechanically (Pierce7). The Curry-Howard correspondence (Howard 19808) puts an equals sign between "proof" and "program." Necula's 1997 proof-carrying code9 takes the move to its limit. Untrusted code carries its own proof of safety, and the host only has to check that certificate quickly, without re-deriving anything itself. Leroy's formally verified CompCert compiler of 200910 and the Z3 solver of de Moura and Bjørner in 200811 turn the same idea into industrial tools.

In numerical computing it is called an error bound. Higham's 2002 backward error analysis12 and Moore's 1966 interval arithmetic13 let you carry "an interval guaranteed to contain the true value" through the computation. What you deliver at the end is then a guaranteed range, not a floating-point number that might be fooling you. In mathematics it is the zeros from Chapter 7, verified up to height $T$: a bound, not a theorem.

What unifies all of these is that a certificate is a local, bounded, independently recheckable guarantee. Its best feature is that it exploits an asymmetry in verification, the one Chapter 7 called the bedrock of mathematics. Producing a certificate can be extremely expensive, while checking one is extremely cheap. Proof-carrying code, the solution to an NP problem, and a mathematical proof all live off this same dividend.

It has only one standard failure mode, but a common one: the vacuous bound. A guarantee can be true and still useless. "The error is at most one hundred percent" and "this model's generalization error is finite" are logically unimpeachable and worthless in practice. A bound is valuable not because it holds, but because it is tight enough to act on.

Optimal Screening: Spend Your Checks Where They Count

The second move in pure form: information has a cost, so put your limited checks where, at the margin, they compress the most uncertainty.

Its forms across fields are just as tidy. In statistics and science it is called design of experiments. Fisher's 1935 The Design of Experiments19 and Box and colleagues' Statistics for Experimenters20 teach how to get the most information out of the fewest trials. In 1956 Lindley gave a measure of "the information an experiment provides"21, and Chaloner and Verdinelli systematized Bayesian experimental design22. Wald's 1945 sequential test23 lets you decide, as the data come in, whether to keep going. Shannon's 1948 information theory14 is the underlying currency of all of it.

In machine learning it is called active learning: deciding which sample is most worth labeling next (Cohn and colleagues33, Settles34). In software testing it is called fuzzing: deciding which inputs to throw computing power at in order to trigger a crash (Miller and colleagues' pioneering 1990 experiment35). The modern scale of this move is striking. Since 2016, Google's OSS-Fuzz has been continuously feeding huge volumes of malformed input into more than a thousand open-source projects, and it has so far turned up tens of thousands of bugs and vulnerabilities. No team of human testers could be that exhaustive. OSS-Fuzz works by pouring computing power, nonstop, into the places most likely to crash. In auditing it is called sampling: choosing which transactions to check so that you are most likely to find a problem. In interface design it is choosing which question to ask the user (Chapter 5).

Behind all of these is one optimization problem: maximize the expected information gain between the answer and the unknown,

$$q^\star=\arg\max_q\ I(\theta;y_q).$$

When the checking has to be repeated, and its results used while it is still going on, the problem becomes the tension between exploration and exploitation. This is the multi-armed bandit problem. Thompson's 1933 sampling method24, Robbins's founding 1952 paper25, Lai and Robbins's 1985 work on optimal allocation26, and Auer and colleagues' 2002 finite-time analysis27 give the optimal answer to "how many trials to spend reducing uncertainty about which option." The regret of these strategies grows only logarithmically over time,

$$\mathrm{Regret}(T)=O(\ln T).$$

Here we need to guard against a kind of path dependence in how the story is told. Optimal screening is a family of methods. Design of experiments, active learning, audit sampling, fuzzing, and the bandit all belong to it. The line that runs from Kushner and Mockus through Jones's 1998 efficient global optimization30 and Srinivas and colleagues' 2010 GP-UCB32 to the Gaussian-process Bayesian optimization surveyed in Shahriari's 2016 review36 is enormously useful. But it is only one implementation within the family, not the whole of "screening." To equate this move with Gaussian processes is to shrink a universal approach down to a single tool.

This move, too, has only one standard failure mode: optimizing a misspecified measure of information. You gather information very efficiently, but about the wrong question. Or the "information" you are maximizing does not track what you actually care about. The cleverer the screening, the faster a misspecified measure leads you astray.

Why These Two Moves Go Together

Put the two moves side by side. The certificate squeezes the uncertainty on one slice into a guaranteed bound. Screening spends an information budget on observing the slice that compresses uncertainty the most. One says "prove it tight where you can check." The other says "spend your checking where it is most needed." They squeeze the same enemy, the unknown, from two ends. That is also the task they share: with a limited information budget, manage where you cut uncertainty and how hard. Chapter 13 will give each of them its precise place.

Sometimes, though, no amount of compressing or screening is enough, because you lack the capacity to make the judgment at all. Then you can no longer shrink the unknown on your own. You have to borrow the judgment from somewhere else. That is what the next pair of moves is about.

Next chapter: 10. Borrowed Judgment →← 8. The Organization Blind to Itself

References

Waypoints: 1. historical scientific judgment; 2. theoretically studied material; 3. how science progresses; 4. how to live in an unverifiable world. This section was checked source by source.

  1. L. Valiant (1984). "A Theory of the Learnable." Communications of the ACM, 27(11), 1134-1142. doi:10.1145/1968.1972 [2] Valiant here proposed the "probably approximately correct" (PAC) learning framework, rigorously defining "learning a concept" as obtaining, with high probability and within polynomial time and samples, a hypothesis whose error is small enough. This paper gave the first provable language for "can it be learned, and how many samples does it take," and is the source of this chapter's "generalization bound" move.
  2. V. Vapnik and A. Chervonenkis (1971). "On the Uniform Convergence of Relative Frequencies of Events to Their Probabilities." Theory of Probability & Its Applications, 16(2), 264-280. doi:10.1137/1116025 [2] This founding work proved the conditions under which empirical frequencies converge uniformly to true probabilities, and from it grew the VC dimension, later named after the two authors, used to characterize the "capacity" of a family of functions. It explains why an error bound proven on finite samples can hold for unseen data, and is the probabilistic and combinatorial bedrock beneath generalization bounds.
  3. V. Vapnik (1995). The Nature of Statistical Learning Theory. Springer. doi:10.1007/978-1-4757-2440-0 [2] In this book Vapnik organized statistical learning theory into a complete system: with structural risk minimization at its core, it trades off empirical error against model complexity and is thereby led to the support vector machine. It is the standard reading for understanding why a "complexity penalty" appears in a generalization bound, laying out the intuition of the first move clearly and coherently.
  4. A. Blumer, A. Ehrenfeucht, D. Haussler, and M. Warmuth (1989). "Learnability and the Vapnik-Chervonenkis Dimension." Journal of the ACM, 36(4), 929-965. doi:10.1145/76359.76371 [2] This paper formally brought the VC dimension into the PAC framework, proving that a concept class is PAC-learnable if and only if its VC dimension is finite, and gave a sample-complexity bound depending on the VC dimension. It is the direct source of the generalization-bound formula in the main text, nailing down the criterion that "finite capacity makes it learnable."
  5. A. Blumer, A. Ehrenfeucht, D. Haussler, and M. Warmuth (1987). "Occam's Razor." Information Processing Letters, 24(6), 377-380. doi:10.1016/0020-0190(87)90114-1 [2] This short paper gives a learning-theoretic version of "Occam's razor": a hypothesis that can compress the training data short enough will, with high probability, generalize. It turns "simplicity makes it learnable" from a philosophical maxim into a provable proposition, providing a clean footnote to this chapter's motif of "compressing the unknown."
  6. D. McAllester (1999). "PAC-Bayesian Model Averaging." Proceedings of the 12th Annual Conference on Computational Learning Theory (COLT), 164-170. doi:10.1145/307400.307435 [2] McAllester here proposed the PAC-Bayes bound: it gives a generalization guarantee for a posterior distribution over a family of hypotheses, with the penalty measured by the KL divergence between posterior and prior. It is a refinement of the PAC bound, which is what the main text means in calling it the "refinement" of the first move, and it often gives tighter results than the classical VC bound.
  7. B. Pierce (2002). Types and Programming Languages. MIT Press. Google Books [2] This textbook of Pierce's systematically explains the theory and construction of type systems, centered on the two properties of type safety, "progress" and "preservation," and on how they are mechanically checked. It is precisely the standard basis for the main text's line that "a type system does not prove a program all correct, only one property, in exchange for something decidable and recheckable," and is the introductory book for understanding the form the certificate move takes in software.
  8. W. Howard (1980). "The Formulae-as-Types Notion of Construction." In J. Seldin and J. Hindley (eds.), To H. B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism, 479-490. Academic Press. Google Books [2] Howard's widely circulated manuscript (written in 1969, formally published in 1980) established the correspondence "formulae as types, proofs as programs": the propositions of intuitionistic logic correspond one-to-one with types, and proofs with terms. It is the classic text of the Curry-Howard correspondence, setting an equals sign between "checking the type of a program" and "checking a proof," which is exactly the logical core of the certificate move.
  9. G. Necula (1997). "Proof-Carrying Code." Conference Record of the 24th ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages (POPL), 106-119. doi:10.1145/263699.263712 [2][4] Necula proposed "proof-carrying code": an untrusted program carries its own formal proof of its safety, and the host need only quickly verify this certificate, without trusting the code's origin or re-deriving anything. It pushes the verification asymmetry of Chapter 7 to its limit, and is the purest engineering embodiment of this chapter's certificate concept.
  10. X. Leroy (2009). "Formal Verification of a Realistic Compiler." Communications of the ACM, 52(7), 107-115. doi:10.1145/1538788.1538814 [2][3] Leroy reported the results of CompCert: a C compiler formally verified in Coq, whose generated code being semantically consistent with the source program is something machine-proven. It shows that "real software with recheckable guarantees" is no fantasy, and is the landmark case of industrializing the certificate move.
  11. L. de Moura and N. Bjørner (2008). "Z3: An Efficient SMT Solver." Tools and Algorithms for the Construction and Analysis of Systems (TACAS), LNCS 4963, 337-340. Springer. doi:10.1007/978-3-540-78800-3_24 [2][4] This paper introduced Z3, an efficient SMT solver that can decide the satisfiability of logical formulas with theories such as arithmetic and arrays, and is widely used in program verification and symbolic execution. It made "automatically producing recheckable certificates" into a ready-to-hand industrial tool, and is one of the engines by which the certificate move spread in practice.
  12. N. Higham (2002). Accuracy and Stability of Numerical Algorithms (2nd ed.). SIAM. doi:10.1137/1.9780898718027 [2] This authoritative work of Higham's systematically treats the error analysis of numerical algorithms, especially backward error analysis: rather than asking "how far the answer departs from the true value," ask "of exactly which perturbed problem this answer is the exact solution." It is the standard reference for the main text's "error bound" move, teaching how to equip floating-point computation with a trustworthy guarantee.
  13. R. Moore (1966). Interval Analysis. Prentice-Hall. Google Books [2] This pioneering work of Moore's established interval arithmetic: each quantity participates in computation carrying an interval guaranteed to contain the true value, so the output is not a number that may deceive but a guaranteed range. It gave numerical computation the idea of "delivering a bound rather than a point," which is exactly the embodiment of the certificate move in that field.
  14. C. Shannon (1948). "A Mathematical Theory of Communication." Bell System Technical Journal, 27(3), 379-423; 27(4), 623-656. doi:10.1002/j.1538-7305.1948.tb01338.x [1][2] This paper of Shannon's, founding information theory, measures uncertainty with entropy and gives the fundamental limits of source coding and channel capacity. It is the ultimate source of the notion that "information has a cost and can be measured," which the main text calls the "underlying currency" of optimal screening; this chapter's entire discussion of information gain and compression takes it as its unit.
  15. J. Rissanen (1978). "Modeling by Shortest Data Description." Automatica, 14(5), 465-471. doi:10.1016/0005-1098(78)90005-5 [2] Rissanen proposed the minimum description length (MDL) principle: the best model is the one that encodes the data together with the model itself most shortly. It turns "compression is understanding" into an operable model-selection criterion, resonating precisely with this chapter's motif of "compressing the unknown," and is also an information-theoretic realization of Occam's razor.
  16. M. Li and P. Vitányi (2008). An Introduction to Kolmogorov Complexity and Its Applications (3rd ed.). Springer. doi:10.1007/978-0-387-49820-1 [2] This standard textbook systematically treats Kolmogorov complexity: the complexity of an object equals the length of the shortest program that can generate it. This is the uncomputable ideal of "compression," contrasted with the operable approximation that is MDL; it provides the theoretical ceiling for this chapter's compression motif, showing that optimal compression is itself a kind of unverifiable limit.
  17. W. Hoeffding (1963). "Probability Inequalities for Sums of Bounded Random Variables." Journal of the American Statistical Association, 58(301), 13-30. doi:10.1080/01621459.1963.10500830 [2] Hoeffding here gives an exponential upper bound on the probability that a sum of bounded random variables deviates from its mean. This inequality is the basic tool for compressing the gap between "the empirical average" and "the true expectation" into a confidence bound, which the main text calls the "probabilistic engine" of generalization bounds; most of the certificate move's concentration arguments begin from it.
  18. E. Candès, J. Romberg, and T. Tao (2006). "Robust Uncertainty Principles: Exact Signal Reconstruction from Highly Incomplete Frequency Information." IEEE Transactions on Information Theory, 52(2), 489-509. doi:10.1109/tit.2005.862083 [2] This founding paper of compressed sensing proves that as long as a signal is sparse enough, it can be reconstructed exactly through convex optimization from far fewer measurements than the classical sampling theorem requires. It is a mathematical exemplar of "spending checks on the cutting edge, wringing all the information from very few observations," echoing this chapter's two threads of compression and optimal screening.
  19. R. Fisher (1935). The Design of Experiments. Oliver and Boyd. Google Books [1][3] This classic of Fisher's established the basic principles of modern experimental design: randomization, replication, and blocking, along with the famous "lady tasting tea" thought experiment. It teaches how to wring the most credible information from the fewest trials, and is the source reading for the optimal-screening move in statistics and science.
  20. G. Box, W. Hunter, and J. Hunter (1978). Statistics for Experimenters: An Introduction to Design, Data Analysis, and Model Building. John Wiley & Sons. Google Books [3][4] This popular practical work by Box and colleagues explains experimental design, data analysis, and model building to the people who actually run experiments, stressing factorial design and the iterative rhythm of sequential learning. It brings Fisher's principles down to the engineering site, and is a practical guide to understanding "how to arrange the checking most economically."
  21. D. Lindley (1956). "On a Measure of the Information Provided by an Experiment." The Annals of Mathematical Statistics, 27(4), 986-1005. doi:10.1214/aoms/1177728069 [2] Lindley used information theory to give a measure of "how much information an experiment provides," namely the expected information gain between prior and posterior. This is precisely the theoretical prototype of the main text's optimal-screening objective $\arg\max_q I(\theta;y_q)$, turning "which experiment to run" into a maximizable quantity.
  22. K. Chaloner and I. Verdinelli (1995). "Bayesian Experimental Design: A Review." Statistical Science, 10(3), 273-304. doi:10.1214/ss/1177009939 [2] This review systematically surveys Bayesian experimental design: with a utility function (often the expected information gain) as the objective, it uniformly derives various optimal-design criteria. It integrates Lindley's measure into a complete framework, and is the reader's entry point for quickly grasping the screening core of "maximizing expected information gain."
  23. A. Wald (1945). "Sequential Tests of Statistical Hypotheses." The Annals of Mathematical Statistics, 16(2), 117-186. doi:10.1214/aoms/1177731118 [1][2] Wald proposed the sequential probability ratio test: judge as the data come in, and once the evidence is strong enough, stop and draw a conclusion, thereby saving much on average over a fixed-sample test. It turns "whether to keep checking" itself into an optimal decision, and is the forerunner of the "use it as you check" branch of optimal screening.
  24. W. Thompson (1933). "On the Likelihood that One Unknown Probability Exceeds Another in View of the Evidence of Two Samples." Biometrika, 25(3-4), 285-294. doi:10.2307/2332286 [1][2] Thompson here proposed the sampling method later named after him: select an option at random according to the posterior probability that "it really is the best," naturally balancing exploration and exploitation. This is one of the earliest solutions to the multi-armed bandit problem, and remains both a simple and a strong strategy for it to this day.
  25. H. Robbins (1952). "Some Aspects of the Sequential Design of Experiments." Bulletin of the American Mathematical Society, 58(5), 527-535. doi:10.1090/s0002-9904-1952-09620-8 [1][2] This paper of Robbins's formally established the multi-armed bandit problem as a mathematical object and proposed the earliest sequential allocation strategy, founding the research direction of "how to trade off exploration and exploitation." It is the starting point of the whole later bandit literature, and this chapter's discussion of "how many trials to spend reducing which uncertainty" begins here.
  26. T. Lai and H. Robbins (1985). "Asymptotically Efficient Adaptive Allocation Rules." Advances in Applied Mathematics, 6(1), 4-22. doi:10.1016/0196-8858(85)90002-8 [2] Lai and Robbins proved the regret lower bound for the multi-armed bandit: the cumulative regret of any reasonable strategy grows at least logarithmically over time, and they constructed asymptotically optimal allocation rules attaining this lower bound. It established that the main text's $O(\ln T)$ is an insurmountable limit, drawing a ceiling for the entire class of problems.
  27. P. Auer, N. Cesa-Bianchi, and P. Fischer (2002). "Finite-time Analysis of the Multiarmed Bandit Problem." Machine Learning, 47(2-3), 235-256. doi:10.1023/a:1013689704352 [1][2] This paper gives simple algorithms such as UCB1, based on "optimism in the face of uncertainty," and proves they have logarithmic regret bounds in finite time (not merely asymptotically). It turns Lai and Robbins's asymptotic result into a concrete, directly usable, analyzable strategy, and is the standard citation for the "upper confidence bound" class of methods.
  28. H. Kushner (1964). "A New Method of Locating the Maximum Point of an Arbitrary Multipeak Curve in the Presence of Noise." Journal of Basic Engineering, 86(1), 97-106. doi:10.1115/1.3653121 [1][2] This early paper of Kushner's characterizes an unknown noisy curve with a probabilistic model, and on that basis selects the next sampling point to seek the maximum, an early form of the Bayesian-optimization idea. It shows that "where to measure once more" can be treated as an optimal decision, and is a pioneering work of optimal screening in global optimization.
  29. J. Mockus, V. Tiesis, and A. Žilinskas (1978). "The Application of Bayesian Methods for Seeking the Extremum." In L. Dixon and G. Szegő (eds.), Towards Global Optimization 2, 117-129. North-Holland. Google Books [2] Mockus and colleagues systematically developed Bayesian global optimization and proposed the acquisition function of "expected improvement": under a probabilistic model, select the point most likely to bring improvement for evaluation. It turned Kushner's intuition into a general method, and is the direct predecessor of today's Bayesian optimization.
  30. D. Jones, M. Schonlau, and W. Welch (1998). "Efficient Global Optimization of Expensive Black-Box Functions." Journal of Global Optimization, 13(4), 455-492. doi:10.1023/a:1008306431147 [2][4] This paper proposed the EGO algorithm, which uses a Gaussian process to build a surrogate model for an expensive black-box function and then picks the next evaluation point by the expected-improvement criterion, drastically reducing the number of evaluations. It is the landmark work generalizing Bayesian optimization, but the main text also cautions: it is only one implementation within the optimal-screening family of methods, not the whole of it.
  31. C. Rasmussen and C. Williams (2006). Gaussian Processes for Machine Learning. MIT Press. doi:10.7551/mitpress/3206.001.0001 [2][4] This standard textbook systematically treats Gaussian processes: a nonparametric Bayesian method that places a prior on functions themselves and can give predictive uncertainty. It is the theoretical foundation of the surrogate model on which Bayesian optimization depends, and is the core reference for readers who want to understand "how uncertainty is modeled and used to decide where to check next."
  32. N. Srinivas, A. Krause, S. Kakade, and M. Seeger (2010). "Gaussian Process Optimization in the Bandit Setting: No Regret and Experimental Design." Proceedings of the 27th International Conference on Machine Learning (ICML), 1015-1022. arXiv:0912.3995 [2] Srinivas and colleagues proposed the GP-UCB algorithm, carrying the "upper confidence bound" idea from the bandit over to Gaussian-process optimization, and gave provable bounds for its regret. It stitches Bayesian optimization together with bandit theory, demonstrating precisely that this chapter's two ends, proving a bound and spending the checking, were two sides of one task all along.
  33. D. Cohn, Z. Ghahramani, and M. Jordan (1996). "Active Learning with Statistical Models." Journal of Artificial Intelligence Research, 4, 129-145. doi:10.1613/jair.295 [2][4] Cohn and colleagues give a statistical framework for active learning: under a statistical model, select for labeling the sample that most reduces the model's variance (or predictive uncertainty). It turns "which sample is most worth labeling next" into a computable criterion, and is the representative work establishing active learning as a branch of optimal screening.
  34. B. Settles (2009). Active Learning Literature Survey. Computer Sciences Technical Report 1648, University of Wisconsin-Madison. link [2][4] This widely cited survey of Settles's systematically organizes the various query strategies of active learning, such as uncertainty sampling, query by committee, and expected error reduction, and compares their applicable settings. It is the standard entry point for a quick overview of "how to spend the labeling budget," setting the various implementations of this move side by side for comparison.
  35. B. Miller, L. Fredriksen, and B. So (1990). "An Empirical Study of the Reliability of UNIX Utilities." Communications of the ACM, 33(12), 32-44. doi:10.1145/96267.96279 [3][4] Miller and colleagues fed randomly generated input into various UNIX utilities, and a fair number of programs crashed or hung as a result, which is the pioneering experiment of fuzzing. It shows that "throwing compute at random or suspect inputs to shake out a fault" is a cheap and effective way of checking, and is the starting point of optimal screening in software testing.
  36. B. Shahriari, K. Swersky, Z. Wang, R. Adams, and N. de Freitas (2016). "Taking the Human Out of the Loop: A Review of Bayesian Optimization." Proceedings of the IEEE, 104(1), 148-175. doi:10.1109/jproc.2015.2494218 [2][4] This review comprehensively surveys Gaussian-process-based Bayesian optimization: the surrogate model, the acquisition function, and their applications in settings such as hyperparameter tuning. It is the standard reading for understanding the current state of the field, but the main text uses it to remind readers not to shrink the universal posture of "optimal screening" down to the single tool of "Gaussian processes."
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