The course so far, where we are

Roadmap

1. Why: calibrated intervals that covered 26% on new chemistry
2. What UQ is: aleatoric, epistemic, and three kinds of interval
3. Calibration: how to check any uncertainty, before choosing a method
4. Gaussian processes: what the band promises, and three ways it fails
5. Bootstrap and ensembles: what the spread measures, and a worked recalibration
6. Conformal prediction: a guarantee, and the one condition it needs
7. Models with uncertainty built in: pycse

Intervals that held, until the chemistry changed

• Neural-network force fields: DFT-quality energies, far cheaper
• Useful only if you know when to trust them
• Conformal prediction + distances in the network's latent space
• In distribution: calibrated and sharp; calibration takes minutes against ~11,000 GPU-hours of training
• Trained on QM9 without fluorine, tested with it: 90% intervals covered 26%

Hu, Musielewicz, Ulissi & Medford, MLST 2022 (CMU ChemE co-authors)

Three numbers from today

All from lectures/l14/figures/make_figures.py

Aleatoric and epistemic

• Practical test: epistemic if the modeler "sees a possibility to reduce" it
• From structural reliability, carried into ML as reducible against irreducible

Der Kiureghian & Ditlevsen, Struct. Saf. 2009 (paywalled) · Hüllermeier & Waegeman, Mach. Learn. 2021

Aleatoric and epistemic, live

Aleatoric and epistemic, the split depends on the model

• Concrete strength scatters at fixed proportions, partly from curing temperature
• No temperature input: that scatter is noise (aleatoric)
• Temperature as an input: the same scatter is learnable (epistemic)
• Two more sources for a surrogate:
  • Model inadequacy: "No model is perfect"
  • Code uncertainty: the simulator was not run at this input

Hüllermeier & Waegeman sec. 2.3 · Kennedy & O'Hagan 2001, sec. 2.1, six sources

Three kinds of interval

• "A prediction interval necessarily encloses the corresponding confidence interval"
• A specimen tested against a 40 MPa spec is a new measurement
• Every coverage number today is for prediction intervals

Heskes, NIPS 1996 · Gelman et al., BDA3 sec. 1.1

Why ML makes this hard

• Training residuals are too small: maximum likelihood "systematically underestimates σ²"
• Big networks are overconfident: ResNet confidence "substantially higher than its accuracy"
• ReLU networks extrapolate in straight lines: proved for classifiers, Theorem 3.1
• No data describes the region without data: every method substitutes an assumption

Laves et al., MIDL 2020 · Guo et al., ICML 2017 · Hein et al., CVPR 2019

Aleatoric and epistemic, a question

Calibration

• Coverage: fraction of held-out points inside their interval
• Reliability diagram: observed against nominal coverage at several levels
• Below the diagonal: overconfident, intervals too narrow

Kuleshov, Fenner & Ermon, ICML 2018, Eq. 3

Calibration, sharpness

• Goal: "maximizing the sharpness ... subject to calibration"
• Climatological forecaster: ignore the inputs, always issue the long-run spread

Gneiting, Balabdaoui & Raftery, JRSS B 2007 (author's copy)

Calibration, proper scoring rules

• Proper: the expected score is best when the forecast is the true distribution
• NLL: ; punishes a confident miss; unbounded
• CRPS: distance between predicted CDF and a step at ; "generalizes the absolute error"
• Report one of them with coverage and width

Gneiting & Raftery, JASA 2007 (author's copy)

Calibration, on average and locally

• 90% overall can be 100% on easy points and 70% on hard ones
• Consistency: calibrated given the predicted σ
• Adaptivity: calibrated given the input
• Good consistency "does not imply a good adaptivity"
• A held-out region of input space is an adaptivity check

Pernot, APL Mach. Learn. 2023 · Levi et al., Sensors 2022

Calibration, recalibration and its limit

• Scaling: multiply every σ by one factor fitted on held-out data
• Isotonic: a monotone map from predicted to observed probability
• Both assume future data resemble the held-out data
  • Kuleshov: needs "enough i.i.d. data"
  • Ovadia: temperature-scaling error "increases significantly as the shift increases"
• Catalysis example: several metrics side by side (Tran et al. 2020, CMU ChemE)

Laves 2020 · Kuleshov 2018 · Ovadia et al., NeurIPS 2019

Gaussian processes

After Rasmussen & Williams 2006, Fig. 2.2 (free online)

Gaussian processes, the predictive variance

• Far from data, : the band returns to the prior
• No in it: depends "only on the inputs" (R&W p. 18)
• At fixed settings, a bad fit gets the same band as a good one
• Fitted settings bring in "implicitly" (Deringer et al. 2021)

Gaussian processes, fitting the settings

Gaussian processes, failure 1: noise that grows

Gaussian processes, fixing heteroscedastic noise

• Fixes: second GP on the log noise (Goldberg et al. 1998); a cheap alternation (Kersting et al. 2007); variational (Lázaro-Gredilla & Titsias 2011)
• Surprise: , so the first try learned noise 0.7× too small

Gaussian processes, a scikit-learn trap

• alpha=array sets a noise variance per training point
• It does not give the GP a noise model at new inputs
• WhiteKernel estimates "the global noise level"
• Input-dependent noise at prediction time needs a model of the noise

scikit-learn user guide, Gaussian processes

Gaussian processes, failure 2: the wrong kernel

Gaussian processes, the shrinking length scale

Duvenaud, kernel cookbook · Capone, Pleiss & Hirche, NeurIPS 2023

Gaussian processes, failure 3: five points

Gaussian processes, how often five points mislead

• 200 data sets each, settings by maximum likelihood, exact coverage
• Little data: "both short and long lengthscales explain the data consistently"
• GP error bars are Bayesian, not frequentist guarantees

Capone, Lederer & Hirche, ICML 2022 · Fiedler, Scherer & Trimpe, AAAI 2021 · R&W sec. 5.4.1

Gaussian processes, cost and a case

• Exact inference ; inducing points: "from to " (Titsias 2009)
• High dimension: the prior "simply says too little" (Binois & Wycoff 2022)
• Case: a silicon GP potential's predicted error "rises notably" on a path to a defect, and it still misses a minimum DFT finds

Deringer et al., Chem. Rev. 2021, sec. 5.2, Fig. 24 (open access)

Gaussian processes, what to do

• Look at the fitted noise; compare with replicate measurements
• Check coverage by region, not only overall
• Watch the length scale as data arrive
• Under about a dozen points: a rough guide, not a 95% statement

Bootstrap

• Estimates the variance of the fit (Efron 1979)
• Misses the noise: a confidence interval, not a prediction interval
• Misses the bias: "the bias component ... is negligible" is an assumption (Heskes 1997)
• "May ... fail entirely when bias is a dominant source of error" (Palmer et al. 2022)

Deep ensembles

• Each sees all the data: "bagging deteriorated performance"
• Each predicts a mean and a variance, trained on Gaussian NLL
• The spread alone is the epistemic part only
• Different starts land in different modes of the loss surface

Lakshminarayanan, Pritzel & Blundell, NeurIPS 2017 · Fort, Hu & Lakshminarayanan 2019

Ensembles, what the spread measures

Ensembles, the toy results

• Inside: members agree to 0.021; the noise is 0.1
• Beyond: spread 0.63 against error 0.61, in this 1-D example

Ensembles, the evidence

• Best baseline: "deep ensembles seem to perform the best across most metrics" (Ovadia 2019); Scalia 2020, Tan 2023 agree for molecules
• Overconfident in level: "consistently underestimates uncertainty" (Hirschfeld 2020); "require to be calibrated for each system" (Kahle & Zipoli 2022)
• Weakest where it matters: scaffold-split coverage "always underestimated" (Scalia)
• Cost: times one model

A useful relative signal; not an interval until it has noise and a check.

Ensembles, a question

Ensembles, recalibrating on concrete

• Concrete strength, Lecture 9's network, five seeds
• Grouped split: random 20% of mixes held out
• Extrapolation split: lowest water/cement 20% of mixes held out
• A quarter of the training mixes set aside to calibrate

Same fix four times: σ scaling (Laves), STD scaling (Levi), Musil et al. 2019, Kellner & Ceriotti 2024 Eq. 8

Ensembles, recalibrating on concrete, live

Ensembles, recalibrating on concrete, results

• Grouped: most of is the missing noise (replicate scatter 5.0 MPa)
• Extrapolation: spread already large; the factor over-widens
• Rank correlation of error with spread: 0.29 and 0.17

Ensembles, what to do

• Never report the spread alone as an interval
• Recalibration costs a third data split
• Fit on data like where the model will be used
• Otherwise report coverage on a held-out region, and expect error in either direction

Split conformal prediction

1. Fit any model on the fitting set
2. Scores on calibration points:
3. = the -th smallest score
4. Interval:

Lei et al., JASA 2018 (CMU) · Angelopoulos & Bates

Split conformal prediction, the guarantee

• Any model, any distribution, any finite
• At you need , or the interval is infinite

Split conformal prediction, live

Conformal prediction, marginal not conditional

• Split conformal: "the width is exactly constant over x" (Lei sec. 5.2)
• Toy, 90% target: 100%, 95%, 82% by third
• Exact conditional coverage cannot be had without assumptions (Vovk 2012; Foygel Barber et al. 2021)
• Normalized scores : 94%, 91%, 90%
• CQR: conformalize quantile regressions, "fully adaptive to heteroscedasticity" (Romano et al. 2019)

Jackknife+ and CV+

• Split conformal spends data on calibration
• Jackknife+: leave each point out, refit, record
  • Interval from the quantiles of
• CV+: the same with folds
• Guarantee , and the 2 is tight; "in practice" about
• Plain jackknife: coverage "may actually vanish"
• Cost: or fits. Small data, cheap model: CV+. Otherwise split

Barber, Candès, Ramdas & Tibshirani, Ann. Stat. 2021

Conformal prediction on concrete

• Grouped: every conformal method meets 90%, as guaranteed
• Extrapolation: split conformal 56%; widths in MPa

Conformal prediction, out of domain

• Strongest mixes are not exchangeable with calibration mixes: held out because different
• The guarantee did not fail; its condition was not met
• Toy, window sliding out: 99%, 94%, 86%, 56%, 16%
• Normalizing rescued concrete only because the spread grew there; toy: 23%
• The fluorine case from the opening is this failure
• Weighted conformal needs the likelihood ratio and overlapping support (Tibshirani et al. 2019); beyond exchangeability, a bound you cannot compute (Barber et al. 2023)

Conformal prediction in code, MAPIE

from mapie.regression import SplitConformalRegressor, CrossConformalRegressor

split = SplitConformalRegressor(model, confidence_level=0.9, prefit=True)
split.conformalize(X_cal, y_cal)
y_pred, y_int = split.predict_interval(X_test)

cvplus = CrossConformalRegressor(model, confidence_level=0.9,
                                 method="plus", cv=GroupKFold(10))
cvplus.fit_conformalize(X_train, y_train, groups=groups_train)

• Version 1 renamed MapieRegressor and alpha; most tutorials use the old names

MAPIE docs · Taquet et al. 2022

Conformal prediction, a question

pycse, the last-layer idea

• Treat the network's last layer as a linear model on learned features
• Use linear regression's uncertainty there; predict(X, return_std=True)
• Bishop's warning: with localized features the model "becomes very confident ... when extrapolating"

pycse, inside and beyond the data

pycse, an import trap

from pycse.sklearn import NNBR            # ImportError in pycse 2.11.1

from pycse.sklearn.nnbr import NeuralNetworkBLR
from pycse.sklearn.llpr_regressor import LLPRRegressor
from pycse.sklearn.dpose import DPOSE
from pycse.sklearn.lr_uq import LinearRegressionUQ

Trade-offs

Recap

• UQ is a claim about coverage; check it on held-out data, with width and a proper score
• GP: honest about missing data only if the prior is right; fails with varying noise, the wrong kernel, few points
• Ensembles: the best common baseline; add noise, then rescale
• Conformal: an exact guarantee that needs exchangeability
• No method checks itself outside the range where it was checked

Standings

Nicknames only. Everyone who skipped one still counted in every bar you saw.

Before next time

• Reading: Angelopoulos & Bates, sections 1 and 2; Palmer et al. 2022
• Practice module for this session: on the course site
• Run l14-uq.ipynb; it installs MAPIE and pycse

Notes for this lecture: lectures/l14/notes.md