uncertainty view markdown

Some notes on uncertainty in machine learning, particularly deep learning.


  • calibration - predicted probabilities should match real probabilities
    • platt scaling - given trained classifier and new calibration dataset, basically just fit a logistic regression from the classifier predictions -> labels
    • isotonic regression - nonparametric, requires more data than platt scaling
      • piecewise-constant non-decreasing function instead of logistic regression
  • confidence - predicted probability = confidence, max margin, entropy of predicted probabilities across classes
  • ensemble uncertainty - ensemble predictions yield uncertainty (e.g. variance within ensemble)
  • quantile regression - use quantile loss to penalize models differently + get confidence intervals


Note: outlier detection uses information only about X to find points “far away” from the main distribution

  • overview from sklearn
    • elliptic envelope - assume data is Gaussian and fit elliptic envelop (maybe robustly) to tell when data is an outlier
    • local outlier factor (breunig et al. 2000) - score based on nearest neighbor density
    • idea: gradients should be larger if you are on the image manifold
    • isolation forest (liu et al. 2008) - lower average number of random splits required to isolate a sample means more outlier
    • one-class svm - estimates the support of a high-dimensional distribution using a kernel (2 approaches:)
      • separate the data from the origin (with max margin between origin and points) (scholkopf et al. 2000)
      • find a sphere boundary around a dataset with the volume of the sphere minimized (tax & duin 2004)
  • detachment index (kuenzel 2019) - based on random forest
    • for covariate $j$, detachment index $d^j(x) = \sum_i^n w (x, X_i) \vert X_i^j - x^j \vert$
      • $w(x, X_i) = \underbrace{1 / T\sum_{t=1}^{T}}{\text{average over T trees}} \frac{\overbrace{1{ X_i \in L_t(x) }}^{\text{is } X_i \text{ in the same leaf?}}}{\underbrace{\vert L_t(x) \vert}{\text{num points in leaf}}}$ is $X_i$ relevant to the point $x$?

uncertainty detection

Note: uncertainty detection uses information about X / $\phi(X)$ and Y, to find points for which a particular prediction may be uncertain. This is similar to the predicted probability output by many popular classifiers, such as logistic regression.

  • rejection learning - allow models to reject (not make a prediction) when they are not confidently accurate (chow 1957, cortes et al. 2016)
  • To Trust Or Not To Trust A Classifier (jiang, kim et al 2018) - find a trusted region of points based on nearest neighbor density (in some embedding space)
    • trust score uses density over some set of nearest neighbors
    • do clustering for each class - trust score = distance to once class’s cluster vs the other classes
  • Understanding Failures in Out-of-Distribution Detection with Deep Generative Models (zhang…ranganath, 2021) - explicit likelihood DGMs (e.g. autoregressive models, normalizing flows) have been shown to assign higher likelihoods to unrelated inputs than even those from the training distribution
    • OOD detection has been defined as the task of identify- ing “whether a test example is from a different distr. from the training data” (Hendrycks & Gimpel, 2017)
    • without any constraints on out-distributions, the task of OOD detection is impossible
    • ood_distr

bayesian approaches

  • epistemic uncertainty - uncertainty in the DNN model parameters
    • without good estimates of this, often get aleatoric uncertainty wrong (since $p(y\vert x) = \int p(y \vert x, \theta) p(\theta \vert data) d\theta$
  • aleatoric uncertainty - inherent and irreducible data noise (e.g. features contradict each other)
    • this can usually be gotten by predicting a distr. $p(y \vert x)$ instead of a point estimate
    • ex. logistic reg. already does this
    • ex. regression - just predict mean and variance of Gaussian
  • gaussian processes

neural networks


directly predict uncertainty

nearest-neighbor methods

bayesian neural networks

conformal inference

  • conformal inference constructs valid (wrt coverage error) prediction bands for individual forecasts
    • relies on few parametric assumptions
    • holds in finite samples for any distribution of (X, Y) and any algorithm $\hat f$
    • starts with vovk et al. ‘90
  • simple example: construct a 95% interval for a new sample (not mean) by just looking at percentiles of the empirical data
    • empirical data tends to undercover (since empirical residuals tend to underestimate variance) - conformal inference aims to rectify this
  • Uncertainty Sets for Image Classifiers using Conformal Prediction (angelopoulos, bates, malik, jordan, 2021)