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SAND2017-6776
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Uncertainty Quantification for

Machine Learning
David J. Stracuzzi, Maximillian G. Chen, Michael C. Darling, Matthew G. Peterson,
Charles Vollmer

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 SAND2017-6776
Unlimited Release
Printed June, 2017

Uncertainty Quantification for Machine Learning

David J. Stracuzzi
Data-Driven and Neural Computing Department
Sandia National Laboratories
djstrac@sandia.gov

Maximillian G. Chen
Mission Analytics Solutions 2 Department
Sandia National Laboratories
mgchen@sandia.gov

Michael C. Darling
Data-Driven and Neural Computing Department
Sandia National Laboratories
mcdarli@sandia.gov

Matthew G. Peterson
Scalable Analytics and Visualization Department
Sandia National Laboratories
mgpeter@sandia.gov
Charles Vollmer
Data-Driven and Neural Computing Department
Sandia National Laboratories
cvollme@sandia.gov

3
 Abstract

In this paper, we assert the importance of uncertainty quantification for machine learning and
sketch an initial research agenda. We define uncertainty in the context of machine learning,
identify its sources, and motivate the importance and impact of its quantification. We then
illustrate these issues with an image analysis example. The paper concludes by identifying
several specific research issues and by discussing the potential long-term implications of
uncertainty quantification for data analytics in general.

4
Acknowledgements

This work was funded by the Sandia National Laboratories Laboratory Directed Research
and Development (LDRD) program. The authors thank Kristina Czuchlewski, Stephen
Dauphin, Rich Field, and Jim Stewart for helpful comments and discussion.

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6
Contents

1 Introduction 9

2 Motivating Uncertainty Quantification 11

2.1 Why Uncertainty Matters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 Probability Versus Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3 Performance Versus Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.4 Uncertainty Quantification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 A Preliminary Data-Driven Uncertainty Analysis Approach 17

3.1 Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2 Gaussian Mixture Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.3 Bootstrap Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4 Demonstration on Optical and Lidar Data 21

5 Discussion 27

5.1 Evaluating Each Data Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

6 Conclusions 31

References 32

7
List of Figures

2.1 Illustration of the difference between probability and uncertainty. . . . . . . . . . . 12

2.2 Relationship between machine learning and uncertainty quantification. . . . . . . 14

4.1 Images of target area captured by optical and lidar sensors. . . . . . . . . . . . . . . . 21

4.2 Most likely categories assigned to each pixel. . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.3 Category confusion plots showing entropy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.4 Plots of pixel category uncertainty. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.5 Violin plots showing change in posteriors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

5.1 Maps of pixel category probabilities and changes. . . . . . . . . . . . . . . . . . . . . . . . 28

8
Chapter 1

Introduction

Machine learning establishes statistical relationships between observable variables and some
unobservable state of the world. Such mappings are important for domains in which theo-
retical mappings are incomplete or missing, ranging from image analysis to cyber security
to climate modeling. However, statistical models are fundamentally stochastic: predicted
and inferred values are random variables and inherently uncertain. A rigorous evaluation of
uncertainty can improve the precision and quality of model predictions, the model sensitivity
to rare or weakly indicated phenomena, and end-user decision making processes.

Uncertainty quantification provides a measure of sufficiency of the available data and the
selected modeling approach for answering a question of interest. Currently, such determina-
tions depend heavily on expert opinion, using a mix of domain and modeling expertise, and
accuracy-based validation metrics, such as precision-recall or ROC curves. Model induction
is an ill-posed problem (Hadamard, 1923) however, meaning that the resulting model parame-
terizations may not be unique and may be highly sensitive to the input data. Accuracy-based
validation metrics do not address questions of uniqueness or stability, so these issues often
go unexamined.

In this paper, we propose uncertainty quantification as a research area that requires at-
tention by the machine learning community. Although many with a background in statistics
are familiar with uncertainty, most machine learning research focuses on model induction
and performance evaluation as opposed to the rigorous evaluation of the suitability of a
learned model to a particular task. Our goal is to demonstrate that uncertainty analysis
of learned models provides insight and information that is not otherwise available, and to
provide an overview of research needed to bring uncertainty quantification into mainstream
practice.

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10
Chapter 2

Motivating Uncertainty Quantification

If machine learning establishes relationships between observable and unobservable variables,

then uncertainty quantification characterizes the variability in those relationships. Variabil-
ity arises from multiple points within the machine learning process, but generally reduces
to a single interpretation: What is the range of possible responses that a model might make
given the available data? Explicitly quantifying variability in a model and its outputs has a
number of implications for the machine learning community, which we highlight throughout
the paper. In this section, we discuss what uncertainty means in a machine learning context,
where it comes from, and why analyzing uncertainty is important.

2.1 Why Uncertainty Matters

Consider spell checking with automatic replacement as implemented in many smartphones,

tablets, and other environments. Most approaches combine limited context and the poten-
tially incorrect or incomplete spelling of the word in question. Spell checkers often do a
mediocre job of providing corrections and completions. The results can turn a simple typo
into completely nonsensical statements, or worse, alter the author’s intent. We argue that
such issues do not stem from poor language modeling — performance is limited given the
available information and computational resources — but from a neglect of uncertainty.

For a given misspelling and context, the number of possible corrections can be large. More
importantly, many of the corrections may be similarly appropriate given the available data.
This case corresponds to high uncertainty — many equally plausible possible solutions —
and automatically selecting a replacement amounts to random guessing among alternatives.
The available information is not sufficient to strongly indicate one word over another. This
is a key point of our discussion. The output of a learned model is a random variable, and the
strength of evidence that supports selection of one value over another is important.

The issue becomes more complex for problems in which data comes from multiple sources.
For example, if we observe a single geospatial location using optical, lidar, and infrared
sensors, we then have a multimodal analysis problem that relies on color, height, and heat
information to distinguish objects in the scene. Ideally, two or more sources may provide
complementary information, which improves performance and reduces variability in detection

11
 Figure 2.1. Illustration of the difference between probabil-
ity and uncertainty. Each panel indicates a different uncer-
tainty distributions that yield similar probability point esti-
mates.

or segmentation analyses. A second possibility is that some sources contribute very little
to the final result. This occurs when a distinguishing feature, such as heat, is not captured
by a given modality, such as optical imagery. In some cases, data sources may disagree on
identification, which, at a minimum, increases the variability in the results. Importantly, the
cases outlined above can only be distinguished by evaluating uncertainty.

2.2 Probability Versus Uncertainty

Quantifying variability and characterizing a range of possible outputs naturally evokes the
notion of probability. However, probability and uncertainty are two separate concepts. Prob-
ability measures the likelihood or belief in an inferred outcome, while uncertainty character-
izes the range of possible outcomes or beliefs due to imperfect information. Many statistical
models are probabilistic, but evaluating uncertainty requires additional analysis.

For example, given a probabilistic classifier M , uncertainty characterizes the variability

in the probability estimates. Figure 2.1 illustrates the difference. The horizontal axes corre-
spond to the probability that a particular example x belongs to class c. The point estimate
shown in panel (a) then corresponds to the probability P (x = c|M ) assigned by M that x is
a member of c.

12
 Now suppose that we repeatedly sample our training data and construct a new classifier,
Mi , for each sample. The vertical axis then indicates the frequency with which the target
example is assigned a given probability (a histogram) across all of the classifiers. Panel (b)
shows one possible distribution P (x = c|Mi ) clustered tightly around the point estimate,
suggesting that the point estimate is a stable approximation of the true probability.

Figure 2.1(c) shows a second distribution over the probability values, this time bimodal.
Some of the Mi produced a high value for P (x = c|Mi ), while others set it low. Importantly,
an ensemble that votes or averages the individual probabilities produces a point estimate
similar to that shown in panel (b). The bimodal curve indicates far more uncertainty and
variability in the response, suggesting at least two plausible interpretations of the example.

Figure 2.1(d) shows a more uniform distribution over probabilities, which indicates a
strong dependence between classifier output and the specific training sample. Again, an
average over P (x = c|Mi ) creates a point estimate similar to the other panels. The voted
probability reasonably estimates the degree to which our target example fits into class c,
but it does not characterize the variability in that estimate. For many practical machine
learning applications, the variability matters as we illustrate throughout the paper.

2.3 Performance Versus Uncertainty

Traditional performance evaluation metrics also differ from uncertainty quantification. Con-
fusion matrices, ROC curves, and F-Scores all use the true and false positive and negative
rates to quantify a classifier’s performance. These methods provide a global measure of a
classifier’s ability to discriminate among examples of different classes. However, none of
these methods accounts for the variability in a classifier’s output.

Returning to Figure 2.1, the models in panels (b-d) all predict the same class assignment
and get the same credit for a correct response. Given a slightly different example or a slightly
different training set, the models in (b) would tend to remain stable in their predictions while
those in (c) or (d) might vary substantially. The outputs in the latter cases resemble (biased)
random guessing. Traditional performance evaluation does not distinguish between a lucky
guess and a consistently correct response. Thus, a model that exhibits high discrimination
and high uncertainty is far less reliable than one that exhibits low uncertainty. Minor changes
in the training or test data can lead to large and unpredictable shifts in performance for an
uncertain model.

2.4 Uncertainty Quantification

Quantifying the uncertainty in a learned model relative to a given input example requires
first identifying the different sources of error and then combining those sources into an
overall uncertainty in the predicted quantity. Figure 2.2 shows a breakdown of the problem

13
 inverse uncertainty quantification problem

classic machine learning problem

model regularization
unobservable inference indirect
induction /
inferred state observations
calibration
(c) (a)
(b)

solution inference model-form regularization measurement

uncertainty = errors
∪
uncertainty
∪ effects ∪ errors

Figure 2.2. The relationship between the classic machine

learning problem and inverse uncertainty quantification. The
steps of the standard machine learning task and their asso-
ciated sources of uncertainty. Uncertainty in model predic-
tions combines the uncertainties associated with each ma-
chine learning processing step.

space. To illustrate the issues involved, we first review the basic steps of a machine learning
application, and then highlight the sources of uncertainty in the context of an example.

Machine learning maps observed data (a) to unobservable properties of interest (c). For
example, seismic event detection maps noisy waveforms into signal onset times by learning
a model (Figure 2.2b) of the data from which the onset time can be estimated. In this case,
one model captures the noise preceding the signal while a second includes both signal and
noise. A learning algorithm optimizes the fit of each model such that the time point at which
the two models meet represents the onset.

Many applications preprocess data, such as by a low-pass filter in the seismic example,
and bias the model, such as by a prior, to push the resulting parameterization toward
specific areas of the solution space (called regularization) to improve model fit. Likewise,
inference procedures map examples through the model into output values. Depending on
the specific model used, inference may only be approximate. Taken together, these steps
comprise the major design decisions of a machine learning problem (inner box of Figure 2.2),
and they mirror the inverse problem frequently solved in natural science and engineering
design domains.

The difference between classical inverse problems and machine learning is that the model
in Figure 2.2b would traditionally be composed of theoretical equations that define the map-
ping from observations to unobservables instead of an induced statistical model. Uncertainty
arises from many sources, as shown in the bottom row of Figure 2.2, and must be quanti-
fied and aggregated to fully characterize the modeled system (outer box), known as the

14
inverse uncertainty quantification problem. The problem is well-studied for cases in which
an equation-based model is known (see Smith, 2014, for example). The switch to stochastic
models, however, raises new questions about uncertainty quantification methods.

Returning to the seismic example, measurement errors arise at the seismograph, which
requires calibration, receives vibrations from non-seismic sources such as wind, and has
limited sensitivity. Regularization effects arise from data preprocessing, which removes ir-
relevant signals, but may also alter the seismic information. Model-form uncertainty arises
from the learning process: many plausible model parameterizations exist and each provides
a slightly different output. The issue is exacerbated if we consider different classes of models,
such as different types of autoregressive models in this case. All of these sources, plus any
inference errors as noted above, combine to produce an uncertainty in the output value. To
fully characterize a seismic onset, we must extract a probability density function over possi-
ble onset times, which reveals alternative plausible hypotheses and their relative likelihood,
as opposed to just the most likely onset.

Finally, note that confidence intervals provide only a weak approximation of uncertainty.
For example, confidence intervals placed around the curves in Figure 2.1 might show that one
distribution has greater variance than another, but still hides the underlying shape. While
sometimes sufficient, we argue that the shape is important in general.

15
16
Chapter 3

A Preliminary Data-Driven
Uncertainty Analysis Approach

To illustrate the importance of uncertainty quantification in machine learning applications,

we consider an image analysis case study. In this section, we outline a simple analysis
framework, demonstrating its use on a specific example in Section 4. Our preliminary work
focuses almost entirely on quantifying model-form uncertainty. In section 5, we return to the
general problem of uncertainty analysis to outline research questions associated with other
sources of uncertainty and longer-term implications for the work.

To quantify uncertainty in unsupervised image classification, we use the Gaussian Mix-

ture Model (GMM) with bootstrap sampling. Our approach builds on the Bayesian pixel
classification methods surveyed by Falk et al. (2015). Importantly, given that the goal is to
illustrate uncertainty quantification, we set aside discussion of which model is most appro-
priate to the task. We choose to use the GMM and bootstrap since they are well known and
allow our discussion to focus on uncertainty analysis rather than the model itself. A variety
of Bayesian methods are capable of producing the types of posterior distributions that we
discuss below. However, this is rarely done in practice, and often requires either additional
processing or even modification of the underlying algorithms.

3.1 Setting

Consider N data sources, Di for i = 1, . . . , N , such that each source consists of a regularly
spaced lattice indexed by j = 1, . . . , n. Each index refers to a specific pixel which may belong
to one of K categories. We use category to refer to unsupervised clusters and class to refer
to the semantic tags used to label the categories. Classes can be assigned either manually
or through a supervised labeling process, which we defer to future work.

For simplicity and without loss of generality, we assume that the N data sources are
combined into observations D = y1 , . . . , yn , such that each yj represents the concatenated
data vectors from all sources. We revisit this assumption in the discussion of future work.
Each yj is a noisy observation of pixel j. The true category for each pixel, zj , is not directly
observable, but is a random variable conditioned on the observations and assumed model.

17
 Our goal is the posterior estimation of the model parameters, and therefore the true
category, conditioned on the observations. To estimate the desired posteriors, we take a
two-step approach. First, we use a bootstrap procedure to resample the original image data,
D = y1 , . . . , yn , and obtain multiple samples that allow us to estimate the distribution of
the probability, P (zj = k|D), that k is the true category of pixel j. Second, we fit a GMM to
each bootstrapped sample, obtaining point estimates for P (zj = k|D) from each sample. The
aggregation of point estimates provides the posterior distribution over category probabilities.

3.2 Gaussian Mixture Model

Given data D with independent observations y1 , ..., yn , the likelihood for a mixture model
with K components is
n X
Y K
LM IX (θ1 , ..., θK ; τ1 , ..., τK |y) = τk fk (yi |θk )
i=1 k=1

where fk and θk are the respective density and parameters of the k th components in the
th
PK and τk is the probability that an observation belongs to the k component (τk ≥
mixture
0; k=1 τk = 1). Most commonly, fk is the multivariate normal (Gaussian) density φk , which
is parameterized by its mean µk and covariance Σk , with probability distribution function
(pdf):

exp{− 21 (yi − µk )T Σ−1

k (yi − µk )}
φk (yi |µk , Σk ) ≡ p . (3.1)
det(2πΣk )

The Expectation-Maximization (EM) algorithm (Dempster et al., 1977; McLachlan and

Krishnan, 1997) finds the maximum-likelihood estimate for unknown parameters in a data
model. The data, which consists of n multivariate observations and is denoted by X =
(xi , i = 1, ..., n), is recoverable from the parameter space (Y, Z), in which Y = (yi , i =
1, ..., n) is observed and Z = (zi , i = 1, ..., n) is unobserved.

If the xi are independent and identically distributed (iid) according to a probability

distribution f with parameters θ, then the complete-data likelihood is
n
Y
LC (xi |θ) = f (xi |θ).
i=1

Further, if the probability that a particular variable is unobserved depends only on the
observed data y and not on z, then the observed-data likelihood, LO (y|θ), can be obtained
by integrating z out of the complete-data likelihood,
Z
LO (y|θ) = LC (x|θ)dz. (3.2)

18
The MLE for θ based on the observed data maximizes LO (y|θ).

The EM algorithm alternates between two steps, an “E step” and an “M step.” In the
“E step,” the conditional expectation of the complete data log-likelihood given the observed
data and the current parameter estimates are computed, i.e. compute
Q(θ|θ(t) ) = EZ|X,θ(t) [log L(θ; X, Z)]. (3.3)
In the “M step,” parameters that maximize the expected log-likelihood from the E step are
determined, i.e. compute
Q(t+1) = arg max Q(θ|θ(t) ). (3.4)
θ

The unobserved portion of the data may involve values that are missing due to nonresponse
and/or quantities that are introduced to reformulate the problem for EM. Under fairly mild
regularity conditions, EM can be shown to converge to a local maximum of the observed-data
likelihood (e.g. (Dempster et al., 1977; Boyles, 1983; Wu, 1983; McLachlan and Krishnan,
1997)). Although these conditions do not always hold in practice, the EM algorithm has
been widely used for maximum likelihood estimation for mixture models with good results.

In EM for GMMs, the data is a set of n iid multivariate Gaussian distributions, which
means xi , i = 1, ..., n, follow the pdf given by (3.1). The “complete” data are considered to
be xi = (yi , zi ), where zi = (zi1 , ..., ziK ) is the unobserved portion of the data. , with
(
1 if xi belongs to category k ,
zik =
0 otherwise.
Assuming that each zi is iid according to a multinomial distribution of one draw from K
categories with
Q probabilities τ1 , ..., τK , and that the density of an observation yi given zi
is given by K φ
k=1 k (y |θ
i k ) zik
, the resulting complete-data log-likelihood that we want to
maximize with the MLEs of θk is
n X
X K
l(θk , τk , zik |x) = zik log[τk φk (yi |θk )].
i=1 k=1

We wish to estimate θk that maximizes l(θk , τk , zik |x), i.e.,

max l(θk , τk , zik |x),
θk ∈Θ

Θ = {(µk , Σk , τk ) : µk ∈ Rd , Σk = ΣK
k > 0, Σk ∈ R
d×d
,
K
X
τk ≥ 0, τk = 1}.
k=1

In EM for GMMs we can estimate zik , the conditional probability that observation i
belongs to category k in the E-step with the estimate:
τ̂k φ(yi ; µk , Σk )
ẑik = PK .
j=1 τ̂j φ(yi ; µk , Σk )

19
 In the M-step, (3.5) is maximized in terms of τk and θk with zik fixed at the values
computed in the E step, ẑik . The resulting estimates for the category probabilities, means,
and covariances are
n Pn
1X ẑik yi
τ̂k = ẑik , µ̂k = Pi=1 n , (3.5)
n i=1 i=1 ẑik
Pn
ẑik (yi − µk )(yi − µk )T
Sk = i=1 Pn . (3.6)
i=1 ẑik

(Fraley and Raftery, 2002)

With the above estimated values, we can compute any of the following parameters that
estimate probability and uncertainty:

∗
1. zik : estimated conditional probability that observation i belongs to category k
∗
2. 1 − zik : measure of the uncertainty that observation i belongs to category k

3. {j|zij∗ = maxk zik

∗
}: classification of observation i to the category with the maximum
estimated conditional probability
∗
4. 1 − maxk zik : measure of the uncertainty in the classification

3.3 Bootstrap Method

The bootstrap method considers the following problem, initially studied by Efron (1979).
Given a random sample Y = (Y1 , Y2 , ..., Yn ) from an unknown probability distribution F ,
estimate the sampling distribution of random variable R(Y, F ) on the basis of the observed
data, y. To do so, we re-sample y with replacement to get a new sample y∗ from the pool
of observed data.

In the image analysis problem, we have only one image from each data source. To compute
the uncertainty of the class probabilities, we can generate many similar images from which
we can estimate a distribution over class probabilities. After using the bootstrap method
to generate the sample of images, we can obtain sampling distributions and statistics for
the model parameters, τk , µk , Σk , and the pixel categories ẑi k. Note that this approach only
captures the model-form uncertainty.

The procedure is as follows. First, obtain S bootstrap samples by sampling the observed
data y = y1 , ..., yn with replacement and obtaining n pixels. Denote the bootstrap samples
y∗1 , y∗2 , ..., y∗S . Then, for each of the S bootstrap samples, fit a GMM and obtain estimates
for the parameter zik using the EM algorithm. The S values of ẑik provide a sampling
distribution for ẑik .

20
Chapter 4

Demonstration on Optical and Lidar

Data

We demonstrate uncertainty quantification on a multimodal imagery analysis task. The

images cover a small region in Philadelphia that contains trees, grass, water, pavement, a
building, and a variety of small, undetermined objects. Figure 4.1a shows the 100x100 pixel
optical image of the target area. Each pixel contains red, green, and blue values scaled
from 0 to 1. Figure 4.1b shows the same region imaged with lidar (Light Radar) which has
been preprocessed into a height map (lighter colors indicate taller data points). We use the
data as prepared by O’Neil-Dunne et al. (2013) and merge the two sources into a single,
four-dimensional vector containing the R, G, B, and height-above-ground values.

We apply the GMM with K = 5 categories and S = 1, 000 bootstrap iterations to the
data shown in Figure 4.1. A common representation of the most likely category for each

(a) (b)

Figure 4.1. 100x100 pixel images of target area captured

by optical (a) and lidar (b) sensors.

21
 (a) (b)

Figure 4.2. Most likely categories assigned to each pixel

for optical only (a) and combined optical with lidar (b). The
arrow identifies a roof pixel that changes in both category
and uncertainty with the addition of lidar data.

pixel assigns a unique color to each category, as illustrated in Figure 4.2. Panel (a) shows
the category estimates when using only optical data while panel (b) shows the categorization
when using both the optical and lidar sources. Note that the two trees in the scene become
clearer while the boats and the water become less defined. The improved tree recognition
follows from lidar’s ability to detect the branches, even without foliage, while the loss of
definition in the water follows from the typically poor performance of lidar over water.

Figure 4.2 provides no information about the probabilities associated with each category
or the uncertainties. We can capture the former by applying Shannon entropy (Shannon,
2001) to the probability distribution over the five categories. For a discrete random variable
X with possible values {x1 , ..., xn } and probability mass function P (X), entropy is: H(X) =
P n
i=1 −P (xi ) ln P (xi ). Entropy is maximized if the probability mass is equally distributed
across all of the available categories and minimized when all of the mass is assigned to a
single category.

We can use per-pixel entropy scores to produce a grayscale image in which the brightness
indicates the entropy value, as illustrated in the top row of Figure 4.3. The same idea
applies to the category-colored images by adjusting the colors toward white based on the
entropy as shown in the bottom row. Lighter shades indicate greater entropy. The resulting
images identify (approximately) the degree to which pixels strongly identify with a particular
category.

Based on the optical image alone (panel a), the largest areas of category confusion (high

22
 (a) (b)

Figure 4.3. Category confusion plots showing the entropy

only (top row) and the entropy overlaid onto the most prob-
able category (bottom row) for optical (a) and combined (b)
imagery.

entropy) center on the concrete near the bottom of the image, while the trees indicate very
little confusion. However, the addition of lidar data in panel b increases the confusion
substantially across the entire image, even while the results in many areas represent an
improvement over the optical data alone. Note in particular the confusion over the roof
and shadow areas. The analysis is struggling with the relatively steep changes in height
associated with these areas without a corresponding change in color, which makes assigning
a category difficult.

23
 (a) (b)

Figure 4.4. Category uncertainty plots showing the stan-

dard deviation of the posterior distribution alone (top row)
and overlaid onto the most probable category (bottom row)
for optical (a) and combined (b) imagery.

Importantly, the images in Figure 4.3 do not indicate variability in the probability values.
The mean category probabilities do not provide information about the shape of the associated
posterior distributions. We can approximate the width of the posteriors by plotting the
standard deviations for each pixel’s most probable category. Increasing variance results in
lighter (grayscale) pixel values. As with the confusion plots, we can overlay the uncertainty
information onto the colored category images as shown in Figure 4.4.

As with the entropy plots, uncertainty in the optical image is mostly confined to the

24
concrete. However, comparing panel (b) of Figures 4.3 and 4.4 shows the difference between
uncertainty and confusion. Although the roof shows high entropy in the category proba-
bilities, the uncertainty is generally low. This implies that several categories received some
probability mass, but the underlying posteriors are not broadly distributed. In other words,
even though none of the categories provides a perfect match, the most likely category as-
signment should be reliable. This is not true of areas around the pavement, water, or the
upper left tree, all of which show high uncertainty.

Each of the described visualizations provides a crude estimate of the confusion or uncer-
tainty associated with pixel category posterior distributions. To fully examine the posteriors,
we create modified violin plots which show kernel density estimations of the posteriors for
each category individually, as illustrated in Figure 4.5. The shaded bars indicate the mean
probability for each category. Though not practical in general, the violin plots provide a
detailed view of important data points, such as those with high uncertainty or those near
category boundaries.

Figure 4.5 shows the violin plots for the roof pixel identified by the arrow in Figure 4.2.
Note the change in most likely category when lidar is included. Panel (a) shows that three
of the posteriors for the optical analysis, categories 1, 2, and 4, have bimodal characteristics.
Category 1, which differs from the other roof pixels, clearly dominates the other categories.
However, category 4, which does agree with the other roof pixels, also includes a secondary
mode that suggests that in some of the bootstrapped samples, category 4 was the best fit.
For cases such as Figure 4.5a, in which some of the categories have a bimodal structure but
the secondary modes are small, the class probabilities are sufficient.

Adding the lidar data in panel b changes the both the result and the utility of the uncer-
tainty results substantially. Now category 4 is the most likely, and the category probability
is higher than the best in panel a. However, the posteriors are distributed broadly, meaning
that even though category 4 has a relatively high probability, the assignment is less certain
than in the optical-only case. The variable posteriors associated with the categories in the
lower panel suggest that the optical and lidar sources do not agree for this pixel.

For the case study, we can apply background knowledge about the physics of the two
sensors and the scene to conclude that although the pixel may be off-color, it still belongs
to the roof. Consider next a case in which we lack the domain knowledge to determine
the correct interpretation manually. Then the role of uncertainty analysis becomes more
important. We can use the uncertainty results to determine the stability and reliability of
the learned model, even in the absence of ground truth. In the next section, we extend this
argument to assessing the value or impact of each input sensor to the output.

25
 (a)

(b)

Figure 4.5. Violin plots for the roof pixel identified in

Figure 4.2 showing the change in posteriors before (a) and
after (b) incorporating lidar data into the analysis.

26
Chapter 5

Discussion

The previous section provides a demonstration of the types of insights that uncertainty
quantification can provide into a statistical model and its associated data. In this section,
we look ahead to using the uncertainty information to identify how each data modality
specifically contributes to the analytic results. We also identify several areas of future work
required to realize these benefits.

5.1 Evaluating Each Data Source

In our case study, we generate an initial analysis based on the optical data alone. When
we add the lidar data and rerun the analysis, some of the pixels get assigned to different
categories, some of them exhibit changes in category confusion, and some show changes in
uncertainty. In financial domains, the value of adding information to an analysis is typically
measured in dollars, such as expected loss or gain. In general however, methods for measuring
the value of adding data to an analysis remain less well defined.

We define data from a given sensor as valuable to the extent that it changes the result
of an analysis. Figure 5.1 shows a simple attempt at using the category probabilities and
uncertainty information to define a notion of value for sensor data. Panel a shows a proba-
bility map for one of the five categories, corresponding roughly to grass, estimated from the
optical data. The shade of each pixel in the image indicates the probability that the pixel
belongs to the category, with dark green indicating high probability and white indicating
zero probability. The grass is visible through the tree branches, so many of the pixels under
the tree are recognized as grass in the absence of height information.

Panel b shows the probability map for the combined data. Notice that the tree has been
mostly assigned to another category, but the region still gets assigned to the green category
with low probability. Panel c shows the absolute value of the difference in probability between
the first two panels, which corresponds to the impact on the green category of adding lidar
to the optical data. Although it does not distinguish between increased and decreased
probabilities, it does indicate which areas of the data are impacted by the new information.

Finally, panel d shows the Kullback-Leibler (K-L) divergence (Kullback and Leibler,
1951) between the posterior distributions for the green category at each pixel. Darker pixels

27
 (a) (b)

(c) (d)

Figure 5.1. Probability maps showing green category pixel

probabilities based on optical data (a), combined optical and
lidar (b), and the difference between the two (c). Panel (d)
shows the K-L divergence between the category posteriors
associated with panels (a) and (b).

indicate greater divergence in the uncertainties due to the lidar. Although similar to the
probability difference map in this case study, they are not equivalent. The degree to which
these or related methods for evaluating the contribution of a data source to an analysis
provide useful information remains to be seen. However, we note that both changes in

28
probability and uncertainty may be useful for determining which data to collect, store, and
analyze during future iterations of an analysis or alternate versions of an analysis (grass
versus tree coverage, for example). This is particularly important in the content of remote
sensing applications in which different sensors can be selected and scheduled to achieve a
desired result, and background knowledge may not always be sufficient to determine the best
sensor combination.

5.2 Future Work

Our initial experiments with uncertainty quantification highlight several areas for future
work, many of which are related to the inverse uncertainty quantification problem problem
depicted in Figure 2.2. The first relates to the Gaussian assumption of the data model. For
example, in a pilot experiment (Authors, 2016) that included synthetic aperture radar data,
we found that GMMs do not perform well because the Gaussian assumption is inappropriate
for the modality. An alternative is to eliminate the assumption by constructing a Bayesian
nonparametric mixture model (see Orbanz and Buhmann, 2008, for example). Other al-
ternatives include the distance-dependent Chinese restaurants process (ddCRP) (Blei and
Frazier, 2011; Ghosh et al., 2011), which considers the distance between pixels, and proba-
bilistic fusion (Simonson, 1998). Extracting the posteriors from these methods is not trivial,
however. A related issue concerns regularization methods and their impact on modeling
results.

A second line of research relates to the sampling process, which has substantial impact
on computational efficiency. For example, biasing each run of the GMM with the result of
a prior run may reduce computation, but may also yield overly optimistic uncertainty esti-
mates. Another issue is the relationship between the number of data observations (pixels),
the number of categories, and the number of bootstrap samples required. We can also con-
sider alternatives to bootstrapping based on the measurement errors. If sensor performance
is well-characterized, we can sample data points from within the known range of errors in-
stead from the actual observations, which may produce more robust estimates of model-form
uncertainty. Other bootstrap alternatives include blockwise updates (Hall et al., 1995; Barbu
and Zhu, 2005). We can also investigate Markov Chain Monte Carlo (MCMC) methods for
sampling the posterior distribution, such as parametric representations (Fan et al., 2007),
min-marginal energies (Kohli and Torr, 2006), and random maximum a posterior (MAP)
perturbation (Hazan et al., 2013; Papandreou and Yuille, 2011), and the differences between
the uncertainty results obtained via methods from sampling the data and the posterior dis-
tribution.

A third area extends from the preceding discussion of source evaluation. Having estimated
the category assignment posterior distributions, an alternate formulation of the multimodal
data integration problem may be possible. Earlier, we merged the data sources (after co-
registration) into a single vector for each location. A more efficient approach may entail
analyzing each source independently using unsupervised methods and then merging the

29
distributions at the supervised stage by marginalizing over data sources and categories.
Generating the appropriate class posteriors requires substantial additional research, but
entails an important benefit. The unsupervised step is task independent, so changes to the
supervised task, or changes to the data sources included in the analysis, do not require
restarting the expensive mixture modeling or sampling steps.

Finally, the simple visualization methods used above require further refinement. We
have argued that uncertainty analysis uncovers information about the data and model not
available elsewhere. However, using this information to improve human-machine interaction
and trust requires visualizations that convey the implications of the analysis results succinctly
and accurately. Very little research has been conducted on uncertainty visualization.

30
Chapter 6

Conclusions

Our goal in this work is to demonstrate the importance of uncertainty quantification in ma-
chine learning, illustrate the implications of uncertainty quantification for the larger problem
of data analysis, and to identify several areas in which additional research is needed. Broadly
speaking, uncertainty analysis provides detailed insights into the variability and reliability
of predictions and results generated by machine learning and statistical models. To be clear,
uncertainty does not speak to the correctness of an output. However, in the absence of
ground truth and supervised data, uncertainty does provide a alternative measure for model
evaluation. Greater variability in output values indicates that the model has very little (or
conflicting) evidence to support it’s result. Importantly, the uncertainty distributions refer
to a particular example; they are not aggregated across many different examples.

Aside from the implications for determining the value of information and source selection
discussed above, uncertainty can also speak to broader questions of resource allocation,
algorithm selection, and so on. For example, given large quantities of data and limited
processing capability, allocating extra cycles to examples that have high uncertainty based
on an initial analysis may be productive. Similarly, we can start considering the trade-off
between the accuracy and variability in results produced by learning algorithms. Finally,
the results of many machine learning problems are used as the basis for decision making
by human data analysts or systems operators. Providing uncertainty information may yield
deeper insight into how a model produced it’s result, and the trustworthiness of that result.
Given these issues, we anticipate that uncertainty quantification will play an increasing role
in data analysis applications as the diversity of data and information available continues to
increase.

31
32
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