# Why have a Unified Predictive Uncertainty?
Disentangling it using Deep Split Ensembles

Utkarsh Sarawgi1, Wazeer Zulfikar1, Rishab Khincha1, 2, Pattie Maes1

###### Abstract

Understanding and quantifying uncertainty in black box Neural Networks (NNs)
is critical when deployed in real-world settings such as healthcare. Recent
works using Bayesian and non-Bayesian methods have shown how a unified
predictive uncertainty can be modelled for NNs. Decomposing this uncertainty
to disentangle the granular sources of heteroscedasticity in data provides
rich information about its underlying causes. We propose a conceptually simple
non-Bayesian approach, deep split ensemble, to disentangle the predictive
uncertainties using a multivariate Gaussian mixture model. The NNs are trained
with clusters of input features, for uncertainty estimates per cluster. We
evaluate our approach on a series of benchmark regression datasets, while also
comparing with unified uncertainty methods. Extensive analyses using dataset
shits and empirical rule highlight our inherently well-calibrated models. Our
work further demonstrates its applicability in a multi-modal setting using a
benchmark Alzheimer’s dataset and also shows how deep split ensembles can
highlight hidden modality-specific biases. The minimal changes required to NNs
and the training procedure, and the high flexibility to group features into
clusters makes it readily deployable and useful. The source code is available
at
https://github.com/wazeerzulfikar/deep-split-ensembles

##  1 Introduction

(a) Unified

(b) Disentangled

(c) Unified

(d) Disentangled

Figure 1: Examples of modelling unified vs disentangled uncertainties on 2
two-dimensional toy regression datasets,
$y=(x_{1}^{3}+\epsilon_{1})(x_{2}^{3}+\epsilon_{2})$ and
$y=(x_{1}^{4}+\epsilon_{1})(x_{2}^{4}+\epsilon_{2})$ where
$\epsilon_{1},\epsilon_{2}\thicksim\mathcal{N}(0,3^{2})$. The red points are
the observed noisy training samples and the black points are their projections
on the respective axes. The grey regions show the predicted mean along with
three standard deviations. The disentangled uncertainties have ‘different’
grey regions on the two dimensions (as opposed to unified), and are able to
‘contain’ the black points better. This illustrates how decomposed
uncertainties can capture disentangled information about the individual noise
in the input features.

Vast developments across a variety of machine learning tasks have led to
extensive deployment of neural networks (NNs) in safety-critical applications
ranging from medical diagnosis to self-driving cars (LeCun, Bengio, and Hinton
2015). For reliable, fair and aware models in many regression tasks, the point
prediction is not sufficient; the uncertainty or the confidence of that
prediction must also be estimated by the model. Understanding what a model
does not know is critical to using machine learning systems and mitigating
plausible biases and risks in decision making (Gal 2016).

Heteroscedasticity can be modelled as the changing variability of the random
disturbance in output values given the input features; in other terms, the
probabilistic variability introduced by the stochastic data generation
processes. A ‘unified’ predictive uncertainty would be a single estimate
modelled for all input features together. Multiple probabilistic methods have
been proposed to quantify the same. Bayesian approximation techniques such as
variational inference (VI) (Graves 2011; Blundell et al. 2015), expectation
propagation (Hernández-Lobato and Adams 2015), dropout-based VI (Kingma,
Salimans, and Welling 2015; Gal and Ghahramani 2016), NNs as Gaussian
processes (Lee et al. 2017), deterministic VI (Wu et al. 2018), Bayesian model
averaging in low-dimensional parameter subspaces (Izmailov et al. 2020), and
approximate Bayesian ensembling (Pearce, Leibfried, and Brintrup 2020) have
been shown to be quite useful in modelling the uncertainties in NNs. Non-
Bayesian approaches (Osband 2016; Lakshminarayanan, Pritzel, and Blundell
2017; Dusenberry et al. 2020; Jain et al. 2020) that involve bootstrapping and
ensembling multiple probabilitic NNs have shown performances comparable to
Bayesian methods with reduced computational costs and modifications to the
training procedure. Ashukha et al. (2020) performed a broad study of
ensembling techniques in context of uncertainty estimation. Qiu, Meyerson, and
Miikkulainen (2020) proposed a framework using residual estimation with an I/O
kernel (RIO) to estimate uncertainty in any pretrained standard NN. Almost all
previous works (MacKay 1992; Kay, Titterington et al. 1999; Welling and Teh
2011; Kendall and Gal 2017; Shridhar, Laumann, and Liwicki 2018; Snoek et al.
2019) including the ones above estimate a unified predictive uncertainty.
However, a single ‘unified’ uncertainty would fundamentally be unable to
distinguish the granular sources of heteroscedasticity in data, which is
critical in applications such as healthcare as it can provide rich information
about the underlying causes. ‘Disentangled’ predictive uncertainties would be
able to separate these tied sources with granular uncertainty estimates.

We propose a conceptually simple non-Bayesian approach, deep split ensemble,
to disentangle the predictive uncertainties using a multivariate Gaussian
mixture model while training NNs with clusters of correlated features. These
correlations can be statistical, or based on different input modalities
(multi-modal), domain knowledge or user needs. Figure 1 shows application on
simple examples using a multi-dimensional toy regression dataset (Section
3.2), highlighting an advantage of modelling disentangled predictive
uncertainties over unified uncertainties.

Recent works have shown how NNs are usually overconfident at predicting
probability estimates representative of the true likelihood (Guo et al. 2017).
One can use the model’s confidence on a target distribution to compare it with
its accuracy and adjust the predictions (Platt 1999; Guo et al. 2017;
Kuleshov, Fenner, and Ermon 2018). However, the distribution over this
observed data may shift and eventually be very different once a model is
deployed in practice. Robustness of uncertainty estimation under these
conditions of distributional shift is necessary for the safe deployment of
machine learning systems (Amodei et al. 2016; Varshney and Alemzadeh 2017;
Kumar, Liang, and Ma 2019; Thiagarajan et al. 2020). Snoek et al. (2019)
recently showed how post-hoc calibration can fail under even a mild shift in
the data. We show that modelling disentangled predictive uncertainties using
our approach produces inherently well-calibrated estimates per cluster of
features, without any post-hoc calibration. We assess it using a granular
feature-wise distributional shift. This helps address the critical and
practical concerns of risk, uncertainty, and trust in a model’s output.

#### Summary of contributions:

  1. 1.

To our knowledge, we are the first to propose a method to disentangle unified
predictive uncertainties with NNs.

  2. 2.

We perform a rigorous and comprehensive evaluation on the inherent calibration
of our models under dataset shifts on benchmark regression datasets.

  3. 3.

To demonstrate the applicability of our method, we extend it to include domain
knowledge, and to a multi-modal setting to highlight the potential hidden
modality-specific biases.

##  2 Deep Split Ensembles: Disentangling predictive uncertainties

###  2.1 Notation and setup

Let $\mathbf{x}\in\mathbb{R}^{d}$ represent a set of $d$-dimensional input
features and $y\in\mathbb{R}$ denote the real-valued label for regression.
Given a training dataset $\mathcal{D}=\\{(\mathbf{x}_{n},y_{n})\\}_{n=1}^{N}$
consisting of N i.i.d. samples, we model the probabilistic predictive
distribution $p_{\theta}(y|\mathbf{x})$ using a neural network with parameters
$\theta$.

We split the set of $d$ input features of $\mathbf{x}$ into $k$ exhaustive
clusters, $k\neq 1$, each containing $m_{i}$ number of features, s.t.
$\forall^{k}_{i=1}m_{i}<d$ and $\sum_{i=1}^{k}m_{i}=d$. Features are non-
overlapping i.e. a particular feature belongs to only one cluster. Let
$\mathbf{c}^{i}_{n}\in\mathbb{R}^{m_{i}}$ denote $i^{th}$ cluster containing
$m_{i}$ input features of $n^{th}$ data point $\mathbf{x}_{n}$. Thus,
$\\{(\mathbf{c}^{i}_{n},y_{n})\\}_{n=1}^{N}$ represents $i^{th}$ input feature
cluster and corresponding label for N samples. Label $y_{n}$ is the same
across any input cluster $c^{i}_{n}$ corresponding to the $n^{th}$ data point
(Equation 2). For clusters with one feature each,
$k=d\implies\forall^{k}_{i=1}m_{i}=1$.

(a)  Figure 2: A representative Deep Split NN model architecture

###  2.2 Defining ‘Deep Split Ensemble’ with multivariate Gaussian mixture

Jacobs et al. (1991); Xu, Jordan, and Hinton (1995) showed how local expert
networks can be explicitly trained on differing input features and used a
gating network to combine into a mixture of experts model. Hinton (1999)
introduced the product of experts model in which several individual
probabilistic models are combined together to model the data. Williams,
Agakov, and Felderhof (2002) further considered each expert as a Gaussian for
a richer structure. Recent works have shown improvements in performance of
ensembles by training them jointly with a unified loss as compared to post-hoc
ensembling of independent models (Lee et al. 2015; Furlanello et al. 2018;
Dutt, Pellerin, and Quénot 2020). Such mixture of experts have been widely
used to predict the target value only. We model each expert to predict an
uncertainty given the corresponding input features, while also contributing to
a single target value prediction (Figure 2), trained with a unified loss
(Section 2.3). This allows for disentangled predictive uncertainties per
cluster of input features, as opposed to a single unified predictive
uncertainty.

Our method forms clusters of input features (Section 2.3), and splits NNs
proportionately. Each split, referred to as a ‘deep split’, takes in a cluster
of input features $\mathbf{c}^{i}_{n}$ and models its output as the predicted
mean $\mu_{\theta}(\mathbf{c}^{i}_{n})$ and the standard deviation
$\sigma_{\theta}(\mathbf{c}^{i}_{n})$ parameterizing a Gaussian distribution.
However, only one common mean $\mu_{\theta}(\mathbf{c}_{n})$ is learnt across
the deep splits (Equation 1). This is because while we aim to disentangle the
predictive uncertainties, we still want the model to learn the regression
value using all the input features together. We combine these Gaussian output
distributions of deep splits to parametrize a multivariate Gaussian (MVN)
$p_{\theta}(\mathbf{y}_{n}|\mathbf{x}_{n})$;
$\mathbf{y}_{n}\thicksim\mathcal{N}\left(\bm{\mu}_{\theta}(\mathbf{x}_{n}),\Sigma_{\theta}(\mathbf{x}_{n})\right)$,
using the common mean and the covariance matrix
$\Sigma_{\theta}(\mathbf{x}_{n})$ represented as a diagonal matrix of the $k$
individual variances (Equations 2 and 3). We call this entire model ‘deep
split NN’ (Figure 2).

$\bm{\mu}_{\theta}(\mathbf{x}_{n})=[\mu_{\theta}(\mathbf{c}^{1}_{n}),\dots,\mu_{\theta}(\mathbf{c}^{k}_{n})]^{T}\text{;
}\\\ \forall_{i=1}^{k}\hskip
5.69054pt\mu_{\theta}(\mathbf{c}^{i}_{n})=\mu_{\theta}(\mathbf{c}_{n})$ (1)

$\Sigma_{\theta}(\mathbf{x}_{n})=\text{diag}(\sigma^{2}_{\theta}(\mathbf{c}^{1}_{n}),\dots,\sigma^{2}_{\theta}(\mathbf{c}^{k}_{n}))\\\
\text{}\mathbf{y}_{n}=[y_{n}^{1},\dots,y_{n}^{k}]^{T}\text{\hskip
5.69054pt}\text{;\hskip 5.69054pt}\forall_{i=1}^{k}\hskip
5.69054pty_{n}^{i}=y_{n}$ (2)

$p_{\theta}(\mathbf{y}_{n}|\mathbf{x}_{n})=\frac{1}{\sqrt{(2\pi)^{k}|\Sigma_{\theta}(\mathbf{x}_{n})|}}\times\\\
\text{
exp}\left(-\frac{1}{2}\left(\mathbf{y}_{n}-\bm{\mu}_{\theta}(\mathbf{x}_{n})\right)^{T}\Sigma_{\theta}(\mathbf{x}_{n})^{-1}(\mathbf{y}_{n}-\bm{\mu}_{\theta}(\mathbf{x}_{n}))\right)$
(3)

The weight update equations of a network modelling a Gaussian output
distribution, as derived by Nix and Weigend (1994), show that the learning
rate $\eta$ is affected by variations in $\sigma^{2}(\mathbf{x})$; $\eta$ is
effectively amplified for input patterns where $\sigma^{2}(\mathbf{x})$ is
smaller than average compared to patterns where $\sigma^{2}(\mathbf{x})$ is
larger than average. While this behavior is noted across the datapoints, it is
further applicable in our network across deep splits modelling different
$\sigma^{2}(\bm{c}^{i})$, assisting cluster-wise gradient backpropagation
through our model. This biases the allocation of the network’s resources
towards lower-noise regions, discounting regions of the input space where the
network is producing larger than average uncertainties, thus implementing a
form of robust regression.

Lakshminarayanan, Pritzel, and Blundell (2017); Snoek et al. (2019);
Dusenberry et al. (2020); Dietterich (2000); Pearce, Leibfried, and Brintrup
(2020) have shown improved performance with an ensemble of an NN initialized
with random parameters, compared to the NN performance. We train a parallel
ensemble of our deep split NNs while uniformly weighing the predictions across
the ensemble. This forms a mixture of uniformly-weighted multivariate
Gaussians.

We call the above method the ‘deep split ensemble’, and train it using a
simple procedure.

Datasets | RMSE | NLL
---|---|---
| RIO | Deep | Anchored | Deep Split | RIO | Deep | Anchored | Deep Split
| Ensembles | Ensembling | Ensembles | Ensembles | Ensembling | Ensembles111NLLs of Deep Split Ensembles in Table 1 are averaged over feature clusters of corresponding datasets. Refer to Appendix D.3 for an exhaustive list of cluster-wise predictive uncertainty estimates for all datasets.
Boston | – | 3.28 $\pm$ 1.00 | 3.09 $\pm$ 0.17 | 2.53 $\pm$ 0.15 | – | 2.41 $\pm$ 0.25 | 2.52 $\pm$ 0.05 | 2.23 $\pm$ 0.04
Concrete | 5.97 $\pm$ 0.48 | 6.03 $\pm$ 0.58 | 4.87 $\pm$ 0.11 | 4.40 $\pm$ 0.10 | 3.24 $\pm$ 0.10 | 3.06 $\pm$ 0.18 | 2.97 $\pm$ 0.02 | 2.85 $\pm$ 0.02
Energy | 0.70 $\pm$ 0.38 | 2.09 $\pm$ 0.29 | 0.35 $\pm$ 0.01 | 0.41 $\pm$ 0.02 | 1.03 $\pm$ 0.35 | 1.38 $\pm$ 0.22 | 0.96 $\pm$ 0.13 | 0.28 $\pm$ 0.11
Kin8nm | – | 0.09 $\pm$ 0.00 | 0.07 $\pm$ 0.00 | 0.19 $\pm$ 0.00 | – | -1.20 $\pm$ 0.02 | -1.09 $\pm$ 0.01 | -0.20 $\pm$ 0.02
Naval | – | 0.00 $\pm$ 0.00 | 0.00 $\pm$ 0.00 | 0.00 $\pm$ 0.00 | – | -5.63 $\pm$ 0.05 | -7.17 $\pm$ 0.03 | -5.28 $\pm$ 0.02
Power | 4.05 $\pm$ 0.12 | 4.11 $\pm$ 0.17 | 4.07 $\pm$ 0.04 | 4.04 $\pm$ 0.05 | 2.81 $\pm$ 0.03 | 2.79 $\pm$ 0.04 | 2.83 $\pm$ 0.01 | 2.78 $\pm$ 0.01
Protein | 4.08 $\pm$ 0.06 | 4.71 $\pm$ 0.06 | 4.36 $\pm$ 0.02 | 4.05 $\pm$ 0.03 | 2.82 $\pm$ 0.01 | 2.83 $\pm$ 0.02 | 2.89 $\pm$ 0.01 | 2.76 $\pm$ 0.00
Wine | 0.67 $\pm$ 0.03 | 0.64 $\pm$ 0.04 | 0.63 $\pm$ 0.01 | 0.60 $\pm$ 0.02 | 1.09 $\pm$ 0.10 | 0.94 $\pm$ 0.12 | 0.95 $\pm$ 0.01 | 0.89 $\pm$ 0.02
Yacht | 1.46 $\pm$ 0.49 | 1.58 $\pm$ 0.48 | 0.57 $\pm$ 0.05 | 0.86 $\pm$ 0.07 | 1.79 $\pm$ 0.88 | 1.18 $\pm$ 0.21 | 0.37 $\pm$ 0.08 | 0.90 $\pm$ 0.09
Table 1: Results on UCI regression benchmark datasets comparing RMSE and NLL

###  2.3 Training procedure

#### Feature clustering and NN splitting:

The input feature space is required to be split into $k$ exhaustive clusters.
For splitting, we use hierarchical clustering based on Pearson correlation
distance, since we want to estimate predictive uncertainties for clusters of
similar features. The dendograms thus obtained upon hierarchical clustering
with complete linkage are thresholded relative to the maximum distance to
obtain feature clusters (details in Appendix A.1). Note that we are clustering
features, which should not be confused with clustering datapoints. The NN is
then split to train using all feature clusters (Section 2.2 and Figure 2). We
also show splitting based on domain knowledge (Section 3.4) and across input
modalities (Section 3.5).

#### Training criterion:

The deep split ensemble is trained with the clusters of input features of all
training datapoints and their corresponding ground truth labels using a proper
scoring rule $l(\theta,\mathbf{x},\mathbf{y})$. We optimize for the negative
log-likelihood (NLL) of the joint distribution
$p_{\theta}(\mathbf{y}|\mathbf{x})$ according to Equation 4.

$-\log
p_{\theta}(\mathbf{y}_{n}|\mathbf{x}_{n})=\frac{1}{2}\big{(}\log\Sigma_{\theta}(\mathbf{x})\\\
+\left(\mathbf{y}-\bm{\mu}_{\theta}(\mathbf{x})\right)^{T}\Sigma_{\theta}(\mathbf{x})^{-1}\left(\mathbf{y}-\bm{\mu}_{\theta}(\mathbf{x})\right)\big{)}+\text{
constant}$ (4)

#### Parallel ensembling:

As discussed in Section 2.2, we train a parallel ensemble of our deep split
models initialized with random NN parameters, while uniformly weighing the
predictions across the ensemble. This forms a mixture of uniformly-weighted
multivariate Gaussians $p_{E}(\mathbf{y}_{n}|\mathbf{x}_{n})$ as shown in
Equation 5, where $E$ is the total number of models in the ensemble. The mean
$\bm{\mu}_{E}(\mathbf{x}_{n})$ and variance $\Sigma_{E}(\mathbf{x}_{n})$ of
such a mixture
$E^{-1}\sum_{e}\mathcal{N}\left(\bm{\mu}_{\theta_{e}(\mathbf{x}_{n})},\Sigma_{\theta_{e}(\mathbf{x}_{n})}\right)$
are shown in Equations 6 and 7 respectively (refer to Appendix B.1 for
derivation). For ease of computing quantiles and predictive probabilities, we
approximate this ensemble prediction as a Gaussian whose mean and variance are
that of the mixture. For a fair comparison with other uncertainty estimation
methods that use ensembles of 5 models (Lakshminarayanan, Pritzel, and
Blundell 2017; Pearce, Leibfried, and Brintrup 2020), we use a value of $E=5$
for our experiments (Section 3). Refer to Appendix D.2 for results with $E=1$
and $E=10$

$p_{E}(\mathbf{y}_{n}|\mathbf{x}_{n})=E^{-1}\sum_{e=1}^{E}{p_{\theta}}_{e}(\mathbf{y}_{n}|\mathbf{x}_{n})$
(5)

$\bm{\mu}_{E}(\mathbf{x}_{n})=E^{-1}\sum_{e=1}^{E}{\bm{\mu}_{\theta}}_{e}(\mathbf{x}_{n})$
(6)

$\Sigma_{E}(\mathbf{x}_{n})=\text{diag}(\sigma^{2}_{E}(\mathbf{c}^{1}_{n}),\dots,\sigma^{2}_{E}(\mathbf{c}^{k}_{n}))\text{\hskip
8.53581pt}\\\ \forall_{i=1}^{k}\hskip
8.53581pt\sigma^{2}_{E}(\mathbf{c}^{i}_{n})=E^{-1}\sum_{e=1}^{E}\big{(}{\sigma_{\theta}}_{e}^{2}(\mathbf{c}^{i}_{n})+{\mu_{\theta}}_{e}^{2}(\mathbf{c}^{i}_{n})\big{)}-\mu_{E}^{2}(\mathbf{c}^{i}_{n})$
(7)

##  3 Experiments and results

###  3.1 Experimental setup and evaluation metrics

For a fair comparison with current state-of-the-art methods for predictive
uncertainty estimation using NNs on benchmark regression tasks, we use the
same experimental setup. The network consists of 50 hidden units with ReLU for
smaller datasets split into 20 train-test folds and 100 units with ReLU for
the larger ‘Protein’ dataset (5 folds). See Appendix C.1 for other training
hyperparameters. We measure the NLL, a proper scoring rule and widely used
metric for evaluating predictive uncertainty (Quinonero-Candela et al. 2005).
We also compute the root mean squared error (RMSE) to measure the performance
of the single mean prediction of our model.

###  3.2 Regression datasets: Toy regression and UCI regression benchmarks

To highlight the need for disentangled uncertainties, we first evaluate the
performance of our method on an extension of the toy regression dataset setup
and used to evaluate probabilistic backpropagation (PBP) (Hernández-Lobato and
Adams 2015), deep ensembles (Lakshminarayanan, Pritzel, and Blundell 2017),
and anchored ensembling (Pearce, Leibfried, and Brintrup 2020) which consists
of 20 examples drawn from $y=(x^{3}+\epsilon)$ where
$\epsilon\thicksim\mathcal{N}(0,3^{2})$. The extended multi-dimensional toy
regression datasets, contain 40 examples each drawn from
$y=(x_{1}^{3}+\epsilon_{1})(x_{2}^{3}+\epsilon_{2})$ and
$y=(x_{1}^{4}+\epsilon_{1})(x_{2}^{4}+\epsilon_{2})$ where
$\epsilon_{1},\epsilon_{2}\thicksim\mathcal{N}(0,3^{2})$. It can be observed
that in the case of the unified uncertainty estimates, the underlying
heteroscedasticity along each input feature can not be captured individually.
However, using the same model architecture, our method can produce ‘different’
uncertainties for each input feature (Figure 1).

We then evaluate and compare our approach to current state-of-the-art methods
for predictive uncertainty estimation using NNs on UCI regression benchmark
datasets (see Appendix C.1 for details on datasets). Table 1 shows the
comparison with the latest and competitive methods - RIO (Qiu, Meyerson, and
Miikkulainen 2020), deep ensembles (Lakshminarayanan, Pritzel, and Blundell
2017) and anchored ensembling (Pearce, Leibfried, and Brintrup 2020); see
Appendix D.1 for a full comparison with other methods - VI (Graves 2011), PBP
(Hernández-Lobato and Adams 2015), MC-dropout (Gal and Ghahramani 2016),
deterministic VI (DVI) (Wu et al. 2018) and subspace inference (SI) (Izmailov
et al. 2020). We observe that our method outperforms the existing methods on
multiple datasets, while also disentangling the predictive uncertainties.

We also highlight the performance of our proposed MVN model trained with a
unified loss, as compared to post-hoc ensembling (vanilla mixture of experts)
of independent cluster-wise NN models. This serves as the baseline for a
comparison of disentangled uncertainty estimates through the corresponding
RMSE and cluster-wise NLL. For each dataset, we train deep ensemble per input
cluster (DEPC) and anchored ensembling per input cluster (AEPC) following
their respective training procedure. This results in a prediction and an
uncertainty estimate per cluster. We then average the predictions across the
clusters for the final prediction. Hence, we have an NLL per cluster and a
single RMSE for the final prediction. Table 3 shows that deep split ensembles
outperform DEPC and AEPC on NLL of every cluster as well as RMSE. We attribute
this to the joint training of cluster-wise experts in deep split ensembles, as
compared to the independent training of cluster-wise experts in DEPC and AEPC.

Datasets | RMSE | Clusters | Cluster-wise NLL
---|---|---|---
| DEPC | AEPC | Deep Split Ens. |  | DEPC | AEPC | Deep Split Ens.
Boston | 5.11 $\pm$ 1.06 | 4.93 $\pm$ 1.03 | 2.53 $\pm$ 0.15 | 1 | 2.91 $\pm$ 0.16 | 3.87 $\pm$ 0.82 | 2.23 $\pm$ 0.04
2 | 2.82 $\pm$ 0.16 | 3.99 $\pm$ 0.94 | 2.20 $\pm$ 0.03
3 | 3.29 $\pm$ 0.10 | 4.23 $\pm$ 1.06 | 2.26 $\pm$ 0.05
Concrete | 10.24 $\pm$ 0.85 | 10.40 $\pm$ 0.93 | 4.40 $\pm$ 0.10 | 1 | 3.77 $\pm$ 0.05 | 5.75 $\pm$ 0.61 | 2.84 $\pm$ 0.02
2 | 3.79 $\pm$ 0.09 | 5.68 $\pm$ 0.60 | 2.85 $\pm$ 0.02
3 | 3.80 $\pm$ 0.05 | 5.83 $\pm$ 0.61 | 2.87 $\pm$ 0.01
Table 2: Results of deep ensemble per input cluster (DEPC), anchored
ensembling per input cluster (AEPC), and deep split ensembles on UCI
regression benchmark datasets333Refer to Appendix D.3 for results on all other
datasets.comparing RMSE and cluster-wise NLL.

(a)  (b)

(c)

(d)

(e)  (f)

(g)

(h)

(i)

(j)

Figure 3: Entropy plots for ‘Boston’ and ‘Concrete’ datasets using
hierarchical clustering (Section 2.3). The first two columns show the kernel
density estimation (KDE) of entropy for in- distribution i.e.
$\mathcal{N}(0,1)$ and out-of-distribution samples, obtained with unified
uncertainty estimation using deep ensemble and anchored ensembling
respetively. The last two columns show ‘cluster-wise’ KDE of entropy for in-
distribution and out-of-distribution samples, obtained with disentangled
uncertainty estimation using deep split ensembles. OOD 1 and OOD 2 refer to
introducing dataset shift by inducing noise sampled from
$\mathcal{N}(6,2^{2})$ into 2 random input features; the features correspond
to different clusters for deep split ensembles. See Appendix E.1 for similar
analyses for all other datasets.

###  3.3 Calibration and uncertainty evaluation

As a consequence of modelling disentangled predictive uncertainties during the
training of NNs using NLL, we observe that our approach produces cluster-wise
inherently well-calibrated models. Moreover, given that our model estimates
disentangled uncertainties, we are able to assess the calibration of our model
in a granular cluster-wise fashion. We assess our models without any post-hoc
calibration. We first demonstrate this using entropy plots with out-of-
distribution samples, and then using cluster-wise calibration curves using
empirical rule.

#### Entropy analyses with out-of-distribution samples:

In real-world settings, there are often dataset shifts where the observed
target data distribution may shift and eventually be very different once a
model is deployed. Subsequently, the predictions need to exhibit higher
uncertainty when this occurs. To assess it, we intentionally introduce a
dataset shift by inducing noise, sampled from Gaussian distributions with
shifted means and variances, into a random feature of a cluster and measure
the corresponding clusters’ predicted entropy (Figure 3). We observe an
increase in the entropy of only the noisy cluster while entropies of other
clusters remain intact with deep split ensembles. However, unified uncertainty
estimation methods like deep ensembles and anchored ensembling show an
increase in the entropy corresponding to all features together. The
disentanglement in OOD behaviour is an inherent characteristic of our method
that cannot be observed in existing methods estimating unified uncertainties.
Consequently, deep split ensembles can help better trace dataset shifts and
pinpoint the noisy feature clusters during test time.

(a)  (b)

(c)

(d)  (e)

(f)

(g)

(h)

Figure 4: ‘Cluster-wise’ calibration curves using empirical rule for ‘Boston’ and ‘Concrete’ datasets using hierarchical clustering (Section 2.3). The columns contain experiments using deep ensemble per input cluster (DEPC), anchored ensembling per input cluster (AEPC) and deep split ensembles respectively. See Appendix E.2 for similar analyses on all other datasets Datasets | RMSE | Clusters ($C$) | Cluster-wise NLL
---|---|---|---
| DEPC | AEPC | Deep Split Ens. |  | DEPC | AEPC | Deep Split Ens.
Power | 4.90 $\pm$ 0.23 | 4.91 $\pm$ 0.22 | 4.07 $\pm$ 0.04 | $C_{p1}$ | 2.99 $\pm$ 0.04 | 3.08 $\pm$ 0.07 | 2.81 $\pm$ 0.05
$C_{p2}$ | 3.13 $\pm$ 0.03 | 3.08 $\pm$ 0.07 | 2.83 $\pm$ 0.05
Wine | 0.64 $\pm$ 0.04 | 0.66 $\pm$ 0.05 | 0.59 $\pm$ 0.02 | $C_{w1}$ | 0.94 $\pm$ 0.06 | 1.02 $\pm$ 0.09 | 0.88 $\pm$ 0.02
$C_{w3}$ | 0.96 $\pm$ 0.06 | 1.03 $\pm$ 0.09 | 0.89 $\pm$ 0.03
$C_{w3}$ | 0.94 $\pm$ 0.07 | 1.03 $\pm$ 0.09 | 0.90 $\pm$ 0.04
Table 3: Results with clusters from human experts for the two datasets below.
See Appendix A.2 for list of clusters ($C$).

#### ‘Cluster-wise’ calibration curves using empirical rule:

It is crucial to have good and stable calibration for reliable uncertainty
estimates. To highlight our inherently well-calibrated models, we further
evaluate to obtain calibration curves using the 68–95–99.7 rule (also called
empirical rule). We first compute the $x\%$ prediction interval for each test
datapoint based on Gaussian quantiles using the predicted mean and variance.
We then calculate the fraction of test observations (true values) that fall
within this prediction interval. For a well-calibrated model, the observed
fraction should be close to $x\%$ calculated earlier. To see how our models
perform in this setting, we sweep from $x=10\%$ to $x=90\%$ in steps of 10,
and consequently a line lying very close to the line $y=x$ would indicate a
well-calibrated model. Here, we further define stability of ‘cluster-wise’
calibration as having similar calibration curves across clusters. As this
experiment aims to test the calibration of the model with respect to each of
the clusters individually, we use DEPC and AEPC to produce more suitable
baselines to compare our method more rigorously. Figure 4 shows the
calibration curves for each feature cluster for the different methods. We
notice that deep split ensembles have a more uniform and stable calibration
across clusters.

###  3.4 Deep split ensembles based on domain knowledge and/or user needs

We illustrate how deep split ensembles allow for modelling predictive
uncertainties using domain knowledge and/or user needs by taking in such
clusters of input feature space. This is important as it brings the human in
the loop and helps better define the task qualitatively. We consulted human
experts, for the ‘Power’ and ‘Wine’ datasets, to qualitatively cluster the
input features based on the uncertainties they would desire from a machine
learning system trained on those datasets (details in Appendix A.2). We then
trained a deep split ensemble, DEPC and AEPC using the same experimental setup
as above. Table 3 shows that the results of deep split ensembles here are
comparable to results in Table 1, while outperforming DEPC and AEPC. The
consistent improved performance upon changing the constituents of the clusters
of input features demonstrates the inherent flexibility available while
training deep split ensembles.

(a)

(b)

(c)

(d)

Figure 5: Entropy plots for Alzheimer’s dataset showing the kernel density
estimation (KDE) for male and female subjects specifically. The first two
plots indicate unified uncertainty estimation using deep ensembles while the
last two plots show disentangled uncertainty estimation using deep split
ensembles. Both the methods are trained on female subjects and evaluated on
male subjects to highlight hidden ‘modality-specific’ gender bias.

###  3.5 Deep split ensembles in multi-modal settings

The split nature of deep split ensembles makes them suitably applicable in
multi-modal settings as heteroscedasticity in data can be highly decoupled due
to the individual nature of the modalities. Each cluster of input features can
be used to represent a particular modality of the input feature space to
obtain a predictive uncertainty per modality.

To demonstrate this in a safety-critical application, we use a multi-modal
Alzheimer’s dementia (AD) dataset, ‘ADReSS’, consisting of speech samples
(audio) and their transcriptions (text), to regress MMSE444Mini-Mental State
Examination (MMSE) scores, ranging from 0 to 30, offers a way to quantify
cognitive function and screen for cognitive loss by testing the individuals’
attention, recall, language and motor skills (Tombaugh and McIntyre 1992).
scores. The standardized dataset contains 108 train and 48 heldout-test
subjects. The train set is further split into 80%-20% train-validation sets.
We first devise a feature engineering pipeline that extracts several multi-
modal cognitive and acoustic feature sets - interventions, disfluency, and
acoustic - based on domain knowledge and context. We then train a deep split
NN with those feature sets. The deep splits corresponding to disfluency and
acoustic feature sets are fully connected and that corresponding to
interventions feature set is an LSTM. See Appendix C.2 for details about the
dataset, feature sets, model architecture and hyperparameters. Our method
shows improved performance over state-of-the-art RMSE results, while also
estimating a predictive uncertainty for each modality (Table 4). We attribute
this observation to the variance-affected learning rate (Nix and Weigend 1994)
which would help stabilize multi-modal training. We further train a deep
ensemble for comparison on predictive uncertainty estimates, and observe that
deep split ensemble achieves a better RMSE as well as a better NLL in each of
the modalities.

To assess the calibration of our model and highlight potential hidden bias in
real-world settings, we then train on only female subjects and evaluate it on
only male subjects from the dataset. This would induce a bias in the model,
which can be illustrated with the predicted uncertainties, as two modes
(disfluency and interventions) are gender neutral by nature, whereas acoustic
features can significantly vary. Consequently, upon experimentation (Figure
5), we observe that the entropy corresponding to all modalities together
increases on the male inputs in case of unified uncertainty estimation using
deep ensemble. However, in case of disentangled uncertainty estimation using
deep split ensemble, only the entropy of the acoustic feature set on the male
inputs significantly increases. The high predicted uncertainty corresponding
to acoustic features for only male subjects highlights the hidden ‘modality-
specific’ gender bias.

Model | RMSE | NLL
---|---|---
Pappagari et al. (2020) | 5.37 | –
Luz et al. (2020) | 5.20 | –
Sarawgi et al. (2020) | 4.60 | –
Balagopalan et al. (2020) | 4.56 | –
Rohanian, Hough, and Purver | 4.54 | –
Deep Ensemble | 4.90 | 3.08
Deep Split Ensemble | 4.37 | 2.94, 2.98, 2.94
Table 4: Test results on multi-modal ADReSS dataset for MMSE score regression.

##  4 Discussion and future work

We have proposed a conceptually simple yet effective non-Bayesian method, deep
split ensembles, to estimate disentangled predictive uncertainties using NNs
for input feature clusters. Disentangling a unified uncertainty allows for
granular information about plausible sources of heteroscedasticity in the
data. This is important in safety-critical settings as it enables improved
risk assessment and decision-making. One can further form clusters containing
one feature each to estimate feature-wise uncertainties. Using thus produced
entropy values, noisy features or clusters can be suppressed while training a
more reliable model for the same dataset with potentially improved
performance. This encourages interoperability between humans and models in a
unique way. Our method also reduces computational costs through sparser
clusterwise connections, requires few changes in the NN, and can be readily
implemented and trained. The nature of the split NN structure facilitates
intuitive model-parallelism training for large models in multi-GPU systems
where each cluster can be placed in separate GPUs. Using domain knowledge from
human experts, deep split ensembles can help satisfy user needs by generating
different combinations of uncertainty estimates desired from a machine
learning system, thus providing a more controllable form of reliability and
awareness with the model. The potential to highlight hidden biases, such as
shown in multi-modal settings, has immediate and apparent real world
applications to mitigate unseen biases in deployed models. This serves as a
motivation for fair and aware systems supporting human-assisted AI.

A direct extension of our work would be to use complex distributions such as
mixture density networks (Bishop 1994) for modelling output distributions.
There are many exciting future directions, such as unsupervised learning to
form deeper representation for the clusters of features (Xie, Girshick, and
Farhadi 2016), using uncertainty attention (Heo et al. 2018; Lee et al. 2018)
to aid in training of the ensemble classifiers, exploring adaptive defer
systems (Madras, Pitassi, and Zemel 2018) along with partial deferring based
on clusters for better calibration of uncertainties coupled with human
experts, and considering uncertainty of human feedback (He, Chen, and An
2020).

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## Broader and ethical impact

Uncertainty estimations are crucial when employing black box neural networks
(NNs) in sensitive and critical applications such as healthcare and self-
driving cars. These NNs often tend to be over confident of their predictions;
the confidence measures are not a true estimate of the model’s performance.
Predicting the uncertainties can help in better understanding how confident
the model is in its predictions and reflect upon the noise introduced by the
stochastic data generation processes. It is important to know what the model
is unsure about. Although current NNs can demonstrate high performance on test
datasets, they sometimes tend to fail when deployed in real world settings due
to noisy real-world data and dataset shifts.

A unified uncertainty estimation helps in providing confidence estimates.
Further disentangling the unified predictive uncertainties give deeper
insights into the various feature clusters and their associated
heteroscedasticity. We believe that anyone who is involved in synergy with a
machine learning system in decision making will benefit from such systems. For
example, when such a system is deployed in a hospital setting to stratify the
risk of a disease/condition, the doctor can understand the uncertainties
associated with each input feature modality, and is able to better interpret
the model’s belief. In such high-risk and safety-critical settings, deploying
a black box NN could be sub-optimal. Additionally, multidisciplinary machine
learning researchers will benefit from this as they will have a tool to better
incorporate domain knowledge and user demands/needs.

Appendix

##  Appendix A Feature Clusters

###  A.1 Feature clusters from hierarchical clustering

As discussed in Section 2.3, the input feature space is split into $k$
exhaustive clusters using hierarchical clustering based on Pearson correlation
distance. The dendograms thus obtained upon hierarchical clustering with
complete linkage are thresholded relative to the maximum distance to obtain
feature clusters (Figure 6); we chose 0.5 and 0.75 to span a variety of number
of clusters and features per cluster. One can change this threshold value to
obtain different sets of feature clusters. Table 5 enlists the clusters thus
obtained for each dataset.

###  A.2 Feature clusters from human experts

As discussed in Section 3.4, we consulted human experts, for the ‘Power Plant
Output’ and ‘Red Wine Quality’ datasets, to qualitatively cluster the input
features based on the uncertainties desired from a machine learning system
trained on those datasets. Table 6 shows the features cluster thus obtained,
and the reasons as mentioned by the human experts are summarized below:

  * •

‘Power Plant Output’: While the Vacuum is collected from and has effect on the
Steam Turbine, the three other ambient variables effect the GT performance.

  * •

‘Red Wine Quality’: Alcohol, pH, fixed acidity, density, and residual sugar
are resultant characteristics of the wine. Volatile acidity and citric acid
are added acidity in the wine-making. Chlorides, free sulphur dioxide, total
sulphur dioxide, and sulphates are preservatives and antibacterials.

(a) Boston Housing

(b) Concrete

(c) Energy Efficiency

(d) Kin8nm

(e) Naval propulsion plant

(f) Power Plant Output

(g) Protein Structure

(h) Red Wine Quality

(i) Yacht Hydrodynamics

Figure 6: Dendograms obtained for UCI datasets upon hierarchical clustering of features based on Pearson correlation distance. The dotted line shows the threshold value used to extract clusters. Dataset | Cluster | Features
---|---|---
Boston Housing | 1 | CRIM, INDUS, NOX, AGE, RAD, TAX, PTRATIO, LSTAT
2 | ZN, RM, DIS, B
3 | CHAS
Concrete | 1 | Fly Ash, Superplasticizer, Fine Aggregate
2 | Water, Age, Blast Furnace Slag
3 | Cement, Coarse Aggregate
Energy Efficiency | 1 | Surface Area, Roof Area, Glazing Area, Glazing Area Distribution
2 | Relative Compactness, Overall Height, Wall Area, Orientation
Kin8nm | 1 | theta1
2 | theta2
3 | theta3
4 | theta4
5 | theta5
6 | theta6
7 | theta7
8 | theta8
Naval Propulsion Plant | 1 | lp, v, ggn
2 | gtt, gtn, ts, tp, t48, t2, p48, p2, pexh, tic, mf
Power Plant Output | 1 | AT, V
2 | AP, RH
Protein Structure | 1 | F1, F2, F4, F5, F6, F7, F8
2 | F3
3 | F9
Red Wine Quality | 1 | chlorides, sulphates
2 | fixed acidity, citric acid, density
3 | volatile acidity, pH
4 | residual sugar, free sulfur dioxide, total sulfur dioxide
5 | alcohol
Yacht Hydrodynamics | 1 | Prismatic coefficient, Beam-draught ratio
2 | Length-displacement ratio, Length-beam ratio, Longitudinal position, Froude number
Table 5: List of feature clusters obtained for UCI datasets using hierarchical clustering. Dataset | Cluster | Features
---|---|---
Power Plant Output | $C_{p1}$ | AT, AP, RH
$C_{p2}$ | V
Red Wine Quality | $C_{w1}$ | alcohol, pH, fixed acidity, density, residual sugar
$C_{w2}$ | volatile acidity, citric acid
$C_{w3}$ | chlorides, free sulphur dioxide, total sulphur dioxide, sulphates
Table 6: List of feature clusters obtained for ‘Power’ and ‘Wine’ UCI datasets
from human experts.

##  Appendix B Derivations and proofs

###  B.1 Derivations of $\mu$ and $\sigma$ of a Gaussian mixture

Given a Gaussian mixture $p_{E}(\mathbf{y}|\mathbf{x})$, where
$p_{E}(\mathbf{y}|\mathbf{x})=E^{-1}\sum_{e}\mathcal{N}\left({\bm{\mu}_{\theta}}_{e}(\mathbf{x}),{\Sigma_{\theta}}_{e}(\mathbf{x})\right)$,
let the mean and the variance of the mixture be $\bm{\mu}_{E}(\mathbf{x})$ and
$\Sigma_{E}(\mathbf{x})$ respectively. Let $t\sim
E^{-1}\sum_{e}\mathcal{N}\left({\bm{\mu}_{\theta}}_{e}(\mathbf{x}),{\Sigma_{\theta}}_{e}(\mathbf{x})\right)$.

### Derivation of the mean of a Gaussian mixture

$t=z+\epsilon$, where $z={\bm{\mu}_{\theta}}_{e}(\mathbf{x})$ with equal
probability $E^{-1}$ for $e=1,2,\dots,E$ and the conditional probability
distribution of $\epsilon$ given $z$ will be
$N(0,{\Sigma_{\theta}}_{e}(\mathbf{x}))$.

$\bm{\mu}_{E}(\mathbf{x})=E[t]$

$\implies\bm{\mu}_{E}(\mathbf{x})=E[E[t|z]]$

$\implies\bm{\mu}_{E}(\mathbf{x})=E\left.\begin{cases}\vdots\\\
{\bm{\mu}_{\theta}}_{e}(\mathbf{x})&\text{with probability }E^{-1}\\\
\vdots\end{cases}\right\\}$

$\implies\bm{\mu}_{E}(\mathbf{x})=E^{-1}\sum_{e=1}^{E}{\bm{\mu}_{\theta}}_{e}(\mathbf{x})$
(8)

### Derivation of the variance of a Gaussian mixture

We have,

$\text{var}(t)=E[\text{var}(t|z)]+\text{var}(E[t|z])$

$\implies\text{var}(t)=E\left.\begin{cases}\vdots\\\
{\Sigma_{\theta}}_{e}(\mathbf{x})&\text{with probability }E^{-1}\\\
\vdots\end{cases}\right\\}\\\ +\text{var}\left.\begin{cases}\vdots\\\
{\bm{\mu}_{\theta}}_{e}(\mathbf{x})&\text{with probability }E^{-1}\\\
\vdots\end{cases}\right\\}$

$\implies\text{var}(t)=E^{-1}\sum_{e=1}^{E}{\Sigma_{\theta}}_{e}(\mathbf{x})\\\
+E^{-1}\sum_{e=1}^{E}({\bm{\mu}_{\theta}}_{e}(\mathbf{x})-\bm{\mu}_{E}(\mathbf{x}))({\bm{\mu}_{\theta}}_{e}(\mathbf{x})-\bm{\mu}_{E}(\mathbf{x}))^{T}$

Using Equation (8) and given our assumption that the outputs of deep splits
(in a deep split NN) are linearly uncorrelated, we have,

$\Sigma_{E}(\mathbf{x})=\text{diag}(\sigma^{2}_{E}(\mathbf{c}^{1}),\dots,\sigma^{2}_{E}(\mathbf{c}^{k}))$

$\sigma_{E}^{2}(\mathbf{c}^{i})=E^{-1}\sum_{e=1}^{E}({\sigma_{\theta}}_{e}^{2}(\mathbf{c}^{i})+{\bm{\mu}_{\theta}}_{e}^{2}(\mathbf{c}^{i}))-\bm{\mu}_{E}^{2}(\mathbf{c}^{i})$

##  Appendix C Details of datasets, model and hyperparameters

Dataset | No. of datapoints | No. of features
---|---|---
Boston Housing | 506 | 13
Concrete | 1,030 | 8
Energy Efficiency | 768 | 8
Kin8nm | 8,192 | 8
Naval propulsion plant | 11,934 | 16
Power Plant Output | 9,568 | 4
Protein Structure | 45,730 | 9
Red Wine Quality | 1,599 | 11
Yacht Hydrodynamics | 308 | 6
Table 7: UCI Benchmark regression dataset details

###  C.1 Benchmark regression datasets

Table 7 shows some statistics of the 9 benchmark regression datasets used in
our experiments (Section 3). We have included all the datasets in the
Supplementary Material provided. The hyperparameters used for training the
deep split ensembles are enlisted in Table 8. For anchored ensembling per
feature cluster (AEPC) and deep ensembles per feature cluster (DEPC), we use
the hyperparameters mentioned in the anchored ensembling and deep ensembles
papers respectively.

Dataset | Learning | Epochs | Batch
---|---|---|---
Rate | Size
Boston Housing | 0.1 | 1000 | 100
Concrete | 0.01 | 1500 | 32
Energy Efficiency | 0.01 | 1500 | 16
Kin8nm | 0.1 | 1000 | 100
Naval Propulsion Plant | 0.01 | 1500 | 32
Power Plant Output | 0.01 | 2500 | 256
Protein Structure | 0.01 | 4000 | 1024
Red Wine Quality | 0.1 | 1000 | 100
Yacht Hydrodynamics | 0.01 | 1500 | 8
Table 8: Hyperparameters used to train deep split ensembles

###  C.2 Alzheimer’s dementia (AD) - dataset, model and hyperparameters

### Dataset

The ADReSS (Alzheimer’s Dementia Recognition through Spontaneous Speech)
dataset, available through the benchmark DementiaBank database upon access
request, is a standardized and balanced dataset of 156 speech samples, each
from a unique subject, matched for age and gender. The dataset consists of
speech recordings and transcripts of spoken picture descriptions elicited from
participants through the Cookie Theft picture from the Boston Diagnostic
Aphasia Exam. The dataset also provides corresponding Mini-Mental Status
Examination (MMSE) scores, ranging from 0 to 30, of the subjects, which offers
a way to quantify cognitive function and screen for cognitive loss by testing
the individuals’ orientation, attention, calculation, recall, language and
motor skills. These scores are used as labels for the regression task. A
standardized train-test split of around 70%-30% (108 and 48 subjects) is
provided by this dataset. The test set is held-out until final evaluation, and
the train set is split into train and validation sets for training models.
This dataset was used for evaluating deep split ensembles in multi-modal
settings (Section 3.5).

### Multi-modal feature engineering pipeline

People with dementia show symptoms of cognitive decline, impairment in memory,
communication and thinking. To include such domain knowledge and context, we
devised an automated feature engineering pipeline that extracts several multi-
modal cognitive and acoustic feature sets - interventions, disfluency, and
acoustic. These three feature sets are then fed to a deep split NN, model
architecture of which is shown in Figure 7. Similar extracted features have
been repeatedly used to propose speech recognition based solutions for
automated detection of mild cognitive impairment from spontaneous speech. The
three extracted feature sets are as follows:

  * •

Interventions features: Cognitive features reflect upon potential loss of
train of thoughts and context. Our system extracts the sequence of speakers
from the transcripts, categorizing it as the subject or the interviewer. To
accommodate for the variable length of these sequences, they are padded or
truncated to length of 32 steps, found upon analyses and tuning of sequence
lengths. These (subject, interviewer, or padding) are then one-hot encoded
resulting in 32x3 input size for this feature set corresponding to every
datapoint.

  * •

Disfluency features: A set of 11 distinct and carefully curated features from
the transcripts; word rate, intervention rate, and 9 different kinds of pause
rates, reflecting upon speech impediments like slurring and stuttering. These
are normalized by the respective audio lengths and scaled thereafter.

  * •

Acoustic features: The ComParE 2013 feature
set555https://dl.acm.org/doi/abs/10.1145/2502081.2502224,
https://dl.acm.org/doi/abs/10.1145/2502081.2502224 was extracted from the
audio samples using the open-sourced openSMILE v2.1 toolkit, widely used for
affect analyses in speech. This provides a total of 6,373 features that
include energy, MFCC, and voicing related low-level descriptors (LLDs), and
other statistical functionals. This feature set encodes changes in speech of a
person and has been used as an important noninvasive marker for AD detection.
Our system standardizes this set of features using z-score normalization, and
uses principal component analysis (PCA) to project the 6,373 features onto a
low-dimensional space of 21 orthogonal features with highest variance. The
number of orthogonal features was selected by analyzing the percentage of
variance explained by each of the components.

### Model architecture

The three feature sets are then fed to a deep split NN, the model architecture
of which is shown in Figure 7.

Figure 7: Model architecture of deep split NN for the ADReSS dataset.

### Setup and hyperparameters

Table 9 shows the setup and hyperparameters for the ADReSS dataset. Best model
was saved upon monitoring negative log-likelihood of the validation set (val
NLL).

Train-val split | 80%-20% (86-22 datapoints)
---|---
Heldout-test set | 48 datapoints
Optimizer | Adam
Learning rate | 0.001
Batch size | 8
Table 9: Setup and hyperparameters for the ADReSS dataset

##  Appendix D Results on benchmark regression datasets

###  D.1 Comparison with state-of-the-art methods

Tables 8 and 11 compare the RMSE and NLL666NLLs of Deep Split Ensembles in
Tables 11 and 12 are averaged over feature clusters of corresponding datasets.
See Appendix D.3 for an exhaustive list of cluster-wise predictive uncertainty
estimates for all datasets. of our method on benchmark regression datasets
with other state-of-the-art methods - particularly VI (Graves 2011), PBP
(Hernández-Lobato and Adams 2015), MC-dropout (Gal and Ghahramani 2016),
deterministic VI (DVI) (Wu et al. 2018) and subspace inference (SI) (Izmailov
et al. 2020).

Datasets | VI | PBP | MC | SI | RIO | Deep | Anchored | Deep Split
---|---|---|---|---|---|---|---|---
Dropout | Ensembles | Ensembling | Ensembles
Boston | 4.32 $\pm$ 0.29 | 3.01 $\pm$ 0.18 | 2.97 $\pm$ 0.85 | 3.45$\pm$0.95 | – | 3.28 $\pm$ 1.00 | 3.09 $\pm$ 0.17 | 2.53 $\pm$ 0.15
Concrete | 7.13 $\pm$ 0.12 | 5.67 $\pm$ 0.09 | 5.23 $\pm$ 0.53 | 5.19 $\pm$ 0.44 | 5.97 $\pm$ 0.48 | 6.03 $\pm$ 0.58 | 4.87 $\pm$ 0.11 | 4.40 $\pm$ 0.10
Energy | 2.65 $\pm$ 0.08 | 1.80 $\pm$ 0.05 | 1.66 $\pm$ 0.19 | 1.59 $\pm$ 0.27 | 0.70 $\pm$ 0.38 | 2.09 $\pm$ 0.29 | 0.35 $\pm$ 0.01 | 0.41 $\pm$ 0.02
Kin8nm | 0.10 $\pm$ 0.00 | 0.10 $\pm$ 0.00 | 0.10 $\pm$ 0.00 | – | – | 0.09 $\pm$ 0.00 | 0.07 $\pm$ 0.00 | 0.19 $\pm$ 0.00
Naval | 0.01 $\pm$ 0.00 | 0.01 $\pm$ 0.00 | 0.01 $\pm$ 0.00 | 0.00 $\pm$ 0.00 | – | 0.00 $\pm$ 0.00 | 0.00 $\pm$ 0.00 | 0.00 $\pm$ 0.00
Power | 4.32 $\pm$ 0.04 | 4.12 $\pm$ 0.03 | 4.02 $\pm$ 0.18 | – | 4.05 $\pm$ 0.12 | 4.11 $\pm$ 0.17 | 4.07 $\pm$ 0.04 | 4.04 $\pm$ 0.05
Protein | 4.84 $\pm$ 0.03 | 4.73 $\pm$ 0.01 | 4.36 $\pm$ 0.04 | – | 4.08 $\pm$ 0.06 | 4.71 $\pm$ 0.06 | 4.36 $\pm$ 0.02 | 4.05 $\pm$ 0.03
Wine | 0.65 $\pm$ 0.01 | 0.64 $\pm$ 0.01 | 0.62 $\pm$ 0.04 | – | 0.67 $\pm$ 0.03 | 0.64 $\pm$ 0.04 | 0.63 $\pm$ 0.01 | 0.60 $\pm$ 0.02
Yacht | 6.89 $\pm$ 0.67 | 1.02 $\pm$ 0.05 | 1.11 $\pm$ 0.38 | 0.97 $\pm$ 0.37 | 1.46 $\pm$ 0.49 | 1.58 $\pm$ 0.48 | 0.57 $\pm$ 0.05 | 0.86 $\pm$ 0.07
Table 10: Results on UCI regression datasets comparing RMSE888RMSE values for dVI are not shown as they were not reported in the paper. Datasets | VI | PBP | MC | dVI | SI | RIO | Deep | Anchored | Deep Split
---|---|---|---|---|---|---|---|---|---
Dropout | Ensembles | Ensembling | Ensembles
Boston | 2.90$\pm$ 0.07 | 2.57$\pm$0.09 | 2.46$\pm$0.25 | 2.41$\pm$0.02 | 2.71$\pm$0.13 | – | 2.41$\pm$0.25 | 2.52$\pm$0.05 | 2.23$\pm$0.04
Concrete | 3.39$\pm$0.02 | 3.16$\pm$0.02 | 3.04$\pm$0.09 | 3.06$\pm$0.01 | 3.00$\pm$0.08 | 3.24$\pm$0.10 | 3.06$\pm$0.18 | 2.97$\pm$0.02 | 2.85$\pm$0.02
Energy | 2.39$\pm$0.03 | 2.04$\pm$0.02 | 1.99$\pm$0.09 | 1.01$\pm$0.06 | 1.56$\pm$1.24 | 1.03$\pm$0.35 | 1.38$\pm$0.22 | 0.96$\pm$0.13 | 0.28$\pm$0.11
Kin8nm | -0.90$\pm$0.01 | -0.90$\pm$0.01 | -0.95$\pm$0.03 | -1.13$\pm$0.00 | – | – | -1.20$\pm$0.02 | -1.09$\pm$0.01 | -0.20$\pm$0.02
Naval | -3.73$\pm$0.12 | -3.73$\pm$0.01 | -3.80$\pm$0.05 | -6.29$\pm$0.04 | -6.54$\pm$0.09 | – | -5.63$\pm$0.05 | -7.17$\pm$0.03 | -5.28$\pm$0.02
Power | 2.89$\pm$0.01 | 2.84$\pm$0.01 | 2.80$\pm$0.05 | 2.80$\pm$0.00 | – | 2.81$\pm$0.03 | 2.79$\pm$0.04 | 2.83$\pm$0.01 | 2.78$\pm$0.01
Protein | 2.99$\pm$0.01 | 2.97$\pm$0.00 | 2.89$\pm$0.01 | 2.85$\pm$0.01 | – | 2.82$\pm$0.01 | 2.83$\pm$0.02 | 2.89$\pm$0.01 | 2.76$\pm$0.00
Wine | 0.98$\pm$0.01 | 0.97$\pm$0.01 | 0.93$\pm$0.06 | 0.90$\pm$0.01 | – | 1.09$\pm$0.10 | 0.94$\pm$0.12 | 0.95$\pm$0.01 | 0.89$\pm$0.02
Yacht | 3.44$\pm$0.16 | 1.63$\pm$0.02 | 1.55$\pm$0.12 | 0.47$\pm$0.03 | 0.225$\pm$0.40 | 1.79$\pm$0.88 | 1.18$\pm$0.21 | 0.37$\pm$0.08 | 0.90$\pm$0.09
Table 11: Results on UCI regression datasets comparing NLL

###  D.2 Results of variants of deep split ensembles

Table 12 shows the RMSE and NLLtab_fn_exhaustive results of variants of deep
split ensemble, with different number of models $E=1,5,10$ in the parallel
ensemble. We observe that $E=5$ is the best of all considering the performance
results and the computational overhead.

Datasets | RMSE | NLL
---|---|---
| $E=1$ | $E=5$ | $E=10$ | $E=1$ | $E=5$ | $E=10$
Boston | 2.76 $\pm$ 1.16 | 2.53 $\pm$ 0.15 | 2.60 $\pm$ 1.31 | 2.33 $\pm$ 0.26 | 2.23 $\pm$ 0.04 | 2.25 $\pm$ 0.28
Concrete | 4.52 $\pm$ 0.55 | 4.40 $\pm$ 0.10 | 4.63 $\pm$ 0.57 | 2.89 $\pm$ 0.12 | 2.85 $\pm$ 0.02 | 2.87 $\pm$ 0.13
Energy | 0.43 $\pm$ 0.06 | 0.41 $\pm$ 0.02 | 0.44 $\pm$ 0.08 | 0.31 $\pm$ 0.19 | 0.28 $\pm$ 0.11 | 0.33 $\pm$ 0.25
Kin8nm | 0.20 $\pm$ 0.00 | 0.19 $\pm$ 0.00 | 0.19 $\pm$ 0.00 | -0.18 $\pm$ 0.03 | -0.20 $\pm$ 0.02 | -0.20 $\pm$ 0.02
Naval | 0.00 $\pm$ 0.00 | 0.00 $\pm$ 0.00 | 0.00 $\pm$ 0.00 | -5.21 $\pm$ 0.07 | -5.28 $\pm$ 0.02 | -5.26 $\pm$ 0.03
Power | 4.06 $\pm$ 0.25 | 4.04 $\pm$ 0.05 | 4.07 $\pm$ 0.26 | 2.80 $\pm$ 0.05 | 2.78 $\pm$ 0.01 | 2.83 $\pm$ 0.06
Protein | 4.14 $\pm$ 0.03 | 4.05 $\pm$ 0.03 | 4.09 $\pm$ 0.04 | 2.79 $\pm$ 0.01 | 2.76 $\pm$ 0.00 | 2.80 $\pm$ 0.02
Wine | 0.63 $\pm$ 0.12 | 0.60 $\pm$ 0.02 | 0.60 $\pm$ 0.05 | 0.95 $\pm$ 0.17 | 0.89 $\pm$ 0.02 | 0.92 $\pm$ 0.10
Yacht | 0.89 $\pm$ 0.41 | 0.86 $\pm$ 0.07 | 0.90 $\pm$ 0.46 | 0.93 $\pm$ 0.25 | 0.90 $\pm$ 0.09 | 0.93 $\pm$ 0.20
Table 12: Results of deep split ensembles with different no. of models $E$ in
the parallel ensemble.

###  D.3 Cluster-wise NLLs

Table 13 shows an exhaustive list of NLLs of all clusters for different number
of models $E$ in the parallel ensemble. Table 12 shows the average of these
cluster-wise NLLs corresponding to each dataset. Table 14 shows NLL per
cluster for DSE, AEPC, DEPC.

Dataset | Cluster | Cluster-wise NLL
---|---|---
$E=1$ | $E=5$ | $E=10$
Boston | 1 | 2.29 $\pm$ 0.07 | 2.23 $\pm$ 0.04 | 2.21 $\pm$ 0.03
2 | 2.33 $\pm$ 0.08 | 2.20 $\pm$ 0.05 | 2.24 $\pm$ 0.03
3 | 2.37 $\pm$ 0.03 | 2.26 $\pm$ 0.04 | 2.30 $\pm$ 0.05
Concrete | 1 | 2.87 $\pm$ 0.01 | 2.84 $\pm$ 0.02 | 2.87 $\pm$ 0.03
2 | 2.89 $\pm$ 0.02 | 2.85 $\pm$ 0.02 | 2.91 $\pm$ 0.04
3 | 2.91 $\pm$ 0.03 | 2.87 $\pm$ 0.01 | 2.93 $\pm$ 0.02
Energy | 1 | 0.28 $\pm$ 0.20 | 0.26 $\pm$ 0.11 | 0.29 $\pm$ 0.27
2 | 0.34 $\pm$ 0.18 | 0.30 $\pm$ 0.11 | 0.37 $\pm$ 0.24
Kin8nm | 1 | -0.18 $\pm$ 0.03 | -0.19 $\pm$ 0.02 | -0.20 $\pm$ 0.02
2 | -0.18 $\pm$ 0.03 | -0.19 $\pm$ 0.03 | -0.20 $\pm$ 0.02
3 | -0.18 $\pm$ 0.03 | -0.19 $\pm$ 0.02 | -0.20 $\pm$ 0.03
4 | -0.18 $\pm$ 0.03 | -0.19 $\pm$ 0.03 | -0.20 $\pm$ 0.02
5 | -0.18 $\pm$ 0.03 | -0.20 $\pm$ 0.03 | -0.20 $\pm$ 0.02
6 | -0.18 $\pm$ 0.03 | -0.19 $\pm$ 0.02 | -0.20 $\pm$ 0.02
7 | -0.18 $\pm$ 0.02 | -0.20 $\pm$ 0.03 | -0.20 $\pm$ 0.02
8 | -0.18 $\pm$ 0.02 | -0.21 $\pm$ 0.02 | -0.21 $\pm$ 0.02
Naval | 1 | -5.19 $\pm$ 0.08 | -5.25 $\pm$ 0.02 | -5.23 $\pm$ 0.04
2 | -5.24 $\pm$ 0.06 | -5.31 $\pm$ 0.02 | -5.29 $\pm$ 0.02
Power | 1 | 2.80 $\pm$ 0.06 | 2.79 $\pm$ 0.01 | 2.83 $\pm$ 0.07
2 | 2.81 $\pm$ 0.05 | 2.80 $\pm$ 0.01 | 2.84 $\pm$ 0.05
Protein | 1 | 2.73 $\pm$ 0.01 | 2.68 $\pm$ 0.01 | 2.72 $\pm$ 0.02
2 | 2.83 $\pm$ 0.00 | 2.80 $\pm$ 0.00 | 2.84 $\pm$ 0.01
3 | 2.83 $\pm$ 0.00 | 2.81 $\pm$ 0.00 | 2.86 $\pm$ 0.02
Wine | 1 | 0.95 $\pm$ 0.05 | 0.91 $\pm$ 0.03 | 0.94 $\pm$ 0.03
2 | 0.99 $\pm$ 0.08 | 0.92 $\pm$ 0.01 | 0.92 $\pm$ 0.08
3 | 0.94 $\pm$ 0.04 | 0.91 $\pm$ 0.02 | 0.91 $\pm$ 0.08
4 | 0.94 $\pm$ 0.05 | 0.90 $\pm$ 0.04 | 0.94 $\pm$ 0.03
5 | 0.95 $\pm$ 0.07 | 0.91 $\pm$ 0.01 | 0.91 $\pm$ 0.08
Yacht | 1 | 1.48 $\pm$ 0.20 | 1.45 $\pm$ 0.10 | 1.49 $\pm$ 0.12
2 | 0.39 $\pm$ 0.10 | 0.36 $\pm$ 0.09 | 0.38 $\pm$ 0.08
Table 13: NLLs of all clusters for different number of models $E$ in the parallel ensemble. Datasets | RMSE | Clusters | Cluster-wise NLL
---|---|---|---
| DEPC | AEPC | Deep Split Ens. |  | DEPC | AEPC | Deep Split Ens.
Boston | 5.11 $\pm$ 1.06 | 4.93 $\pm$ 1.03 | 2.53 $\pm$ 0.15 | 1 | 2.91 $\pm$ 0.16 | 3.87 $\pm$ 0.82 | 2.23 $\pm$ 0.04
2 | 2.82 $\pm$ 0.16 | 3.99 $\pm$ 0.94 | 2.20 $\pm$ 0.05
3 | 3.29 $\pm$ 0.10 | 4.23 $\pm$ 1.06 | 2.26 $\pm$ 0.04
Concrete | 10.22 $\pm$ 0.82 | 10.40 $\pm$ 0.93 | 4.40 $\pm$ 0.10 | 1 | 3.77 $\pm$ 0.05 | 5.75 $\pm$ 0.61 | 2.84 $\pm$ 0.02
2 | 3.79 $\pm$ 0.10 | 5.68 $\pm$ 0.60 | 2.85 $\pm$ 0.02
3 | 3.81 $\pm$ 0.05 | 5.83 $\pm$ 0.61 | 2.87 $\pm$ 0.01
Power | 7.51 $\pm$ 0.20 | 7.53 $\pm$ 0.18 | 4.04 $\pm$ 0.05 | 1 | 3.90 $\pm$ 0.09 | 4.20 $\pm$ 0.09 | 2.79 $\pm$ 0.01
2 | 3.61 $\pm$ 0.02 | 4.18 $\pm$ 0.09 | 2.80 $\pm$ 0.01
Protein | 5.04 $\pm$ 0.01 | 5.00 $\pm$ 0.02 | 4.05 $\pm$ 0.03 | 1 | 3.83 $\pm$ 0.10 | 3.04 $\pm$ 0.00 | 2.68 $\pm$ 0.01
2 | 3.04 $\pm$ 0.00 | 3.05 $\pm$ 0.00 | 2.80 $\pm$ 0.00
3 | 3.06 $\pm$ 0.00 | 3.05 $\pm$ 0.00 | 2.81 $\pm$ 0.00
Wine | 0.67 $\pm$ 0.05 | 0.69 $\pm$ 0.04 | 0.60 $\pm$ 0.02 | 1 | 1.02 $\pm$ 0.07 | 1.08 $\pm$ 0.10 | 0.91 $\pm$ 0.03
2 | 1.01 $\pm$ 0.06 | 1.08 $\pm$ 0.09 | 0.92 $\pm$ 0.01
3 | 1.01 $\pm$ 0.06 | 1.08 $\pm$ 0.09 | 0.91 $\pm$ 0.02
4 | 1.01 $\pm$ 0.06 | 1.09 $\pm$ 0.09 | 0.90 $\pm$ 0.04
5 | 0.99 $\pm$ 0.07 | 1.09 $\pm$ 0.09 | 0.91 $\pm$ 0.01
Table 14: RMSE and Cluster-wise NLL values for DEPC, AEPC, DSE on UCI datasets

##  Appendix E Experiments

###  E.1 Entropy analyses

Figures 8 and 9 show the entropy plots for several datasets using hierarchical
clustering and clusters from human experts. The first two columns show the
kernel density estimation (KDE) of entropy for in- distribution i.e.
$\mathcal{N}(0,1)$ and out-of-distribution samples, obtained with unified
uncertainty estimation using deep ensemble and anchored ensembling
respetively. The last two columns show ‘cluster-wise’ KDE of entropy for in-
distribution and out-of-distribution samples, obtained with disentangled
uncertainty estimation using deep split ensembles. OOD 1 and OOD 2 refer to
introducing dataset shift by inducing noise sampled from
$\mathcal{N}(6,2^{2})$ into 2 random input features; the features correspond
to different clusters for deep split ensembles. We used the entire datasets by
accumulating fold-wise test results. The shift in the KDE (for OOD samples) of
the cluster-wise entropy shows that the deep split ensembles are well-
calibrated.

(a)  (b)

(c)

(d)

(e)

(f)  (g)

(h)

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(o)

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(r)

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(u)  (v)

(w)

(x)

(y)

Figure 8: Entropy plots for ‘Boston’, ‘Concrete’, ‘Energy’, ‘Kin8nm’, and
’Naval’ datasets using hierarchical clustering (Section 2.3). The first two
columns show the kernel density estimation (KDE) of entropy for in-
distribution i.e. $\mathcal{N}(0,1)$ and out-of-distribution samples, obtained
with unified uncertainty estimation using deep ensemble and anchored
ensembling respetively. The last two columns show ‘cluster-wise’ KDE of
entropy for in-distribution and out-of-distribution samples, obtained with
disentangled uncertainty estimation using deep split ensembles. OOD 1 and OOD
2 refer to introducing dataset shift by inducing noise sampled from
$\mathcal{N}(6,2^{2})$ into 2 random input features; the features correspond
to different clusters for deep split ensembles.

(a)  (b)

(c)

(d)

(e)

(f)  (g)

(h)

(i)

(j)

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(m)

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Figure 9: Entropy plots for ’Protein’ and ‘Yacht’ datasets using hierarchical
clustering (Section 2.3), and ’Power Plant’ and ’Wine’ datasets using clusters
from human experts (Section 3.4). The first two columns show the kernel
density estimation (KDE) of entropy for in- distribution i.e.
$\mathcal{N}(0,1)$ and out-of-distribution samples, obtained with unified
uncertainty estimation using deep ensemble and anchored ensembling
respetively. The last two columns show ‘cluster-wise’ KDE of entropy for in-
distribution and out-of-distribution samples, obtained with disentangled
uncertainty estimation using deep split ensembles. OOD 1 and OOD 2 refer to
introducing dataset shift by inducing noise sampled from
$\mathcal{N}(6,2^{2})$ into 2 random input features; the features correspond
to different clusters for deep split ensembles.

###  E.2 65-95-99.7 rule (empirical rule) to assess calibration

We evaluate to obtain calibration curves, where first, we compute the $x\%$
prediction interval for each test datapoint based on Gaussian quantiles using
the predicted mean and variance. Then, we calculate the fraction of test
observations (true values) that fall within this prediction interval. For a
well-calibrated model, the observed fraction should be close to $x\%$
calculated earlier. To see how our models perform in this setting, we sweep
from $x=10\%$ to $x=90\%$ in steps of 10, and consequently a line lying very
close to the line ($y=x$) would indicate a well-calibrated model. Figure 10
shows the calibration curves for each feature cluster of DEPC, AEPC, and deep
split ensembles on UCI datasets.

(a)  (b)

(c)

(d)

(e)  (f)

(g)

(h)

(i)  (j)

(k)

(l)

(m)  (n)

(o)

(p)

(q)  (r)

(s)

(t)

Figure 10: ‘Cluster-wise’ calibration curves using empirical rule for
‘Boston’, ‘Concrete’, ‘Power’, ‘Protein’, and ‘Wine’ datasets using
hierarchical clustering (Section 2.3). The columns contain experiments using
deep ensemble per input cluster (DEPC), anchored ensembling per input cluster
(AEPC) and deep split ensembles respectively.