Output-Constrained Bayesian Neural Networks

Wanqian Yang * 1 Lars Lorch * 1 Moritz A. Graule * 1 Srivatsan Srinivasan 1 Anirudh Suresh 1 Jiayu Yao 1

Melanie F. Pradier 1 Finale Doshi-Velez 1

AbstractBayesian neural network (BNN) priors aredefined in parameter space, making it hardto encode prior knowledge expressed infunction space. We formulate a prior thatincorporates functional constraints aboutwhat the output can or cannot be in regionsof the input space. Output-ConstrainedBNNs (OC-BNN) represent an interpretableapproach of enforcing a range of constraints,fully consistent with the Bayesian frame-work and amenable to black-box inference.We demonstrate how OC-BNNs improvemodel robustness and prevent the predic-tion of infeasible outputs in two real-worldapplications of healthcare and robotics.

1. Introduction

BNNs combine powerful function approximatorswith the ability to model uncertainty, making themuseful in domains where (i) training data is expen-sive or limited, or (ii) inaccurate predictions are pro-hibitively costly and decision-making must be in-formed by our level of confidence (MacKay, 1995;Neal, 1995). Domain experts often have prior knowl-edge about the modeled function and the ability toencode such information on top of training data canthus improve performance. However, BNNs defineprior distributions over parameters, whose high di-mensionality and lack of interpretability make the in-corporation of functional beliefs close to impossible.

We present an interpretable approach for incorporat-ing prior functional information into BNNs in theform of constraints, while staying consistent with theBayesian framework. We then apply our method to

*Equal contribution 1Harvard University. Correspon-dence to: Wanqian Yang <yangw@college.harvard.edu>,Lars Lorch <lars.lorch@gmail.com>.

Presented at the ICML 2019 Workshop on Uncertainty andRobustness in Deep Learning and Workshop on Under-standing and Improving Generalization in Deep Learning.Long Beach, CA, 2019. Copyright 2019 by the author(s).

two domains where the ability to encode such con-straints is crucial: (i) prediction of clinical actions inhealth care, where constraints prevent unsafe actionsfor certain physiological inputs, and (ii) human mo-tion prediction, where joint positions are constrainedby anatomically feasible ranges.

Our contributions are: (a) we introduce constraintpriors, capable of incorporating both negative con-straints (where the function cannot be) and positiveconstraints (where the function should be), applica-ble with any black-box inference algorithm normallyused with BNNs, and (b) we demonstrate the applica-tion of constraint priors with a variety of suitable in-ference methods on toy problems as well as two largeand high-dimensional real-world data sets.

2. Related Work

Most closely related to our work, (Lorenzi & Filip-pone, 2018) considered function-space equality andinequality constraints of deep probabilistic models.However, they focused on deep Gaussian processes(DGPs) rather than BNNs, and on low-dimensionaldata from simulated ODE systems, whereas we con-sider high-dimensional real-world settings. They alsodo not consider classification settings.

(Hafner et al., 2018) specify a Gaussian functionprior with the goal of preventing overconfident BNNpredictions out-of-distribution. In contrast, we use”positive constraints” to guide the function where itshould be. Also related are functional BNNs by (Sunet al., 2019), where variational inference is performedin function-space using a stochastic process model.Their view is more general—and accordingly, morecomplex to optimize—while we focus on constraintsin specific regions of the input-output space.

3. Background

A conventional BNN, operating in the function (orinput-output) space X ×Y , typically has a prior overparameters p(W), where W are the neural networkweights and biases. Given data D = {xn, yn}N

n=1, we

Output-Constrained Bayesian Neural Networks

perform inference to obtain the posterior p(W |D) ∝p(W)p(D |W). The posterior predictive for the out-put y′ for some new input x′ is obtained by integrat-ing over the posterior distribution ofW :

p(y′ | x′,D) =∫W

p(y′ | x′,W)p(W |D)dW (1)

The space ofW is high-dimensional and the relation-ship between the weights and the function is non-intuitive. As such, the prior p(W) is often triviallychosen as an isotropic Gaussian:

p(W) = ∏iN (W i; 0, σ2

p) (2)

4. Output-Constrained BNNs

We consider two kinds of “expert knowledge”: posi-tive constraints define regions where a function shouldbe, and negative constraints define regions where afunction cannot be. This delineation is not arbitrary— the level of prior knowledge (strongly vs. weaklyinformative) and the task (regression or classification)may suggest the use of different prior constraints.

Defining constrained regions Formally, a positiveconstrained region C+ is a set of input-output tu-ples (x, y) defining where outputs given certain in-puts should be. Conversely, a negative constrained re-gion C− is a set of tuples (x, y) defining where out-puts given certain inputs cannot be. We will use Cwhen describing properties of constrained regions ofboth kinds and denote Cx for all x in C and Cy for all yin C. Given this formulation, it is our goal to enforce∫W

p(y′ /∈ C+y | x′ ∈ C+x ,W)p(W | C+,D)dW ≈ 0∫W

p(y′ ∈ C−y | x′ ∈ C−x ,W)p(W | C−,D)dW ≈ 0(3)

Note that (3) is simply the posterior predictive distri-bution conditioned on C. The generality of this ap-proach allows for the incorporation of very compli-cated yet interpretable constraints a priori, such as forexample arbitrary equality, inequality and logical (if-then and either-or) constraints.

Constraint prior We connect the weight space ofthe BNN with constraints through the distribution:

g(W| C) = g(φ(Cx;W); Cy, θ) (4)

where φ(x;W) is the BNN forward pass and θ is theset of tuneable hyperparameters of g. Accordingly, aconstraint prior pC(W) can then be constructed as:

pC(W) := p(W) g(W | C), (5)

achieving the goal of expressing prior function

knowledge in weight space while retaining theweight-space prior p(W). Intuitively, g(W | C) mea-sures the BNN’s adherence to the constrained region.

It remains to describe how g is defined. For positiveconstraints C+, g measures how close φ(C+x ;W) liesto Cy, for which natural choices of distributions ex-ist for both regression and classification. For nega-tive constraints C−, we define g as the expected viola-tion of C−y given φ(C−x ;W) using a classifier function.Complete definitions of g for positive and negativepriors are provided in Appendix A; details on infer-ence procedures are provided in Appendix B.

5. Demonstrations on Synthetic Data

This section provides proof of concepts of OC-BNNsusing 2-dimensional synthetic examples. Refer to Ap-pendix C for experimental details and Appendix Dfor additional results. For regression, the posteriorsare visualized in black/gray for baseline BNNs, andblue for OC-BNNs. Negative constrained regions arered; positive (Gaussian) constraints are green. Forthe classification example, the three classes are color-coded red, green and blue.

OC-BNNs model uncertainty in a manner that re-spects constrained regions while explaining train-ing data. Figure 1 demonstrates this for both theregression and classification setting. Correct predic-tions are maintained with similar uncertainty lev-els as the baseline while constraints are correctly en-forced with uncertainty levels changing to reflect that.These examples demonstrate how OC-BNNs enforceconstraints without sacrificing predictive accuracy.

Figure 1. On both tasks, OC-BNNs reduce uncertainty inconstrained regions while fitting data well. (left) 1D regres-sion. The constraint is composed of two negative regions(red) separated by a small gap. Uncertainty of OC-BNNs(blue) drops sharply in the constrained region comparedto the baseline (gray). (right) 2D classification with threeclasses. Constrained region enforces the prediction of greenclass in the green rectangle (baseline depicted in inset).

OC-BNNs encourage correct out-of-distribution be-havior. Figure 2 (left) depicts sparse data, along without-of-distribution positive constrained regions. Theposterior predictive in-distribution closely mimics the

Output-Constrained Bayesian Neural Networks

baseline, while the posterior out-of-distribution (OOD)learns to avoid the constrained region. This demon-strates that OC-BNNs function well away from thedata, which is important because we typically wantto enforce functional constraints when there is a lackof observed training data for the model to learn from.

OC-BNNs can capture posterior multimodality.While negative constraints C− do not explicitly definemultimodal posterior predictives, a bounded con-strained region does imply that the posterior predic-tive might have probability mass on either side of thebounded region (i.e. for all d dimensions of Y = Rd).Figure 2 (right), demonstrates that we capture chal-lenging posterior predictives.

Figure 2. OC-BNNs capture important posterior qualitiessuch as correct OOD behavior and multimodality. (left)OC-BNNs (blue) maintain the same in-distribution uncer-tainty as the baseline (gray) while adhering to OOD positiveconstraints (green) on either side of the plot. (right) OC-BNNs (blue) posterior samples go both below and abovethe negative constraint box (red).

6. Applications

6.1. Clinical action prediction

MIMIC-III (Johnson et al., 2016) is a benchmarkdatabase containing time series data of various physi-ological measurements and clinical actions prescribedbelonging to > 40, 000 intensive care patients whostayed at the Beth Israel Deaconess Medical Centerbetween 2001 and 2012.

Problem Formulation From the raw time-seriesdata, we construct a balanced dataset for a time-independent classification task of hypotension man-agement. There are 9 features representing variousphysiological states, such as mean blood pressure andlactate levels. The goal is to predict if clinical action(either vasopressor or IV fluid) should be taken.

Constraints The constraint imposed is that formean blood pressure less than 65 units, some ac-tion should be taken, which is physiologically realis-tic. We apply the positive (Dirichlet) constraint prior(Appendix A), as well as the weights-only prior base-line. In the given data, some training points fall

filtered unfilteredBNN OC-BNN BNN OC-BNN

Trai

n ACC 0.745 0.741 0.881 0.878F1 0.805 0.801 0.882 0.880VIOL 0.151 0.149 N/A N/A

Test

ACC 0.660 0.665 0.647 0.649F1 0.746 0.748 0.725 0.736VIOL 0.132 0.126 0.117 0.039

Table 1. Results for the MIMIC experiments with and with-out filtering out the points in the constrained region. Ac-curacy and F1 score remain unchanged when using OC-BNNs. For the experiment with filtration, the violation fac-tor decreases by a factor of 3 when using OC-BNNs.

within the constrained region. We train our modelboth with and without artificially filtering out allpoints within the positive constrained region.

OC-BNNs maintain classification accuracy while re-ducing physiologically infeasible constraint viola-tions. Table 1 displays experimental results, withstatistics computed from the posterior mean. In ad-dition to standard accuracy (ACC) and F1 score, wemeasure the violation fraction (VIOL), which is thefraction of predictions on held-out points that vio-late the constraints. The results show that OC-BNNsmatch standard BNNs on all predictive accuracy met-rics, with significantly lower violation of the con-strained region for the case where points originallyin the constrained region are filtered out.

6.2. Human motion prediction

We evaluate OC-BNNs on data of humans conduct-ing various motions available at (Kratzer, 2019) as de-scribed in (Kratzer et al., 2018). This data contains hu-man upper body poses across many reaching tasks ata frame rate of 120Hz. The poses are provided in theform of upper body joint angles.

Problem formulation Given a subset of trajecto-ries in (Kratzer, 2019), our goal is to predict joint an-gles 20 frames in the future from angles at the currenttime frame and the numerically computed joint veloc-ities and accelerations. In the following, we limit our-selves to abduction and flexion (further denoted as Y-and Z-rotation to match the nomenclature in the orig-inal data (Kratzer, 2019)) of the left and right shoulderduring right-handed reaching motions.

The joint angles in the test data were perturbed withnormally distributed noise (µ=0, σ=2 degrees) tosimulate a scenario in which a human motion predic-tion model is trained on data recorded in a high-endmotion capture lab, and then used to predict motionfrom data obtained by noisy wearable sensors.

Output-Constrained Bayesian Neural Networks

Constraints Several anatomical feasibility or func-tional range constraints for each of the joint anglescould be applied, e.g. as described in (Namdari et al.,2012). We derived constraints on the joint limits fromthe reaching motions provided in (Kratzer, 2019) asthe empirically observed extrema across all motions,which is modeled using the negative constraint prior.

OC-BNNs prevent infeasible predictions. We com-pare a BNN and OC-BNN using the negative priorand the empirical bounds on joint angles. Both mod-els are compared in (i) RMSE using the posterior pre-dictive mean (RMSE) [1·103], (ii) held-out data loglikelihood of N (µpp, σ2

pp) with posterior predictivemean µpp and variance σpp (HO-LL), and (iii) poste-rior predictive violation defined as the percentage ofprobability mass in an infeasible constrained region(PP-VIOL) [%], each evaluated at all target points.

These metrics are summarized in Table 2. We find thatOC-BNNs reduce the possibility of making an infea-sible prediction to less than 0.001%, substantially im-proving on BNNs. Figure 3 shows exemplary motionpredictions obtained with both BNN and OC-BNNfor five consecutive points in a test trajectory.

BNN OC-BNN

Trai

n RMSE 0.929 1.252HO-LL 1718.409 1342.602PP-VIOL 0.046 0.000

Test

RMSE 7.320 12.127HO-LL 101.129 -683.697PP-VIOL 18.447 0.000

Table 2. Results for human motion prediction. While pre-dictive performance and held-out log likelihood are similar,OC-BNNs (negative prior) reduce the chance of predictingan infeasible position to 0.0 % while BNNs make infeasiblepredictions in 18.4% of cases.

Figure 3. Consecutive predictions of right Z-rotation duringan exemplary test trajectory with 10 frame gaps. The pos-terior predictive of BNN (black) and OC-BNN (blue) giventhe input state 20 frames earlier are plotted along the y-axis.While both BNN and OC-BNN do not perfectly generalizeto the test set, the expert knowledge of valid joint positionsenforces feasible and thus more robust predictions.

7. Discussion

OC-BNNs prevent constraint violation while fittinglow- and high-dimensional data. Our results high-light that incorporating expert knowledge into OC-BNNs helps enforcing feasible and thus more ro-bust predictions. Results for both datasets in Sec-tion 6 demonstrate that constraint violation metricsare reduced significantly, whereas accuracy metricsare nearly unchanged. This affirms the behavior ob-served in the synthetic examples in Section 5.

Training data in constrained region can outweighprior effect. The clinical dataset results show that thepresence of data in C reduces the effect of constraintpriors. This is expected and in accordance with theBayesian framework, where the likelihood effect willcrowd out the prior given enough training data, andalso suggests that the practitioner can use OC-BNNseven for situations where the constraints themselvesmay not be fully satisfied.

OC-BNNs can facilitate data imputation. The factthat OC-BNNs model uncertainty correctly in con-strained regions without losing predictive accuracy,even for high-dimensional datasets, show that OC-BNNs can encode imputation in input regions with-out training data. Rather than directly modifying thetraining set through imputation, prior beliefs aboutmissing data can instead be formulated as constraints.

When to use which prior? In the regression setting,negative priors are weakly informative whereas pos-itive priors tend to be strongly informative – one orboth of the prior types can be used depending ondomain knowledge. While the negative prior for-mulation does not apply to classification cases, thisdoes not pose a problem as negative and positive con-straints are complements in discrete space.

8. Conclusion and Outlook

We describe OC-BNNs, a formulation to incorporateexpert knowledge into BNNs by prescribing positiveand negative (i.e., desired and forbidden) regions,and demonstrate their application to synthetic andreal-world data. We show that OC-BNNs generallymaintain the desirable properties of regular BNNswhile their predictions follow the prescribed con-straints. This makes them a promising tool for set-tings like healthcare, where models trained on sparsedata may be augmented with expert knowledge. Inaddition, OC-BNNs may find applications in safe re-inforcement learning, e.g. in tasks where certain ac-tions are known to have catastrophic consequences.

Output-Constrained Bayesian Neural Networks

Acknowledgements

MG and FDV acknowledge support from AFOSR FA9550-17-1-0155. LL and WY acknowledge supportfrom the John A. Paulson School of Engineering andApplied Sciences at Harvard University.

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Davidson, J. Reliable uncertainty estimates in deepneural networks using noise contrastive priors. Ineprint arXiv:1807.09289, 2018.

Johnson, A. E., Pollard, T. J., Shen, L., Li-wei, H. L.,Feng, M., Ghassemi, M., Moody, B., Szolovits, P.,Celi, L. A., and Mark, R. G. Mimic-iii, a freely acces-sible critical care database. Scientific data, 3:160035,2016.

Kratzer, P. mocap-mlr-datasets. https://github.com/charlespwd/project-title, 2019.

Kratzer, P., Toussaint, M., and Mainprice, J. Towardscombining motion optimization and data drivendynamical models for human motion prediction. In2018 IEEE-RAS 18th International Conference on Hu-manoid Robots (Humanoids), pp. 202–208. IEEE, 2018.

Liu, Q. and Wang, D. Stein variational gradient de-scent: A general purpose bayesian inference algo-rithm. In Advances In Neural Information ProcessingSystems, pp. 2378–2386, 2016.

Lorenzi, M. and Filippone, M. Constraining the dy-namics of deep probabilistic models. arXiv preprintarXiv:1802.05680, 2018.

MacKay, D. J. C. Probable networks and plausiblepredictions – a review of practical bayesian meth-ods for supervised neural networks. In Network:Computation in Neural Systems, 6:3, 469-505, 1995.

Namdari, S., Yagnik, G., Ebaugh, D. D., Nagda, S.,Ramsey, M. L., Williams Jr, G. R., and Mehta, S.Defining functional shoulder range of motion foractivities of daily living. Journal of shoulder and el-bow surgery, 21(9):1177–1183, 2012.

Neal, R. M. Bayesian Learning for Neural Networks. PhDthesis, Graduate Department of Computer Science,University of Toronto, 1995.

Neal, R. M. Mcmc using hamiltonian dynamics. InHandbook of Markov Chain Monte Carlo, 2012.

Sun, S., Zhang, G., Shi, J., and Grosse, R. Func-tional variational bayesian neural networks. arXivpreprint arXiv:1903.05779, 2019.

A. Constraint Priors

In this section, we describe the detailed functionalforms of our positive and negative constraints andpriors for both classification and regression settings,noting aspects important for inference.

A.1. Positive constraint prior

Since C+ describes the set of points that the learnedfunction should model, g(φ(Cx;W); Cy, θ) has thestraightforward interpretation of measuring howclosely φ(Cx;W) lies to Cy. Most common prob-ability distributions as well as (possibly improper)user-defined distributions are amenable, though dif-ferentiability may be a condition for certain inferencemethods. In particular, natural choices of distribu-tions exist for both regression and classification.

Regression In the simplest setting, for which thereis a known ground-truth function described perfectlyby C+, the Gaussian distribution is a natural choice:

g(W | C+) = ∏x,y∼π(C+)

N (φ(x;W); y, σ2+) (6)

where π(C+) is a sampling distribution for C+, whichis necessary for tractability if C+ is large or infinite.π(C+) itself can be user-defined as the domain al-lows, allowing for flexibility in sampling. σ+ is thetuneable standard deviation of the Gaussian, control-ling strictness of deviation from C+. More generally, itis possible that there exists multiple y ∈ C+y for somex ∈ C+x . This can be expressed using multimodal dis-tributions, for example:

g(W | C+) = ∏x,{y}K

k ∼π(C+)

K

∑k=1

ωkN (φ(x;W); yk, σ2+) (7)

where ωk are the user-defined mixture weights.

Classification C+y describes the classes that theBNN is constrained to for the corresponding C+x . Inthe discrete setting, the natural distribution is theDirichlet. For K classes,

g(W | C+) = ∏x,y∼π(C+)

Dir(φ(x;W); α) (8)

where αk =

{1 if yk = 11− ασ otherwise

for some control-

lable penalty ασ.

A.2. Negative constraint prior

The negative constraint prior enforces the infeasibil-ity of regions in function space and is constructed byplacing little prior probability on high expected vio-

Output-Constrained Bayesian Neural Networks

lation of C−:

g(W |C−)= exp

(E

x∼π(C−x )

[−γ c

(x, φ(x,W); C−

)])(9)

In (9), c(x, y; C−) is a classifier function that encodessoftly whether or not (x, y) is in C−, which allowsblack-box use with any inference technique:

c(x, y; C−) =J

∑j=1

Kj

∏k=1

στ0,τ1

(f j(

x, y)

k

)(10)

The definition of c(x, y; C−) assumes that the nega-tive region C− is defined by J sets of Kj inequality

constraints f j(x, y) ≤ 0, i.e. C− =⋃J

j=1 C−j with

C−j = {(x, y) | f j(x, y) ≤ 0}, which can define arbi-trary linear and nonlinear shapes in the input-outputspace. στ0,τ1(z) is a soft indicator of whether a con-straint of the form z ≤ 0 is satisfied, a more generally-parameterizable sigmoidal activation defined as

στ0,τ1(z) =(

tanh(−τ0z) + 1)(

tanh(−τ1z) + 1)

(11)

If all constraints for at least one infeasible region C−jare satisfied, our prior knowledge is violated andc(x, y; C−) is far from 0. Otherwise, at least one con-straint of all infeasible regions is violated and ourprior beliefs satisfied; c(x, y; C−) is close to 0. Con-trary to other classification functions, the product oftwo tanh functions with different scales τ0, τ1 enablesa sharp and steep overall classification of violatingvalues in z > 0 and a smoother and flatter classifica-tion for satisfying values in z ≤ 0, making gradientsless vanishing for constraint-satisfying, i.e. region-violating inputs. We use τ0 = 15, τ1 = 2.

B. Inference

Constraint priors can be substituted for the tradi-tional prior term p(W) with any black-box samplingor variational inference (VI) algorithm. Here, we pro-vide a summary of the algorithms we use and de-scribe the trivial modifications used to incorporateconstraint priors pC(W). Note that the general formof pC(W) is not normalized, which does not pose aproblem for inference in practice.

Hamiltonian Monte Carlo (HMC) HMC (Neal, 2012)is a MCMC method considered to be the “gold stan-dard” in posterior sampling even though not beingscalable. We substitute p(W) by pC(W) in the poten-tial energy term U(W) computed at each samplingiteration:

U(W) = − log pC(W)− log p(D |W) (12)

As the presence of g(W | C) increases the magnitudeof the prior pC(W), empirical performance typicallyimproves by using a smaller step-size than with p(W)for the same dataset.

Stein Variational Gradient Descent (SVGD) SVGD(Liu & Wang, 2016) is a VI method where a set ofS particles (in our case, {W s}S

s=1) are optimized viafunctional gradient descent to mimic the true poste-rior. SVGD combines the efficiency of VI methodswith the ability of MCMC methods to capture moreexpressive posterior approximations. p(W) is substi-tuted by pC(W) in the computation of the functionalgradient:

φ̂∗(W) =1S

S

∑s=1

[k(W s,W)∇W s [log pC(W s) (13)

+ log p(D |W s)] +∇W s k(W s,W)]

Our implementation of SVGD uses the weighted RBFkernel k(x, x′) = exp(− 1

h ||x − x′||22) and adaptingbandwith h as suggested in (Liu & Wang, 2016) aswell as mini-batched data D for tractability.

C. Experimental Details

C.1. Synthetic Examples

For all experiments, the BNN used comprises a singlehidden layer with 10 nodes, and Radial Basis Func-tion (RBF) activations σ(x) = exp{−x2}.

All regression plots show the posterior mean function(bold line) as well as the confidence intervals for σ =1 (dark shading) and σ = 2 (light shading).

Figure 1: (left) The constrained regions are y < 2.5and y > 3 for x ∈ [−0.3, 0.3]. The function gener-ating the training points is y = −x4 + 3x2 + 1. Thenegative prior formulation is used. (right) The inputspace is 2-dimensional and there are 3 classes (color-coded) with 8 training points in each class, generatedfrom the Gaussian means (−3, 1), (0,−3) and (2, 3).The constrained region is [1, 3]× [−2, 0] and definedsuch that points within the box should be classified asgreen. The positive prior is used. HMC (10000 burn-in, 1000 samples collected at intervals of 10) is usedfor both examples.

Figure 2: (left) The positive constraints are y = −x +5 for x ∈ [−5.0,−3.0] and y = x + 5 for x ∈ [3.0, 5.0].Both constraints are Gaussian with the σ = 0.5. The3 training points are arbitrarily defined. HMC (10000burn-in, 1000 samples collected at intervals of 10) isused. (Right) The constrained boxed region is x ∈[−1.0, 1.0] and y ∈ [−5.0, 3.0]. The function generat-

Output-Constrained Bayesian Neural Networks

ing the training points is y = −x4 + 3x2 + 1. SVGDwith 75 particles is used with Adagrad.

C.2. Clinical action prediction

For all experiments, the BNN used comprises a 2 hid-den layers of 200 nodes each and RBF activations.SVGD is used for inference with 50 particles, 1500 it-erations, Adagrad optimization, and a suitable batchsize. The size of the full dataset is 298K; this reducesto 125K when points in the constrained region are fil-tered out. Details on the prior formulation for can befound in A. The Dirichlet parameter is set to 10 forallowed classes and 0.01 for forbidden classes.

C.3. Human motion prediction

For these experiments, the BNN used comprises a 2hidden layers of 100 nodes each and RBF activations.For inference, we again used SVGD and Adagradwith 50 particles and 1000 iterations. The negativeprior used 50 samples from π(C−x ) and γ = 10, 000,see Eq. 9.

We randomly chose a subset of 10 right-handedreaching trajectories from (Kratzer, 2019). This datawas randomly split into 5 training and 5 test tra-jectories, which amounts to 243 train Markov statesof sensors for training and 142 states for evalua-tion. Given this problem setting, the regression taskhad 12-dimensional inputs and 4-dimensional tar-gets. The number of training trajectories was kept lowto increase sparsity and the difficulty of successful ro-bust generalization.

D. Additional Results

D.1. Additional Synthetic Examples

Figure 4 shows additional examples for out-of-distribution and multimodal behavior. (left) Out-of-distribution negative constraints. The negativeconstraints are y > −x + 7 and y < −x + 2 forx ∈ [−5.0,−3.0] and y > x + 7 and y < x + 2 forx ∈ [3.0, 5.0]. The training points are identical tothose in the left plot of Figure 2. HMC (10000 burn-in, 1000 samples collected at intervals of 10) is used.(right) Multimodal positive constraints. The two pos-itive functions are y = −0.2x3 + 0.5x2 + 0.7x − 0.5and y = 0.2x3 − 0.15x2 + 3.5, both for the domainx ∈ [−1.0, 1.0]. The training points were arbitrarilydefined. An equally-weighted mixture of two Gaus-sians with σ = 0.5 is used as the positive constraintprior. SVGD with 75 particles and Adagrad are used.

Figure 4. (left) The same training set as Figure 2 (left), butwith negative constraints defined out-of-distribution. OC-BNNs fit the sparse data while avoiding the constraints.(right) Positive prior with mixture of two Gaussians. Us-ing SVGD, individual OC-BNN samples (blue) capture bothmodes.