# P-Age: Pexels Dataset for Robust Spatio-Temporal Apparent Age Classification

Abid Ali*
Inria, France
Université Cote d’Azur, France
<EMAIL_ADDRESS>Ashish Marisetty*
IIIT Naya Raipur, India
<EMAIL_ADDRESS>Francois Bremond
Inria, France
Université Cote d’Azur, France
<EMAIL_ADDRESS>
###### Abstract

Age estimation is a challenging task that has numerous applications. In this
paper, we propose a new direction for age classification that utilizes a
video-based model to address challenges such as occlusions, low-resolution,
and lighting conditions. To address these challenges, we propose AgeFormer
which utilizes spatio-temporal information on the dynamics of the entire body
dominating face-based methods for age classification. Our novel two-stream
architecture uses TimeSformer and EfficientNet as backbones, to effectively
capture both facial and body dynamics information for efficient and accurate
age estimation in videos. Furthermore, to fill the gap in predicting age in
real-world situations from videos, we construct a video dataset called Pexels
Age (P-Age) for age classification. The proposed method achieves superior
results compared to existing face-based age estimation methods and is
evaluated in situations where the face is highly occluded, blurred, or masked.
The method is also cross-tested on a variety of challenging video datasets
such as Charades, Smarthome, and Thumos-14. The code and dataset is available
at https://github.com/Ashish013/AgeFormer. ††footnotetext: * authors
contributed equally

##  1 Introduction

Age estimation is a challenging task that has many applications in various
domains such as social robotics, video surveillance, business intelligence,
social networks, and demography. Typically, the goal of age estimation
involves predicting the age of a person by his/her facial appearance, which
can be affected by many factors such as resolution, lighting condition, pose,
expression, occlusion, and makeup. In recent years, numerous attempts have
been made to estimate the age of a person from facial images [27, 35, 33, 53,
55, 51, 25, 41].

Age estimation models can be broadly classified into two categories:
regression-based and classification-based. Regression-based models directly
output a numerical value for the age, whereas classification-based models
assign the image to one of the predefined age groups. Recent advances in age
estimation are mainly driven by the development of deep learning techniques,
especially Convolutional Neural Networks (CNNs). Several methods adopted 2D
CNNs to estimate age from a single-face image. Furthermore, researchers
recently combined 2D-CNNs with attention-mechanism [45], adversarial learning
[31], domain adaptation [39], and multi-task learning techniques [49, 20] to
improve the robustness of age estimation models. However, estimating the age
of a single-face image is still an open challenge due to several factors such
as occlusions, low-resolution, and lighting conditions. In addition to that,
face-based age estimation models fail in situations where the face is not
visible all the time or blurred due to privacy concerns. We argue that a
single-face image is not enough to overcome these challenges in real-world
situations. Therefore, a new technique based on video that captures the
semantics of the entire body is needed to address these challenges.

3D-CNNs have been widely used in domains such as video analysis [15, 42],
action recognition [47, 4, 22], and medical image segmentation [29], due to
their ability to capture spatial and temporal information together. A video-
based model can be useful to capture the spatio-temporal features of the
frames to predict the age of a person in real-world situations such as
surveillance and or medical assessment, where the face is not visible all the
time. In addition, modern techniques, such as transformers [44] can be used to
further handle long-range dependencies in an environment where the face is
highly occluded. Furthermore, we believe that using only facial features is
not enough to predict age. In addition, other characteristics of the body,
such as the  body dynamics of a person (face, head, and limbs, etc.) can
provide significant cues to estimate the age of the person. Therefore, we
propose a video-based age estimation model that captures significant spatio-
temporal features of the entire body to improve the age estimation task. We
reuse existing video models for a completely new problem (age predictions).
Our results on unseen data such as Charades, Smarthome, and Thumos-14 Figure 5
demonstrate the novelty of utilizing such video models for a completely
different problem such as age classification. This opens up new research
directions for problems other than age classification, such as gender, ethnic
groups, and other demographic classifications. Many works that rely on faces
can be extended to video-based models to capture body dynamics for robust
recognition.

In summary, in this paper, we propose a new direction for age estimation
utilizing video-based models to address the above challenges. We introduced a
new dataset, P-Age, for video-based age classification. Furthermore, we
propose a novel two-stream architecture to improve age prediction in real-
world scenarios. Our architecture uses TimeSformer [6] in combination with
EfficientNet [40] as the backbone. The features of both streams are fused
using a multihead-attention module for age classification. Some of the main
contributions of this paper are as follows:

  * •

To the best of our knowledge, we are the first to propose a video-based method
for age classification in challenging situations. Our novel AgeFormer
architecture achieves superior results compared to existing face-based age
estimation methods.

  * •

We propose a new dataset called Pexels Age (P-Age), scrapped from Pexels (a
royalty-free stock footage website) for age classification into four distinct
groups. Additionally, we provide baseline results on this dataset and compare
our method with existing face-based methods on the P-Age-Face dataset.

  * •

Keeping privacy preservation a priority, we evaluate our method in situations
where the face is i) extremely occluded, ii) blurred, and iii) masked. In
addition, we validate the efficiency of our method in real-world situations by
testing it on a variety of challenging video datasets such as Charades [37],
Smarthome [12], and Thumos-14 [3].

Figure 1: An overview of the P-Age dataset.

##  2 Related Work

Age Estimation: Deep learning-based methods use convolutional neural networks
(CNNs) to automatically learn features from large-scale face datasets [25, 36,
28, 34]. Age-pattern learning methods can be categorized into classification-
based and regression-based approaches. Classification-based methods treat age
estimation as a multi-class problem, where each class corresponds to an age
group or a single year [18, 25, 24]. AGNet [27] uses CNNs network with a
softmax layer to classify the images into 8 age groups. Recently, PML [13]
proposes a progressive marginal loss approach for the classification of
unconstrained facial age.

Regression-based methods treat age estimation as a continuous problem, where
the output is a real-valued age estimate. DEX [33] uses CNNs to learn the
mapping of image pixels to age labels and then computes the expected value of
the age distribution as the final prediction. OR-CNN [28] converts the age
estimation problem into an ordinal regression problem, where each neuron in
the output layer represents an ordinal number, and the model learns to
activate neurons that correspond to the true age or lower. SSR-Net [48] adopts
a multi-stage design that progressively refines age prediction from coarse to
fine. More recently, [36] proposed a new ordinal regression algorithm called
Moving Window Regression (MWR) that introduces the notion of relative rank
(rho-rank) and develops global and local relative regressors (rho-regressors)
to predict rho-ranks within entire and specific rank ranges, respectively, for
age estimation from a single face image.

Furthermore, several datasets have been proposed for age estimation in recent
years, including UTKFace [8], MORPH [32], CACD [9], FG-NET [30]. However,
these datasets are image-based mainly focusing on face-crops. Therefore, the
methods introduced in the past are limited to face-crops only. In contrast, we
propose a video-based model to learn spatial and temporal features from the
semantics of the entire body.

Video Classification: CNNs have shown great success in learning 3D spatio-
temporal representations to recognize human actions [7, 42, 43, 19]. Two-
stream techniques, frequently combined RGB with optical flow [17, 38], to
classify videos. SlowFast network [16] has shown that action recognition can
be enhanced by mixing representations of various temporal resolutions (i.e.
frame rates). More recently, with the introduction of Transformers, numerous
systems have improved action recognition by including attention [5, 26, 46,
11, 10]. TimeSformer [6] introduced a divided space-time attention mechanism
to capture spatio-temporal dependencies throughout the video.

To go beyond Video Classification, our architecture utilizes action
recognition architectures for age estimation tasks. We combine an action
recognition (video) model with a 2D-CNNs method (image) to create a two-stream
network for robust age classification.

Age Group | Downloaded Videos | Filtered Videos | # of Frames
---|---|---|---
Min. | Avg. | Max.
Baby / Toddler | 1760 | 401 | 4 | 315 | 1708
Adolescent | 880 | 531 | 2 | 393 | 2436
Adult | 1048 | 416 | 18 | 485 | 3523
Elderly | 880 | 324 | 2 | 426 | 2106
Total | 4568 | 1672 | - | - | -
Table 1: Statistics of the P-Age dataset.

##  3 P-Age Dataset

In this section, we discuss the key choices in creating the Pexels Age (P-Age)
classification dataset. First, we discuss the data scraping process. Then, we
explain the annotation procedure and the quality control checks to refine the
annotations.

###  3.1 Data Preparation

The P-Age dataset is sourced from Pexels (a royalty-free stock footage
website) [2] using search keywords such as baby, toddler, teen, man, woman,
elderly, old, etc.. Furthermore, our dataset represents different ethnic
groups (African, European, etc.). An overview of the dataset is given in
Figure 1. We downloaded more than 4,500 videos with 27-fps and a high-quality
1080p resolution. The average video has more than 400 frames and 5.7% of the
videos in the P-Age dataset do not have a face appearance. Furthermore, most
of the videos in our dataset have occluded faces (for example, a cinematic
video of a child playing initially from his/her legs and slowly changing the
camera to reveal the face).

###  3.2 Annotation

The P-Age dataset is divided into four age groups: baby/child, teen, adult,
and elderly based on the searched query. For each class, we downloaded at
least 880 videos per class and made sure that the gender ratio was balanced by
using gender-specific keywords such as baby-girl, teen-boy, etc.. To further
save time, we annotated the dataset based on the searched query type (child,
teen, adult, elderly).

Figure 2: Proposed AgeFormer architecture. The main-stream learns spatio-temporal features of body dynamics, while the supporting-stream provides additional face features to improve age prediction. | # videos | # images | occlusions | # videos having no faces
---|---|---|---|---
MORPH [32] | $\times$ | 55.1k | $\times$ | $\times$
UTKFace [8] | $\times$ | 23.7k | $\times$ | $\times$
P-Age | 1672 | 675.5k | $\checkmark$ | 95
Table 2: Comparison with age estimation benchmark datasets.

###  3.3 Quality Control

The downloaded videos are then cleaned with their brief video filenames to
ensure that the video actually belongs to the class to which it is assigned,
and videos that do not belong to any class are removed. Several videos have
more than one person. We use existing object detection and tracking tools to
filter out videos that have multiple people or no people. We utilize Yolov5
[21] with OSNet [54] to filter out such videos. Furthermore, three experts
visually cross-check the annotation to correct any incorrect videos and remove
outliers. After filtering, we have obtained 1672 videos; per-class statistics
of the dataset are given in Table 1. A comparison is noted in Table 2 with
existing age estimation datasets.

###  3.4 P-Age-Face

Furthermore, to compare with other age estimation methods, we have created a
face-only variant of the P-Age dataset called P-Age-Face by randomly sampling
10 frames in a stride of 4 from each video. Following the general pre-
processing convention outlined in [25], we extract the faces using MTCNN [52]
and then use the eye landmark coordinates as a reference to align the face
crops. Since 5.7% of the videos in our dataset lack faces, the P-Age-Face
dataset accounts for 94. 3% of the size of the P-Age dataset. Upon acceptance,
both datasets will be made available to the general public.

##  4 AgeFormer

In this section, we discuss our proposed architecture for robust age
classification in videos. Our design is inspired by action recognition
networks and how they handle the spatio-temporal information to recognize an
action type. In contrast to action recognition methods, where the model learns
gestures and body movements over time, we learn the dynamics of the entire
body (face, head, height and width of limbs, etc.) over time. Here, the time
information helps the model to learn the appearance of the person at different
frames in a video snippet. For example, in the starting frames of a video,
only the lower body is visible to the camera, but after a few seconds, the
entire person appears (face or upper body appearance). Thus, the full
appearance information is compatible with partially visible frames. The
overall architecture is shown in Figure 2.

###  4.1 Video-stream

The main-stream is a video transformer network that uses TimeSformer [6] as
the backbone. TimeSformer [6] is a convolution-free method to classify videos
that relies solely on individual attention in space and time. By allowing
spatio-temporal features to learn directly from a series of frame-level
patches, it applies the traditional Transformer architecture to videos. The
main-stream takes an input clip $X\in\mathbb{R}^{H\times W\times 3\times T}$
having $T$ frames of size $H\times W$. The input frames are decomposed into
$N$ patches of size $P\times P$. The patches are flattened into vectors
$X_{(p,t)}\mathbb{R}^{3P^{2}}$, where $p=1,...,N$ denotes spatial locations
and $t=1,...,T$ defines the frame index. The readers can refer to the
TimeSformer [6] paper for more details. The main-stream captures the spatio-
temporal dynamics of the entire body.

###  4.2 Image-Stream

The supporting stream is a 2D-CNN network that uses EfficientNet B2 [40] as
the backbone. EfficientNet is a robust and high-performing image
classification network that has fewer parameters and is less computationally
expensive, making it the perfect choice to complement the parameter-heavy
video backbone. The supporting stream takes the face-crops extracted using
MTCNN [52], aligned by the eye landmarks from the P-Age-Face dataset as input
$X\in\mathbb{R}^{H\times W\times 3}$, where $H$ and $W$ represent height and
width. This stream learns additional facial features that are known to be good
indicators to further improve age estimation. In addition, a zero matrix is
provided as input to the supporting stream when there is no face information
available. More details of EfficientNet can be found in [41].

###  4.3 Attention Mechanism

A 768 feature vector from the main-stream, and a 1408 feature vector from
supporting-stream are passed through an MLP layer obtaining 512 dimensional
vectors for each stream after applying dropout and layer-normalization. The
two streams are concatenated using a multi-head-attention (MHA) module having
8 attention heads. Query $Q$ is obtained from the main-stream attended with
the Key and Value {$K,V$} pair acquired from the supporting-stream. A skip
connection is added between the main-stream output and the MHA module output
and passes through a linear layer with a softmax activation function to
predict the age class. This architecture allows our model to effectively
capture both facial and body dynamics information in the input for efficient
and accurate age estimation using videos.

##  5 Experiments

Our experiments are categorized into four folds. First, we compare different
video-based methods on the P-Age dataset, providing baseline results. Second,
we compare our AgeFormer with existing face-based SOTA methods on the P-Age-
Face dataset. Third, we evaluate the quality of our method in robust
situations such as privacy preservation (occluded, blurred, or blacked-out
faces). Finally, we cross-test AgeFormer on different challenging video
datasets such as THUMOS-14 [3], Smarthome [12] and Charades [37]. Further
details such as analysis of low-resolution, the effect of different sampling-
rates, and situations where our model did not perform well are provided in the
Supplementary Materials.

###  5.1 Experimental Details

All our models are implemented using the PyTorch library. We leverage the pre-
trained weights from Kinetics-400 and ImageNet to initialize the models for
the video and image streams, respectively. The main-stream receives 32
(temporal length) 224 $\times$ 224 resolution frames sampled at a stride of 4,
while the supporting-stream takes in a 288 $\times$ 288 resolution face crop.
The proposed architecture and video baselines are trained for 25 epochs with a
batch size of 16 using an AdamW optimizer with a learning rate of $3e^{-5}$
and an exponential learning rate scheduler with a gamma of 0.9 using a cross-
entropy loss. On the other hand, the face classifier benchmark training is
conducted for 200 epochs, with a batch size of 512 and a learning rate of
$2e^{-3}$. These hyper-parameter settings have been configured to ensure the
convergence of all models. Furthermore, all of the performance analysis
experiments involve only changing one parameter in the study (resolution,
face-blur, stride) at a time, while the rest of the model remains unchanged
and in inference mode.

Moreover, The P-Age dataset is split in a stratified way into train,
validation, and test sets that have a proportion of 76%, 12%, and 12%,
respectively.

Method | Acc. | Precision/Recall | F1-Score
---|---|---|---
X3D [15] | 77.61% | 0.80 / 0.76 | 0.77
Slow-Fast [16] | 78.11 % | 0.79 / 0.78 | 0.78
I3D [7] | 81.59% | 0.82 / 0.82 | 0.82
MViT [14] | 86.00% | 0.86 / 0.87 | 0.86
TimeSformer [6] | 87.56% | 0.88 / 0.88 | 0.88
Proposed | 89.55% | 0.90 / 0.90 | 0.90
Table 3: Baseline results of different video-based methods on the P-Age
dataset.

###  5.2 Results and Discussion

We divide our experiments into two categories. First, we create a baseline for
video-based age classification. We evaluate different action classification
architectures for the age-estimation task on the P-Age dataset. We compare
both 3D-CNNs and transformer-based methods for this task, as shown in Table 3.
Our AgeFormer achieves SOTA results. Furthermore, the results indicate that
video-based methods, specifically transformer architectures, are good enough
to estimate the age of an individual. Adding an additional face-branch as in
our AgeFormer, can further improve accuracy.

Method | Acc. | Precision/Recall | F1-Score
---|---|---|---
C3AE[50] | 37.70% | 0.21 / 0.30 | 0.21
MWR [36] | 43.39% | 0.40 / 0.42 | 0.39
Kim [23] | 57.67 % | 0.68 / 0.58 | 0.59
ResNet-18 [1] | 61.38 % | 0.68 / 0.60 | 0.61
Levi [25] | 70.37% | 0.70 / 0.69 | 0.69
EfficientNet B2 | 53.97% | 0.66 / 0.55 | 0.52
Proposed* | 77.25% | 0.79 / 0.77 | 0.77
Proposed | 89.42% | 0.89 / 0.90 | 0.89
Table 4: Comparison of existing face-classifiers with proposed video-based
classifiers on P-Age-Face dataset. * means a video-based model (only main-
stream) that utilizes the whole body with blurred faces as input. EfficientNet
B2 results are from our supporting-stream (face model). Figure 3: Spatio-
temporal attention visualization of AgeFormer on P-Age dataset. Figure 4:
Comparison of SOTA face-based age classifier [25] with our proposed video-
based method. Our method takes a video input with blurred faces.

Second, we utilize the Face variant of the P-Age dataset to make use of the
SOTA face-based methods for comparison with our AgeFormer. AgeFormer achieves
new SOTA results compared to face-based methods on the P-Age dataset. We
compare our method with face-based methods in two ways as noted in Table 4.
The results indicate that our proposed method is superior to existing methods,
even when the faces are blurred. This validates our idea of using space-time
dependencies of the entire body. A more qualitative comparison between
Ageformer and SOTA method is provided in the Supplementary Materials.

###  5.3 Important Cues for Predicting Age

We visualize the divided space-time attention weights of the video backbone to
provide insight into which part of the input is the most important when
predicting age. The body of a person (limbs, chest, torso) and its dynamics
hold important cues to differentiate one age group from the other. Our model
learns these cues as shown in the Figure 3. Using the attention roll-out
method, we can visualize that the model is attentive to not just the face, but
also other body parts, especially in the case where the face is not visible
Figure 3 (second row, first example). Furthermore, the prediction of age from
the body dynamics of a person has been validated in unseen videos in Figure 5
(Smarthome), where the faces are barely visible, but our model can accurately
predict age in such situations.

Augmentation | Acc. | Pre./Recall | F1-Score
---|---|---|---
face-blur | 77.61 % | 0.80 / 0.78 | 0.78
face blacked-out | 76.88% | 0.79 / 0.76 | 0.76
Table 5: Effectiveness of the proposed approach in the situation of extreme
privacy preservation in the P-Age dataset. Figure 5: Each row depicts eight
randomly selected videos from a specific dataset, the annotated label is
predicted by AgeFormer. We observe that despite having never been trained on
them, the model performs reasonably well. Figure 6: (a) illustrates the
confusion matrix of the proposed method and (b) shows bad cases of our
proposed method. White color indicates ground-truth, and green indicates the
predicted label.

###  5.4 Privacy Preservation

In this section, we discuss the privacy preservation aspect of our method. In
real-world applications such as activities of daily living, surveillance, and
medical evaluations, most of the data are sensitive to privacy concerns, where
faces are often obscured, blurred, or blacked-out. Furthermore, in cases such
as autism assessment for children [4], the movements of the child cannot be
controlled, and therefore their faces are not visible all the time, or, in
other words, the upper-body is highly occluded. In such cases, existing face-
based methods do not categorize the age of the participants.

Therefore, our AgeFormer is superior to prior face-based methods by
incorporating spatio-temporal features of the entire body. This helps the
model classify the age more efficiently, as demonstrated in Figure 4, even in
scenarios where the faces are blurred-out. In these experiments, we use only
the main-stream of our architecture. We further evaluate our model on two
scenarios, i) blurred, and ii) blacked-out faces. The results in Table 5
validate our idea of using a video-based model with the whole body as input to
preserve privacy.

###  5.5 Qualitative Cross Dataset Performance

In this section, we evaluate the generalizability of our proposed AgeFormer.
We perform a qualitative cross-dataset performance analysis on real-world
video (action recognition) benchmarks such as THUMOS-14 [3], Smarthome [12],
and Charades [37] without training on these datasets. These datasets are
selected because together they contain people from all age groups with
significant variations in facial expressions, poses, lighting conditions, and
occlusions. To this extent, we first randomly sample 30 different videos from
each dataset and perform person detection and tracking to obtain the various
person videos (tracklet). Following that, we use AgeFormer to predict the age
class of each person (tracklet). As there are no ground-truth labels available
for these datasets, we assign only the predicted labels to each person, as
illustrated in Figure 5.

Our results show that, although the model was never trained on low-resolution
data, the proposed approach is resilient in most scenarios, except for a few
cases where the video quality is severely degraded (Figure 5 (1,1)), extreme
low light conditions (Figure 5 (2,8)), or the apparent age of the person is
uncertain and falls between two categories (Figure 5 (3,8)). If a facial crop
is achievable, the latter issue could be potentially resolved by leveraging
the facial feature context of the supporting stream, leading to more
consistent predictions and better performance. In general, the proposed two-
stream method is an effective approach with broad generalizability by
extracting the relevant contextual features from the inputs.

###  5.6 Bad Cases

The proposed method may not perform well in tricky situations, where it is
even difficult for humans to predict the age of an individual, as shown in
Figure 6b. The confusion matrix in Figure 6a demonstrates that the model has a
hard time classifying adults. In such difficult cases, the proposed model
mostly confuses the teen and adult classes. Furthermore, the visual appearance
of the individuals in the first column of Figure 6b illustrates why the model
incorrectly predicts age.

##  6 Conclusion

Age-classification of an individual in real-world situations is a challenging
task. In this paper, we propose a new direction to predict the age of an
individual in a video. Our novel video-based model named AgeFormer achieves a
precise age classification in challenging situations. The proposed
architecture utilizes spatio-temporal information of the dynamics of the
entire body dominating face-based methods for age classification. Experiments
illustrate that video-based models are robust against situations where the
face is obscured (privacy preservation) to predict the age of an individual.
Additionally, we built the first video dataset (P-Age) for age classification,
opening the door for researchers to explore video-based methods for age
classification.

## Acknowledgments

The COFUND BoostUrCAreer program has received funding from the European
Union’s Horizon 2020 research and innovation program under the Marie Curie
grant agreement No 847581, from the Région SUD Provence-Alpes-Côte d’Azur and
IDEX UCAjedi.

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# Posterior predictive distributions of neutron-deuteron cross sections

Sean B. S. Miller  Andreas Ekström  Christian Forssén  Department of Physics,
Chalmers University of Technology, SE-412 96 Gothenburg, Sweden

###### Abstract

We quantify the posterior predictive distributions (PPDs) of elastic neutron-
deuteron ($nd$) scattering cross sections using nucleon-nucleon ($NN$)
interactions from chiral effective field theory ($\chi$EFT) up to and
including next-to-next-to-next-to-leading order (N3LO). These PPDs quantify
the spread in $nd$ predictions due to the variability of the low-energy
constants (LECs) inferred from $NN$ scattering data. We use the wave-packet
continuum discretization method to solve the Alt-Grassberger-Sandhas form of
the Faddeev equations for elastic scattering. We draw 100 samples from the
PPDs of $nd$ cross sections up to 67 MeV in scattering energy, i.e., in the
energy region where the effects of three-nucleon forces are expected to be
small. We find that the uncertainty about $NN$ LECs inferred from $NN$
scattering data, when assuming uncorrelated errors, does not translate to
significant uncertainty in the low-energy $nd$ continuum. Based on our
estimates, the uncertainty of $nd$ predictions are dominated by the $\chi$EFT
truncation error, at least below N3LO. At this order, the 90% credible
interval of the PPD and the truncation error are comparable, although both are
very small on an absolute scale.

##  I Introduction

Chiral effective field theory ($\chi$EFT) [1, 2, 3] promises a systematically
improvable description of the nuclear interaction grounded in the symmetries
of low-energy quantum chromodynamics. Two-nucleon ($NN$) and three-nucleon
($NNN$) interactions from $\chi$EFT are used extensively in modern ab initio
predictions of atomic nuclei and nuclear matter, see, e.g., Refs. [4, 5, 6]
for recent overviews. To make quantitative predictions of the properties of
nuclear systems, the numerical values of the low-energy constants (LECs) that
govern the strengths of the pion-nucleon ($\pi N$) and nucleon-contact
couplings must first be inferred from low-energy data. For this, the Bayesian
approach to statistics [7] provides a natural framework since it yields a
(posterior) probability density function (PDF) that quantifies our uncertainty
about the values of the LECs. Propagating this uncertainty when making
theoretical predictions amounts to averaging the distribution of predictive
samples over the LEC posterior PDF. The result of this is called a posterior
predictive distribution (PPD). This type of distribution sits at the center of
the scientific process whereby we try to predict future data based on previous
data and theory.

There are existing efforts to quantify Bayesian PPDs for various nuclear
observables, e.g., $NN$ scattering cross sections [8] and scattering lengths
[9], few-nucleon [10] and many-nucleon [11, 12, 13] energies, radii, and
decays, as well as nuclear mass models [14, 15]. These probability
distributions quantify our degree-of-belief, and facilitate a meaningful
comparison with experimental data. For example, the PPD finds use in model
checking [7], such as posterior predictive checks. There one simulates data,
using a fitted model, and compares to observed data. The simulated data
corresponds to draws from the PPD and it should look roughly like the observed
data if the model did indeed contain all relevant physics and there has been a
sufficient amount of calibration data.

In this work, we sample the PPDs of selected neutron-deuteron ($nd$)
scattering cross sections arising from the variability of the LEC posterior
when conditioned on $NN$ scattering data. We use $\chi$EFT descriptions of the
$NN$ interaction at all orders up to next-to-next-to-next-to-leading order
(N3LO) in Weinberg power counting. To the best of our knowledge there exists
only frequentist statistical analyses encompassing a subset of nucleon-
deuteron ($Nd$) scattering cross sections and scattering lengths [16, 17, 18],
for which various estimates for dispersion have been quantified. Our analysis
is rooted in Bayesian methodology and therefore provides probability densities
for the predicted observables of interest. As such, the results of this work
facilitates a quantitative measure of the predictive power in the low-energy
$NNN$ continuum using $\chi$EFT interactions carefully calibrated using $NN$
scattering data. This work is part of an ongoing effort towards a full
Bayesian analysis of $\chi$EFT conditioned also on experimental data in the
$Nd$ continuum [19].

To sample the PPDs of elastic $nd$-scattering cross sections, we repeatedly
solve the Alt-Grassberger-Sandhas [20] (AGS) form of the Faddeev equations
using the wave-packet continuum discretization (WPCD) method [21, 19]. This
method is parallelizable with respect to the scattering energy, denoted with
$E_{\text{Lab}}$, in the laboratory frame of reference. Therefore, it is
particularly suitable for sampling PPDs across a range of $E_{\text{Lab}}$
values. Still, the collection of samples from the PPDs is limited by the
number of times we can solve the AGS equation. For this reason, we currently
neglect $NNN$ forces (3NFs) and focus our analysis on cross sections and
polarization observables with $E_{\text{Lab}}\leq 67$ MeV, for which $NN$-only
models typically perform well [22, 23, 24, 16]. The low-energy vector
analyzing power, $A_{y}$, is a possible exception to this statement and we
therefore place a special focus on the analysis of this polarization
observable.

In addition to the inherent uncertainty of inferred LEC values, there are also
other sources of theoretical uncertainty. The model discrepancy due to the
omission of higher chiral orders is an obvious one. Neglecting this
uncertainty can lead to biased and over-confident inferences and predictions
[25]. Fortunately, $\chi$EFT is designed to be an order-by-order improvable
description of the nuclear interaction, and as such the theory itself provides
valuable information about the magnitude of the truncation error [26]. Indeed,
there exists several efforts to quantify the truncation error in effective
field theory predictions of nuclear systems, see, e.g., Refs. [27, 28, 29, 30,
31, 10, 8]. Although our focus is to quantify the PPDs of $nd$ scattering
observables due to variability in the $NN$ LECs, we will also contrast our
findings with estimates of the truncation error.

In Sec. II we define the general structure of the PPDs we sample in this work.
In Sec. III we present the essential elements of the WPCD method we use to
produce elastic $nd$ cross sections. In Sec. IV we present the strategy for
sampling the PPDs, with particular focus on achieving computational speedup,
and the results of the sampling. We also compare the degree-of-belief
intervals of the PPDs with some of the other components of the total error
budget; the $\chi$EFT truncation error in particular. We end with a summary
and outlook in Sec. V.

##  II Setting up the posterior predictive distribution

The PPD is a PDF $\text{pr}(y|D,M,I)$ for a quantity $y$ as predicted by a
model $M$. This distribution quantifies the uncertainty about $y$ given
previous data $D$ and any other assumptions or information $I$. Here, we focus
on the uncertainty of the numerical values of the LECs, denoted
$\vec{\alpha}$, present in the underlying $\chi$EFT $NN$ interaction. As such,
we must marginalize over the LECs by evaluating the following integral

$\displaystyle\begin{split}\text{pr}(y|D,M,I){}&=\int_{\Omega}\text{pr}(y|\vec{\alpha},D,M,I)\text{pr}(\vec{\alpha}|D,M,I)\,d\vec{\alpha}\\\
{}&\propto\int_{\Omega}y(\vec{\alpha})\text{pr}(\vec{\alpha}|D,M,I)\,d\vec{\alpha}.\end{split}$
(1)

In the second line we introduced a short-hand $y(\vec{\alpha})$ for a
deterministic model prediction given numerical values for $\vec{\alpha}$ from
some parameter domain $\Omega$. We also used that $y$ is conditionally
independent of $D$. The proportionality indicates that we are only interested
in the width and shape of the PPD, and not the overall normalization constant.

We will refer to the $\chi$EFT description of the $NN$ interaction at a chiral
order $\nu$ as a “model”, and denote this as $M_{\nu}$. The chiral orders are
defined according to Weinberg power counting with $\nu=0,2,3,4$, and as is
common, we refer to them to as leading order (LO), next-to-leading order
(NLO), next-to-next-to-leading order (N2LO), and N3LO, respectively. The
values of $\vec{\alpha}$ depend on the chiral order $\nu$, but to simplify
notation we do not index $\vec{\alpha}$ by $\nu$.

The PPD is a probabilistic generalization of the familiar point-estimate value
$y_{\star}=y(\vec{\alpha}_{\star})$, obtained by evaluating the model
$M_{\nu}$ at some preferred parameter value $\vec{\alpha}_{\star}$, such as a
local parameter-optimum. We will in some cases resort to evaluating the PPD at
the maximum a posteriori (MAP) value of the LEC posterior

$\vec{\alpha}_{\star}\equiv\underset{\vec{\alpha}}{\mathrm{argmax}}\,\,\text{pr}(\vec{\alpha}|D,M_{\nu},I).$
(2)

Note that the PPD does not necessarily attain its maximum for
$\vec{\alpha}_{\star}$. Indeed, the evaluation of $y(\vec{\alpha})$, through
the AGS equation, is neither linear nor monotonic.

Evaluating the integral in Eq. (1) requires knowledge about the PDF,
$\text{pr}(\vec{\alpha}|D,M_{\nu},I)$. We utilize the available LEC posteriors
up to and including N3LO published in Ref. [9]. These posteriors were sampled
using Hamiltonian Monte Carlo (HMC) while accounting for uncorrelated
$\chi$EFT truncation errors, and were conditioned on the Granada database [32,
33] of $NN$ scattering cross sections for scattering energies
$E_{\text{Lab}}\leq 290$ MeV. The leading neutron-neutron ($nn$) isospin-
breaking LEC was inferred using an empirical value for the $nn$ scattering
length in the ${}^{1}S_{0}$ partial-wave channel. We note that other methods
accounting for correlated $\chi$EFT truncation errors exist, see, e.g., Ref.
[34], which may change the inferred, and rather narrow, distributions of LEC
values we use here.

The HMC algorithm is particularly well-suited for sampling high-dimensional
PDFs and yields virtually uncorrelated draws from
$\text{pr}(\vec{\alpha}|D,M_{\nu})$. A detailed analysis [8] suggests that the
HMC chains we employ in this work to represent the LEC posteriors are
sufficiently converged at all orders, unimodal, and rather concentrated in
parameter space. As such, we have in-depth knowledge of the location of the
posterior mass, which helps tremendously when evaluating the integral in Eq.
(1).

##  III Wave-packet continuum discretization

In this section we summarize the WPCD method [21] for solving the AGS equation
in momentum space. Our results are based on the implementation presented in
Ref. [19]111The implementation, named “Tic-tac”, is available under a GNU
open-source license (GPLv3) on https://github.com/seanbsm/Tic-tac. The AGS
equation for $nd$ scattering, without 3NFs, can be written as

$\hat{U}=\hat{P}\hat{G}_{0}^{-1}+\hat{P}\hat{t}_{1}\hat{G}_{0}\hat{U}\>,$ (3)

where $\hat{U}$ is the transition matrix between asymptotic scattering states,
$\hat{G_{0}}\equiv\frac{1}{E-\hat{h}_{0}\pm i\epsilon}$ is the resolvent of
the free $NNN$ Hamiltonian $\hat{h}_{0}$, $E$ is the total energy,
$\hat{t}_{1}$ denotes the scattering $T$-matrix for the pair-system $(23)$ as
written in standard odd-man-out notation, and $\hat{P}\equiv 2\hat{P}_{123}$
where $\hat{P}_{123}$ is the permutation matrix acting on partially-
antisymmetric $NNN$ states222There is an erroneous extra term $+1$ in the
definition of $\hat{P}$ in Ref. [19].. The large dimensionality of the $NNN$
Hilbert space makes it challenging to apply matrix-inversion type methods to
solve Eq. (3). Instead, one usually resorts to expanding the AGS equation in a
Neumann series that is subsequently re-summed using a Padé approximant [35] to
handle the divergence originating from the integral-kernel
$\hat{G}_{0}\hat{v}_{1}$ with Weinberg eigenvalues [36] outside the unit
circle.

It is well understood how to obtain converged solutions for $U$ in a standard
plane-wave basis, see, e.g., Ref. [37]. In this basis, $\hat{U}$ is obtained
for a specific value of the on-shell energy $E$, and the resolvent
$\hat{G_{0}}$ and $NN$ $T$-matrix $\hat{t}_{1}$ depend explicitly on $E$. This
dependency inflicts several complications such as moving singularities in the
resolvent operator, and a requirement for antisymmetrizing $NN$ $T$-matrices
at many energies when evaluating the AGS integral kernel, which is typically
handled using splines [38].

In this work, we use the WPCD method [21] for solving the AGS equation. This
is one of many bound-state approaches [39] for describing scattering
processes. In WPCD, we discretize the continuum using a wave-packet basis.
Doing so simplifies the numerical analysis of the AGS equation. First, one can
derive a closed-form expression of the channel-resolvent, treating the
associated singularities analytically. Second, the $P$-matrix has no need for
splining. Third, it factorizes the on-shell energy dependence out of the
matrix multiplications associated with the terms of the Neumann series
expansion, providing significant speedup of the most time-consuming parts of
the numerical solution.

As a downside, the WPCD method entails large matrix dimensionalities compared
with the plane-wave representation. However, scattering amplitudes can be
calculated at multiple scattering energies with minor extra computational cost
per energy. This makes WPCD particularly suitable for sampling Bayesian PPDs
across ranges of energies. In fact, we find that calculating scattering
amplitudes at multiple scattering energies only doubles the computational cost
compared to computing the amplitude at a single energy [19]

We define a wave packet $|x\rangle$ as a finite integral of continuum states
$|p\rangle$, e.g., plane-wave states, within a momentum “bin”
$\mathcal{D}\equiv[p,p+\Delta p]$,

$|x\rangle\equiv\frac{1}{\sqrt{N}}\int_{\mathcal{D}}f(p^{\prime})|p^{\prime}\rangle\>p^{\prime}\>dp^{\prime}\>,$
(4)

where $f(p)$ is a weighting function and $N$ is the normalization constant. An
$A$-body wave packet can be straightforwardly defined using wave-packet
discretization for each Jacobi coordinate. A $NNN$ wave packet is given by
$|X\rangle\equiv|x\rangle\otimes|\bar{x}\rangle$, where $|x\rangle$
corresponds to the pair-system $p$-momentum and $|\bar{x}\rangle$ corresponds
to the spectator $q$-momentum.

The eigenstates of the $NN$ Hamiltonian $\hat{h}_{1}$ in a (plane-wave) wave-
packet basis can be used to approximate “scattering” $NNN$ wave packets rather
well. In this basis, it is also possible to evaluate the channel-resolvent
$\hat{G}_{1}\equiv\frac{1}{E-\hat{h}_{1}\pm i\epsilon}$ analytically.
Furthermore, using that $\hat{t}_{1}\hat{G}_{0}\equiv\hat{v}_{1}\hat{G}_{1}$
and $\hat{G}_{0}^{-1}=\hat{v}_{1}$ (on-shell), we can rewrite Eq. (3) to
obtain

$\hat{U}=\hat{P}\hat{v}_{1}+\hat{P}\hat{v}_{1}\hat{G}_{1}\hat{U}\>,$ (5)

where $\hat{U}$ now depends on $E$ only via $\hat{G}_{1}$. This is the
starting point for solving the AGS equation in the WPCD method. Here, as in
Ref. [19], we use an equal number of wave packets, $N_{\text{WP}}$, to
discretize the $p$ and $q$ continua, yielding matrices in Eq. (5) that scale
in size as $\mathcal{O}(N_{\text{WP}}^{4})$. We find that the runtime of the
code follows this quartic scaling with $N_{\text{WP}}$ quite closely. Note,
however, that the calculations at N3LO are $\sim 10\%$ more costly since the
Padé resummation of the Neumann series typically requires more terms to
converge.

##  IV Evaluating posterior predictive distributions

We sample the PPD of a scattering observable by evaluating Eq. (1)
numerically. This is done by computing the $nd$ scattering observable of
interest for a finite set of LEC values drawn from the posterior PDF,
$\text{pr}(\vec{\alpha}|D,M_{\nu})$. In practice, we use the Markov chains
obtained in Ref. [9].

For every sample that we draw from the PPD we must solve the AGS equation.
Fortunately, with the WPCD method we get access to all scattering cross
sections at all angles and energies without any significant computational
overhead. Also, since the permutation operator $\hat{P}$ does not depend on
the LECs, we only have to compute this once and re-use it throughout the
sampling process. However, we have to setup the Neumann series for every new
sample, and this is the most time-consuming part.

In all calculations done here, we use a spin-angular basis of $NNN$ partially-
antisymmetric partial-waves with total angular momentum $\mathcal{J}\leq
17/2$, using both parities, and using $NN$ total angular momentum $J\leq 3$.
We also explicitly account for the charge dependence of the strong $NN$
interaction in the 1S0 channel. This state space provides sufficiently
converged $U$-matrix elements for $E_{\text{Lab}}\leq 100$ MeV when using the
chiral potentials defined in Ref. [8, 9]. Note that our study is limited to
$E_{\text{Lab}}\leq 67$ MeV due to the omission of 3NFs. It has been shown
that, at low scattering energies, the scattering amplitudes are likely
dominated by $NN$ forces [22, 23, 24, 16].

We discuss our general strategy to quantify the PPD in Sec. IV.1, present
results for the PPDs of the differential $nd$ cross section in Sec. IV.2,
relate this to estimates of the $\chi$EFT truncation errors in Sec. IV.3, and
discuss spin-polarization observables, focusing on $A_{y}(n)$, in Sec. IV.4.

###  IV.1 Trading wave-packets for computational speedup

In the limit $N_{\text{WP}}\rightarrow\infty$, the WPCD results converge
towards the results from an exact calculation, e.g., a continuous plane-wave
solution [37] of Eq. (5). However, the computational cost increases
quartically with $N_{\text{WP}}$, and larger values for $N_{\text{WP}}$ will
significantly increase the PPD sampling cost. Balancing cost and accuracy, we
found it sufficient to draw $N=100$ samples from each PPD that we study, since
we are quantifying univariate distributions. Also, we noticed that the shapes
and widths of the PPDs studied here did not change visibly when varying
$N_{\text{WP}}$, and as such we could limit ourselves to $N_{\text{WP}}\leq
75$ and extrapolate to larger values. This will be discussed in the next
section.

At present, using $N_{\text{WP}}=75$, it takes roughly 12 node-hours (384
core-hours333Using two Intel Xeon Gold 6130 CPUs per node, amounting to 32
cores per node.) to compute all necessary scattering amplitudes at $\sim$50
scattering energies below $100$ MeV for a single configuration of values for
the LECs at a specific chiral order $\nu$. This translates to roughly 150k
core-hours to compute all scattering amplitudes for 100 different LEC values
at four chiral orders. The same calculation with $N_{\text{WP}}=150$ would be
16 times more expensive and cost roughly 2.5M core-hours. To monitor the
reduced method accuracy at $N_{\text{WP}}=75$, we repeat the PPD sampling,
with copies of the same LEC samples, at every chiral order with
$N_{\text{WP}}=30$ and $50$. We also use a restricted set of 10 posterior
samples with $N_{\text{WP}}=100$. In addition, we evaluate the PPD at the MAP
value $\vec{\alpha}_{\star}$ of the LEC PDF using
$N_{\text{WP}}=30,\>50,\>75,\>100$, and $150$. The $N_{\text{WP}}=75,150$ MAP
predictions will be used for extrapolation in the next section.

We had little cost-related reason to restrict calculations to
$E_{\text{Lab}}\leq 67$ MeV. Instead, we computed the on-shell $U$-matrices at
all wave-packet $NN$ Hamiltonian eigenenergies below $E_{\text{Lab}}=100$ MeV,
which was roughly two thirds of the wave-packet basis size. Between these
energies we perform linear interpolation of the $U$-matrix elements to
virtually any $E_{\text{Lab}}<100$ MeV. Consequently, we obtained 100 samples
from the PPD of any elastic scattering cross section at every order up to, and
including, N3LO. Of course, with the neglect of 3NFs, we consider our
predictions above $E_{\text{Lab}}=67$ MeV to be incomplete and have therefore
been omitted from the present study. Nonetheless, they allowed us to check on
the width and shape of PPDs all the way to $E_{\text{Lab}}=100$ MeV.

Although the HMC-chains of LEC posterior samples are virtually uncorrelated,
this does not imply that ensuing samples from the $nd$ cross section PPD are
equally uncorrelated. Unfortunately, a chain of 100 samples is typically too
short to quantify, e.g., an integrated autocorrelation time or reliably
determine the autocorrelation itself. Nevertheless, an inspection of the trace
plots of the PPD samples, as shown in Fig. 1, does not indicate any hints of
strong correlation between samples.

Figure 1: Trace plot of the differential cross section PPD at N2LO, for
$E_{\text{Lab}}=12$ MeV and $\theta_{\text{c.m.}}=120$ degrees, using 100
samples from the HMC-chain of samples from the LEC posterior at this order.

In the event of observing strongly correlated samples, the information content
of the PPD chain, as measured by its effective sample size, will drop
inversely to the integrated autocorrelation time and we would have to increase
the number of samples accordingly [8]

###  IV.2 The differential cross section

The convergence of the differential elastic $nd$ cross section at
$E_{\text{Lab}}=12$ MeV with respect to $N_{\text{WP}}$ is shown in Fig. 2.

Figure 2: The differential $nd$ cross section at $E_{\text{Lab}}=12$ MeV
computed using the MAP values for the LECs at chiral orders from LO to N3LO.
The dotted, dash-dotted, dashed, and solid lines at each order show the
results obtained from the WPCD methods with $N_{\text{WP}}=50,\>75,\>100$, and
$150$, respectively.

Clearly, with $N_{\text{WP}}\approx 100$, the results begin to stabilize, at
least for subleading orders. The somewhat reduced convergence rate for the LO
results might be caused by the rather coarse wave-packet representation of the
$NN$ potential for low relative momenta [40]. To remedy this one should either
re-distribute the discretization boundaries to improve the coverage of the
lower momentum region, or simply increase $N_{\text{WP}}$ if possible. Since
we detect a sufficient convergence at subleading orders, we opt for keeping
the discretization mesh the same throughout all calculations and at all chiral
orders.

Next, we study the convergence of the PPD with respect to $N_{\text{WP}}$. In
Fig. 3 we show a histogram of 100 samples of the PPD of the $nd$ differential
cross section at $E_{\text{Lab}}=12$ MeV and $\theta_{\text{c.m.}}=120$
degrees at N2LO using $N_{\text{WP}}=30,\>50$, and 75, as well as 10 samples
at $N_{\text{WP}}=100$ and the location of the MAP prediction using
$N_{\text{WP}}=150$.

Figure 3: The PPDs of the differential $nd$ cross section at
$\theta_{\text{c.m.}}=120$ degrees and $E_{\text{Lab}}=12$ MeV using the N2LO
$NN$ interaction. The three different distributions shown are, from left to
right, for $N_{\text{WP}}=100,\>75,\>50$, and 30, with $N=10,\>100,\>100$, and
100 samples, respectively. For comparison, we also indicate with vertical
lines the locations of the cross section for the MAP LEC point obtained in a
WPCD calculation based on $N_{\text{WP}}=150,\>100,\>75,\>50$, and 30. The
means of the distributions coincide almost with the MAP predictions.

The PPDs based on $N_{\text{WP}}=30,\>50$, and $75$ are very similar in terms
of shape and width. In fact, for all observables that we study in this
work444We study the differential cross section, $\frac{d\sigma}{d\Omega}$, the
neutron vector analyzing power, $A_{y}(n)$, and the spherical tensor analyzing
powers $iT_{11}$, $T_{20}$, $T_{21}$, and $T_{22}$, at angles
$\theta_{\text{c.m.}}=60$ and 120 degrees and scattering energies
$E_{\text{Lab}}=10$-$12,\>35$-$36$, and 65-67 MeV., the width and shape of the
PPD remains approximately constant as we vary $N_{\text{WP}}$, and the main
effect is a shift of the entire distribution. Therefore, we shift the mean of
the samples obtained with $N_{\text{WP}}=75$ using the difference between the
MAP predictions obtained with $N_{\text{WP}}=75$ and $N_{\text{WP}}=150$. This
makes a comparison with experimental data more meaningful.

We did not detect a robust exponential or power-law convergence pattern with
respect to $N_{\text{WP}}$ and leave further analysis of the
$N_{\text{WP}}$-convergence and the WPCD method uncertainty to future work. As
such, there might be additional corrections to the PPDs when using
$N_{\text{WP}}>150$ that we do not account for. However, assuming that the
widths and shapes of the PPDs remain unchanged, our main conclusions in this
work will not be affected.

After shifting the differential cross section obtained with $N_{\text{WP}}=75$
to $N_{\text{WP}}=150$ we obtain the result shown in Fig. 4.

Figure 4: The PPD of the $nd$ differential cross section at
$E_{\text{Lab}}=12,\>36$, and 65 MeV at all orders up to N3LO in the $NN$
interaction. The legends display the average 90% credible intervals (HPDI),
see text for details. The experimental data (markers) are retrieved from the
EXFOR database (at $E_{\text{Lab}}=12$ and 36 MeV) and Ref. [41] (at
$E_{\text{Lab}}=64.5$ MeV). Diamond (cyan) and circle (colorless) markers
represent $nd$ and $pd$ cross sections, respectively.

At all chiral orders and energies we study, the PPD is rather narrow. At LO,
the PPD width is comparable to the experimental uncertainty, while at
subleading orders the experimental uncertainty is typically greater than the
width of the PPD.

To quantify the width of the PPDs, we compute the 90% highest posterior
density interval (HPDI), normalize it to the mean of the PPD, and average over
$\theta_{\text{c.m.}}$. This way, we find that the average HPDI for the
differential cross section at $E_{\text{Lab}}=12$ MeV is, 5.7%, 2.3%, 0.7%,
and 0.5% at LO, NLO, N2LO, and N3LO, respectively. The decreasing values
reflects the increasingly narrow LEC posterior densities obtained at higher
chiral orders [8, 9]. Moving to higher scattering energies we find that the
PPDs remain very narrow still. Apart from LO, the average HPDI values are
comparable to frequentist estimates of dispersion quantified in Ref. [17],
where a similar increase in uncertainty was noted at higher scattering
energies.

Recently it was shown that N2LO potentials with 3NFs yield an excellent
description of differential cross section data [24]. It was suggested in Ref.
[42] that 3NFs are necessary to reproduce the differential cross section
minimum in the vicinity of $E_{\text{Lab}}=65$ MeV. Here, however, we see
similar reproduction of data at NLO and N2LO without 3NFs. Going to N3LO, the
reproduction of experimental data deteriorates. As shown in Ref. [9], the 3H
and 3He ground state energies and radii at N3LO are also markedly worse
compared to N2LO. This trend is a testament to the importance of inferring
LECs in the $NN$\- and $NNN$-sectors of $\chi$EFT simultaneously [43].

We conclude, based on the inference of $NN$ LECs made in Ref. [9], that the
discrepancies between experimental low-energy $nd$ cross section data and
theoretical predictions are not due to the uncertainties stemming from the LEC
variability. Given the very narrow PDFs for the LECs, an opposite finding
would be a testament to a tremendous fine tuning of scattering observables in
the $NNN$ continuum relative to the $NN$ continuum.

###  IV.3 The EFT truncation error

The truncation of the $\chi$EFT expansion used to describe the nuclear
interaction leads to a model discrepancy referred to as an EFT truncation
error. Following Ref. [26], we assume that the theoretical prediction at
chiral order $\nu$ for some observable $y$ can be written as

$y^{(\nu)}(\vec{\alpha};\vec{x})=y_{\text{ref}}(\vec{x})\sum_{k=0}^{\nu}c_{k}(\vec{\alpha};\vec{x})Q^{\nu}(\vec{x})\>,$
(6)

where $\vec{x}$ denotes the kinematic variables $E_{\text{Lab}}$ and
$\theta_{\text{c.m.}}$ and $y_{\text{ref}}$ is a reference value for the
observable in question. This expression renders the expansion coefficients
$c_{k}$ dimensionless quantities, which we also expect to be of natural size,
i.e., $c_{k}\sim\mathcal{O}(1)$. We assume a $\chi$EFT expansion parameter of
the form

$Q=\text{max}\left(\frac{q}{\Lambda_{b}},\frac{m_{\pi}}{\Lambda_{b}}\right),$
(7)

and set the $\chi$EFT breakdown scale to $\Lambda_{b}=600$ MeV as in Ref. [9]
from where we also obtain the LEC posteriors. We set the c.m. momentum, $q$,
according to the kinetic energy, $E_{\text{Lab}}$, of the incoming nucleon.
The $\chi$EFT truncation error, $\delta y_{\nu}$, is the expected magnitude of
the sum of contributions from terms beyond the order $\nu$. Under the
assumption of having independent and normally distributed expansion
coefficients, $c_{k}$, it is shown in, e.g., Ref. [34], that $\delta y_{\nu}$
is also normally distributed and given by

$\delta
y_{\nu}\sim\mathcal{N}\left(0,y_{\text{ref}}^{2}\frac{Q^{2(\nu+1)}}{1-Q^{2}}\bar{c}^{2}\right),$
(8)

where $\bar{c}^{2}$ denotes the variance of the expansion coefficients. Thus,
knowing $\bar{c}^{2}$ enables us to quantify the (variance of the) $\chi$EFT
truncation error. For this purpose, we follow the procedure of, e.g., Ref. [8]
and employ the root-mean-square (RMS) value of order-by-order differences to
estimate $\bar{c}^{2}$. The order-by-order differences are computed from the
mean values of the PPDs at each order $\nu$, thus averaging over a possible
LEC dependence.

We wish to compare the magnitude of the $\chi$EFT truncation error with the
theoretical error in $y$ stemming from the uncertainty about the numerical
values of the LECs. Let us take the differential cross section at
$E_{\text{Lab}}=12$ MeV as an example and inspect it closer. Limiting
ourselves to this low value of $E_{\text{Lab}}$, the effect of 3NFs are
expected to be small [42]. Therefore, we retain the expansion in Eq. (6) and
use Eq. (8) to quantify the $\chi$EFT truncation error. We set
$y_{\text{ref}}$ to the LO prediction. At this scattering energy, we also have
$Q=m_{\pi}/\Lambda_{b}\approx 0.23$. An RMS estimate from the expansion
coefficients at $\theta_{\text{c.m.}}=30,\>90$ and $150$ degrees (omitting LO
results due to their role in the definition of $y_{\text{ref}}$) gives
$\bar{c}=14.8$. This is a fairly unnatural value which arises from an
oscillating convergence when including higher chiral orders.

The PPDs due to the LEC variabilities and the $\chi$EFT truncation errors are
compared in Fig. 5.

Figure 5: Comparison of the PPD due to LEC variability and an estimate of the
normally distributed $\chi$EFT truncation error for the differential cross
section for NLO (green), N2LO (blue), and N3LO (red). The $\chi$EFT truncation
is shown for two different variances; $\bar{c}=14.8$ (solid line) and
$\bar{c}=1$ (dashed line). To set the absolute scale, we included the
experimental measurement (gray) from Ref. [44]

Clearly, the truncation error (solid lines) is typically much greater than the
error due to the uncertain values of the $NN$ LECs (histograms) up to an
including N2LO. Therefore, we find it unnecessary to account for a possible
LEC variability in the expansion coefficients in Eq. (6). At N3LO, the two
errors are becoming comparable. However, at this order, both of the errors are
tiny, $\lesssim 4\%$, on an absolute scale. In fact, they are both smaller
than typical experimental errors, indicated as the gray area in Fig. 5. In
addition to the RMS estimate of $\bar{c}^{2}$ we also show the truncation
error (dashed line) based on a naturalness assumption where we set
$\bar{c}^{2}=1$. In this limit, the two errors become comparable for this
observable already at NLO.

At higher energies, we see in Fig, 4 that the predictions at N3LO deviates
from the ones at NLO and N2LO. When analyzing the truncation errors at
$E_{\text{Lab}}=36$ MeV, we obtain $\bar{c}=65.1$, which signals the presence
of an unnaturally large contribution in the $\chi$EFT expansion. We find that
omitting the shift between N2LO and N3LO has a significant impact and yields a
more reasonable value of $\bar{c}=15.1$. Doing the same at $E_{\text{Lab}}=12$
MeV yields $\bar{c}=11.3$, i.e., a relatively small change from when including
the shift. The truncation error is expected to increase with the on-shell
energy, and thus it should become greater than the LEC uncertainty for
$E_{\text{Lab}}>12$ MeV, but we leave a more detailed study for future work.

###  IV.4 Spin-polarization observables

There are many different possibilities to form observables related to spin-
polarization in the initial and/or final states of the $Nd$ reactants [45].
The fine details of the angular dependence of these observables can depend
sensitively on the spin structure of the $NN$ and $NNN$ interactions. A well-
known example is the low-energy vector analyzing power $A_{y}$. This
observable depends sensitively on the 3P partial waves of the $NN$-interaction
[46, 47]. There are indications that it also depends sensitively on parts of
the subleading 3NF [24]. It has turned out to be very challenging to reproduce
the experimental data for this observable at laboratory scattering energies
$E_{\text{Lab}}\lesssim$ 30 MeV [37, 48].

Figure 6: The PPD of the $nd$ neutron analyzing power $A_{y}(n)$ at
$E_{\text{Lab}}=10,\>35$, and 67 MeV up to N3LO in the $NN$ interaction. The
legends display the average 90% credible intervals (HPDI), see text for
details. The experimental data (markers) are retrieved from the EXFOR database
(at $E_{\text{Lab}}=10$ and 66.6 MeV) and Ref. [49] (at $E_{\text{Lab}}=35$
MeV). Diamond (cyan) and circle (colorless) markers represent $nd$ and $pd$
cross sections, respectively.

Given the possibly fine-tuned nature of $A_{y}$, it is particularly
interesting to quantify the PPD due to the variability in the $NN$ LECs of
$\chi$EFT. In Fig. 6, we show the PPDs for $A_{y}$ at NLO, N2LO, and N3LO as
well as the average 90% credibility intervals. At $E_{\text{Lab}}=10$ MeV we
do not reproduce the experimental data at any chiral order. We note that the
N3LO calculation appears to improve the description of the data at the
polarization maximum. However, the low-angle description is markedly worse
compared to the result at N2LO. For $E_{\text{Lab}}\approx 35-67$ MeV it
appears sufficient to use $NN$-only forces at N2LO to describe presently
available data.

It is clear that the variability due to the LECs inferred from $NN$ data does
not give rise to any significant uncertainty nor does it explain discrepancies
between theory and data. We refrain from quantifying the $\chi$EFT truncation
error for this observable since our calculation omits 3NFs, which may very
well play a significant role in explaining the low-energy $A_{y}$ values.
Nevertheless, a crude estimate to account for the $\chi$EFT truncation error
with missing 3NFs can be obtained by pulling out factors of $Q$ in Eq. (8),
starting at N2LO [50]. We found that this procedure induced rather large
$\chi$EFT uncertainties that covered the experimental data at all orders.

As for the remaining spin-polarization observables, their $NN$ PPDs exhibit
similar patterns and widths as presented above for the differential cross
section and $A_{y}(n)$, i.e., the vastly dominating source of uncertainty is
the $\chi$EFT truncation error, at least below N3LO.

##  V Summary and outlook

We sampled the PPDs for the $nd$ differential cross section $d\sigma/d\Omega$
at $E_{\text{Lab}}=12$, 36, and 65 MeV scattering energy, and neutron
analyzing power $A_{y}(n)$ at $E_{\text{Lab}}=10$, 35, and 67 MeV. The
underlying samples from the LEC posterior were obtained from a previous
analysis of $NN$ data [9]. The HMC algorithm used in that analysis yields
virtually uncorrelated samples which we find most likely persists for the
elastic $nd$ observables. The main conclusion from this work is that the
uncertainty about $NN$ LECs, when conditioned on $NN$ scattering data and
uncorrelated estimates of the $\chi$EFT truncation errors, does not entail
significant uncertainties in the low-energy $nd$ continuum. Although we only
show results for selected observables, we find them to be representative of
all elastic $nd$ scattering observables, at least for $E_{\text{Lab}}\lesssim
67$ MeV.

When compared with estimates of the $\chi$EFT truncation error, we find that
the uncertainty stemming from the numerical values of the $NN$ LECs are
negligible, at least up to (and including) N2LO in Weinberg power counting. At
N3LO, the width of the PPD and the credible interval of the truncation error
are starting to become comparable. However, these uncertainties are very small
and, in fact, are comparable to typical experimental errors.

In this work we have not quantified the errors due to having a finite number
of wave-packets in the WPCD method. Instead, we extrapolated all results to
$N_{\text{WP}}=150$ and relied on the fact that the widths and shapes of all
studied PPDs remain the same when using fewer wave-packets, i.e.,
$N_{\text{WP}}=50$ and 75. Future work should be dedicated to understanding
the scaling of the WPCD method-error with respect to the discretization of the
continuum.

Throughout our analysis, the PPDs were conditioned on $NN$ scattering data.
For the predicted differential cross section, we find reasonable agreement
with experimental $Nd$ scattering data. The same observation was made for many
polarization observables, not shown explicitly in this paper. However, less
accuracy is observed in the low-energy $A_{y}$ analyzing power. A natural next
step would therefore be to simultaneously infer the $NN$ and $NNN$ LECs from
$NN$ plus $Nd$ scattering data. This would shed more light on the necessity of
including 3NFs to explain this data.

The inference of LECs in $\chi$EFT is not restricted to use only scattering
observables. In fact, any low-energy nuclear data can be utilized (and will be
relevant given that it has a high information content). On the other hand, the
abundant sets of experimentally measured $NN$ [32, 33], $\pi N$ [51], and $Nd$
[23] scattering cross sections provide data where theoretical predictions do
not rely on many-body interactions beyond 3NFs. In addition, a scattering
cross section can be tied to a well-defined (external) momentum, providing a
clear interpretation of the soft scale entering the $\chi$EFT expansion
parameter $Q$ and the associated truncation error. This identification of a
soft scale is more ambiguous in bound states of nuclear many-body systems.

A Bayesian analysis of LECs in $\chi$EFT conditioned on $Nd$ data requires
efficient solutions to the AGS equations. Indeed, traversing larger domains of
the multi-dimensional LEC parameter-spaces would require orders of magnitude
more samples than what we employed in this work. Fortunately, recent advances
in model reduction methods [52], utilizing singular value decomposition [53]
and eigenvector continuation [54, 55, 56] methods, show great promise in
delivering accurate and fast solutions to the Faddeev equations. Some of these
methods appear compatible with our existing implementation for solving the AGS
equations with the WPCD method.

## Acknowledgments

This work was supported by the European Research Council (ERC) under the
European Unions Horizon 2020 research and innovation programme (Grant
agreement No. 758027). The work of C. F. was supported by the Swedish Research
Council (dnr. 2017-04234 and 2021-04507). The computations and data handling
were enabled by resources provided by the Swedish National Infrastructure for
Computing (SNIC), partially funded by the Swedish Research Council through
grant agreement no. 2018-05973.

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