# Shoupa: An AI System for Early Diagnosis of Parkinson’s Disease

Jingwei Li1, Ruitian Wu1, Tzu-liang Huang1, Zian Pan1, Ming-Chun Huang1,2

(2022)

###### Abstract.

Parkinson’s Disease (PD) is a progressive nervous system disorder that has
affected more than 5.8 million people, especially the elderly. Due to the
complexity of its symptoms and its similarity to other neurological disorders,
early detection requires neurologists or PD specialists to be involved, which
is not accessible to most old people. Therefore, we integrate smart mobile
devices with AI technologies. In this paper, we introduce the framework of our
developed PD early detection system which combines different tasks evaluating
both motor and non-motor symptoms. With the developed model, we help users
detect PD punctually in non-clinical settings and figure out their most severe
symptoms. The results are expected to be further used for PD rehabilitation
guidance and detection of other neurological disorders.

  1. 1.

Jingwei Li, Ruitian Wu, Tzu-liang Huang, Zian Pan, and Mingchun Huang are with
Duke Kunshan University, China. (Equal Contribution with alphabetical order)
E-mail: { jl957, rw219, th270, zp45<EMAIL_ADDRESS>
  2. 2.

Ming-Chun Huang is also with Suzhou Huanmu Intelligence Technology Co., Ltd,
China

Permission to make digital or hard copies of part or all of this work for
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party components of this work must be honored. For all other uses, contact the
owner/author(s). CHASE’ 22, November 17-19, 2022, Washington, DC, USA  ©2022
Copyright held by the owner/author(s). ACM ISBN 978-1-4503-9476-5/22/11.
https://doi.org/10.1145/3551455.3564708

††journalyear: 2022††copyright: rightsretained††conference: ACM/IEEE
International Conference on Connected Health: Applications, Systems and
Engineering Technologies; November 17–19, 2022; Washington, DC,
USA††booktitle: ACM/IEEE International Conference on Connected Health:
Applications, Systems and Engineering Technologies (CHASE’ 22), November
17–19, 2022, Washington, DC, USA††doi: 10.1145/3551455.3564708††isbn:
978-1-4503-9476-5/22/11

##  1\. Introduction

Parkinson’s disease (PD) is a progressive nervous system disorder associated
with a decline in motor and non-motor physical function. In 2019, PD resulted
in 5.8 million disability-adjusted life years and is estimated to influence
14.2 million people by 2040 (dorsey2018parkinson, ). Although PD cannot be
cured entirely, early detection and treatment can be beneficial for people
with PD. Detection in the early stages of PD not only helps relieve symptoms
(jankovic2008parkinson, ) but also prolongs life expectancy
(pinto2004treatments, ).

However, the early detection of PD is often difficult since PD’s early
symptoms are subtle and it’s similar to other neurological disorders of the
brain (postuma2015mds, ). Although a traditional solution of getting a
complete evaluation with the help of a neurologist or PD specialist at an
early stage is helpful, more than 40% of individuals older than 65 have fewer
opportunities to meet a PD specialist, even in wealthy countries
(dorsey2018parkinson, ). The lack of opportunities may result from the strict
testing environment and the time-consuming nature of the diagnostic procedures
to screen out other possible diseases.

With the spread of smart mobile devices, early PD diagnosis can be conducted
more easily and more accessible to patients. There are mainly three advantages
of mobile detection. First, PD testing will no longer be influenced by time,
space, and facility. With the help of mobile devices, the test can be
conducted in a non-clinical setting. Second, mobile PD detection can provide
good accuracy with the assistance of computers together with some existing
techniques. Third, the use of mobile health allows a continuous collection and
monitoring of patient data. As a chronic disease, constantly collected data
would help trace the effectiveness of follow-up medication and physical
rehabilitation for various symptoms.

Figure 1. Shoupa System’s Overall Workflow

##  2\. System Design

In response to the above-mentioned inconveniences of the current Parkinson’s
disease early diagnosis, we develop a mobile Parkinson’s diagnosis software
enabling users to know whether they are at risk of having PD even at home.

Shoupa is developed based on the Flutter framework and can be run on Android
5.0+ or IOS 9.0+ devices. Its Back-end is based on an Amazon AWS server. The
system supports both motor and non-motor tests. In motor tests, we focus on
the users’ finger movement and body balance, in which finger movement is
examined by rhythm and drawing tests, while body balance is examined by gait
and tremor tests. Meanwhile, in non-motor tests, we measure users’ memory and
vocal ability and also collect user’s general information through three
surveys: MDS-UPDRS, MMSE, and demographic surveys. We designed visual memory
and auditory memory tests to check users’ memory and used pronunciation and
reading tests for vocal ability examination.

Since we have multiple categories of tests, the modality of data we receive
also varies greatly, including gyroscope data (gait and tremor tests), audio
data (pronunciation and reading tests), image data (drawing test), and
numerical/categorical data (surveys). We will process data of the same
modality in the same way to convert all the attributes into standardized 1-D
vectors.

After pre-processing the data, we will fuse attributes describing the same
body function into a scorer. Based on the scorer, the system will give an
evaluation result to that certain function (For example, finger movement
only). Since our system tests multiple functions of the body, we will finally
have several scorers. Finally, all the scorers will be ensembled with certain
weights to give a comprehensive diagnosis of PD.

Figure 1 shows the overall workflow of our system.

##  3\. Data Collection and Analysis

In this section, methods that we used to collect and analyze data is
introduced in greater detail.

To collect data on PD patients in China, We collaborate with Guangzhou First
People’s Hospital. Patients conduct our tests under the guidance of medical
professionals so that the data is guaranteed to be real and valid. All the
data being collected is desensitized and is only used for research purposes.

With the collected data, we build a detection model which could return how
likely the users will be affected by PD and what their most severe symptoms
are. The essence of the model is based on multi-modal and multi-category data:

  1. (1)

Multi-modality: For all collected attributes, there could be data of different
modalities, including numerical/ categorical data (surveys), image data
(drawing test), gyroscope data (gait and tremor tests), and audio data
(pronunciation and reading tests).

    1. (a)

For the basic numerical/categorical data, they are supposed to be standardized
to mitigate the impacts of the magnitude of data.

    2. (b)

For the image data, we employ basic CNN with Residual Layers(7780459, )
appended to transform into a 1-D vector.

    3. (c)

For the gyroscope data and audio data, since they are both temporal-based, we
process them in the same way. We transform it firstly into 2D time-frequency
spectrogram representation and then process them similarly to how we process
2-D data.

  2. (2)

Multi-dimension: We have data describing different functions, including
speech, memory, body balance, etc. We directly fuse the multi-modal variables
describing the same function as a whole dimension. The diverse dimensions of
our model ensure that our model gives a comprehensive evaluation to the users.
For each dimension, a scorer is trained based on Neural Network or Support
Vector Machine. We expect to implement the multi-dimensional structure with
XGBoost(XGboost, ) to optimize weights for each scorer and employ it to
reflect how each symptom contributes to the final diagnosis.

By integrating all different dimensions and modalities, we expect to give a
final scorer giving a reliable diagnosis of the PD. Also, since each dimension
is ensembled with corresponding weights, we could give a very detailed outcome
pinning which functions the users are mostly weak in.

##  4\. Conclusion & Future Work

To wrap up, we have developed the framework for a system that offers
convenient and accurate PD early-stage diagnosis for the elderly. In the
future, we are planning on several things to polish our system:

  1. (1)

Split data collection into two rounds and use first-round data to do some
correlation analysis. Combined with direct feedback from users, we are
planning to reiterate our APP and model to make it fit into real-world
scenarios.

  2. (2)

Add a rehabilitation recommendation system that could arrange rehabilitation
plans for the users. The system could also dynamically trace the
rehabilitation progress.

  3. (3)

Extend Shoupa to more other cerebral diseases, including Alzheimer’s and
Epilepsy. Since these diseases share similar symptoms, we can use transfer
learning to directly transfer data and trained models to other diseases.

###### Acknowledgements.

This work was supported by the 2021-2022 Kunshan Government Research Fund
(21KKSGR050). The funders had no role in the study design; data collection,
analysis, or interpretation; in the writing of the report; or in the decision
to submit the article for publication. This work also obtained IRB Approval
from Duke Kunshan University (FWA00021580) and Guangdong General Hospital
(KY-Q-2021-208-02).

## References

  * [1] E Ray Dorsey and Bastiaan R Bloem.  The parkinson pandemic—a call to action.  JAMA neurology, 75(1):9–10, 2018.
  * [2] Joseph Jankovic.  Parkinson’s disease: clinical features and diagnosis.  Journal of neurology, neurosurgery & psychiatry, 79(4):368–376, 2008.
  * [3] Serge Pinto, Canan Ozsancak, Elina Tripoliti, Stéphane Thobois, Patricia Limousin-Dowsey, and Pascal Auzou.  Treatments for dysarthria in parkinson’s disease.  The Lancet Neurology, 3(9):547–556, 2004.
  * [4] Ronald B Postuma, Daniela Berg, Matthew Stern, Werner Poewe, C Warren Olanow, Wolfgang Oertel, José Obeso, Kenneth Marek, Irene Litvan, Anthony E Lang, et al.  Mds clinical diagnostic criteria for parkinson’s disease.  Movement disorders, 30(12):1591–1601, 2015.
  * [5] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun.  Deep residual learning for image recognition.  In 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pages 770–778, 2016.
  * [6] Tianqi Chen and Carlos Guestrin.  Xgboost: A scalable tree boosting system.  pages 785–794, 08 2016.


# Self-supervised learning for segmentation

###### Abstract

Self-supervised learning is emerging as an effective substitute for transfer
learning from large datasets. In this work, we use kidney segmentation to
explore this idea. The anatomical asymmetry of kidneys is leveraged to define
an effective proxy task for kidney segmentation via self-supervised learning.
A siamese convolutional neural network (CNN) is used to classify a given pair
of kidney sections from CT volumes as being kidneys of the same or different
sides. This knowledge is then transferred for the segmentation of kidneys
using another deep CNN using one branch of the siamese CNN as the encoder for
the segmentation network. Evaluation results on a publicly available dataset
containing computed tomography (CT) scans of the abdominal region shows that a
boost in performance and fast convergence can be had relative to a network
trained conventionally from scratch. This is notable given that no additional
data/expensive annotations or augmentation were used in training.

Index Terms—  Self-Supervised learning, Deep Learning, kidney, CT segmentation

##  1 Introduction

In nature, human beings learn cumulatively. We build upon previous knowledge
to learn new concepts quickly and efficiently. One way to achieve this in
neural networks has been through transfer learning where a pre-trained (on a
large dataset) network is used for training for the new task with the
associated data. Generally, a neural network learns basic features such as
edges and lines in the first few layers. The complexity of features learned by
the network increases from the input layer to the final layer with the later
ones being more task-specific. Hence, only the weights of the later layers are
typically modified during transfer learning. Ideally, the similarity level of
the pre-training task and main task can yield a good transfer in performance.
However, in the case of medical images where labeled data is scarce, this is
difficult. Nevertheless, there have been attempts to do transfer learning
across domains, for example, disease classification with the VGG network [1]
whereas such attempts are sparse for the segmentation task.

Self-supervised learning is another recent paradigm that can offer a solution
to the data sparsity problem. Here, a model is pre-trained on a proxy task
based on a dataset that is also used for the main task. This proxy task is
chosen such that it is easy to solve, but requires learning some form of image
semantics. It is formulated such that implicit information in the same dataset
can be used as labels, thus removing the need for additional annotation.

Many examples of such tasks are found in the recent literature in computer
vision. Agrawal et al. [2] tasked the network to predict the camera
transformation of moving targets for the main tasks of scene recognition and
recognition. Doersch et al. [3] predicted the relative position of patches for
object detection in the Pascal VOC dataset [4]. Recently, Vondrick et al. [5]
used the colorization of videos as a proxy task for visual tracking.

Self-supervision is a promising idea in medical image analysis since medical
data is often rich in anonymous metadata such as patient age, medical history,
and longitudinal information but often lacking manual annotations by medical
experts. On longitudinal MR data, Jamaludin et al. [6] predicted the
degradation level of vertebral disks using self-supervision after using a
proxy task which classified pairs of images as belonging to the same or a
different patient. Li et al. [7] maximized the distance between registered
images at multiple scales as a proxy task for non-rigid image registration.
However, the use of self-supervision for improving segmentation in medical
volumes is relatively unexplored.

In this work, we explore self-supervision for medical image segmentation. The
selection of an appropriate proxy task is critical to the effectiveness of
self-supervised learning and needs careful design. We demonstrate this for
kidney segmentation from abdominal CT volumes. The proposed proxy task
exploits the anatomical asymmetry between the left and right kidneys. We show
that segmentation with self-supervision allows for faster convergence and a
better performance relative to training from scratch while using the same
amount of data.

##  2 Method

The left and right kidney differ slightly in size, shape, and spatial location
[8] (see fig 1). Thus, we propose a proxy task to classify whether a given
pair of kidneys belong to the same side. To accomplish this, the network needs
to develop an understanding of the structure and sizes of the kidneys. We
hypothesize that this knowledge is useful in the main task of kidney
segmentation. We begin by dividing the abdominal CT volume into two halves
along the sagittal plane, thus obtaining one kidney in each half. We observed
that when complete halves of the CT scan are passed, the network learns to
perform this task very easily, based on the spatial location of the two
kidneys. In such cases when the task is too easy and does not require
considerable semantics, it is observed that transfer learning to do a main
task is poor. It is desirable to force the network to learn to do the task,
based only on the shape and size, rather than the spatial location.
Accordingly, each kidney section is extracted using the kidney annotation mask
and a fixed region of 64x112x112 pixels around the kidney is passed to the
network.

Fig. 1: Axial view of right and the left kidney in a CT volume

Thus, only the kidney part of the scan is passed to the network for the proxy
task. A siamese network, composed of two identical sub-networks with shared
weights [9], is employed to perform this proxy task. Each of the two branches
of the network is given the region of one kidney. Label 1 (0) is assigned if
both the kidneys are of the same (opposite) side. The siamese network for the
proposed proxy task is shown in fig. 2

Fig. 2: Siamese network for proxy task

The network is trained using contrastive loss, defined as

$L_{C}=\sum^{N}_{n=1}(y)d^{2}+(1-y)\text{max}(0,m-d)^{2}$

where, $y$ is the label; $d=||a-b||^{2}$ is the L2 distance between the
embeddings $a$ and $b$ of the final layers of each branch of the siamese
network; $m$ is the margin. This loss works by minimizing the distance when
the label is 1 but maximizing it when the label is 0. After the embeddings of
opposite pairs cross $m$, the loss flips to 0. The value of margin was set to
1 in our implementation.

On completion of training for the proxy task, one of the branches of the
siamese network is used as an encoder for the main kidney segmentation task.
Since the siamese network uses shared weights for both branches, any of the
branches can be used as the encoder. Thus, only the encoder part of the
segmentation network is pre-trained. The rest of the network uses randomly
initialized weights. The segmentation network is trained using a weighted sum
of soft dice loss [10] and weighted binary cross-entropy (BCE). The weights
for BCE and dice loss are initialized as 0.6 and 0.4 respectively. As training
progresses, the weight for BCE is decreased while the weight for dice loss is
increased by the same amount. The intuition behind this is that BCE improves
voxelwise performance while dice loss improves segmentation overlap.

##  3 Experiments and results

###  3.1 Implementation details

We employ an encoder-decoder system with U-Net like skip connections for the
segmentation. The encoder branch consists of 4 blocks with densely connected
layers. It has been shown that densely connected layers control the vanishing
gradient problem while keeping the number of parameters low [11]. Each of
these blocks consists of 2, 2, 4 and 8 densely connected layers respectively.
After each block, downsampling is performed with the help of a 3D
convolutional layer with stride 2. This is followed by another convolutional
layer with kernel size 1x1x1 to combine and reduce the number of feature maps.

To reduce complexity, the decoder is smaller and consists of plain 3D
convolutional blocks connected serially. Each block in the decoder comprises
of 2 3D convolutional layers of kernel size 3x3x3. After each block,
upsampling is performed using a transpose convolutional layer. We also connect
an additional reconstruction section to the encoder to form an autoencoder.
This reconstruction part is structurally similar to the decoder except for the
last layer but does not employ any skip connections. This block serves as a
regularizer during training and controls overfitting [12]. Fig. 3 shows the
network architecture.

As 3D convolution is memory intensive, we have trained with batch size 1.
Group norm is better suited for regularization in the case of small batch
sizes [13]. Hence, all convolutional layers are preceded by a Group norm
layer. We use Adam optimizer with an initial learning rate $10^{-3}$ and
weight decay $10^{-2}$. The code is implemented with the PyTorch framework in
Python and was run on NVIDIA GTX 1080GPU with 11GB RAM.

Fig. 3: Network architecture for segmentation

###  3.2 Dataset

The CT dataset from the KiTS 2019 challenge [14] was used for assessment. It
consists of abdominal CT scans of 210 cases with manual annotations and 90
cases without annotation. The data was first resampled to have a uniform voxel
spacing of 3.22x1.62x1.62 mm, resulting in a median size of 128x248x248
voxels. The voxels were clipped to a window of [-80,300] by a windowing
operation to improve the contrast of the kidney region. Data normalization was
done by subtracting the mean and dividing by the standard deviation. Rather
than resizing the volumes to a fixed size, the volumes were zero-padded to
obtain a size closest to the nearest multiple of 16 to permit some variable
size input to the network.

###  3.3 Experiments and results

Fig. 4:  Sample segmentation results at the end of 200 epochs for 3 cases. Top
to bottom: cases 1-3. Left to Right: results of models trained without (i.e.
trained from scratch) and with self-supervision, overlaid on the ground truth.
Green: Ground truth; Red: Prediction.

The 210 volumes were split randomly into 2 sets: 80% for training while
reserving the remaining 20% for validation. Two identical networks were taken
and one was trained using self-supervision and the resulting model is
henceforth referred to as a model with self-supervision or MwS. The second
network was trained from scratch and results in a model referred to as a model
without self-supervision or MwoS. The initial training settings for both
networks were identical and training was done for a fixed number of epochs.
The effectiveness of self-supervised learning was evaluated both qualitatively
and quantitatively.

Fig. 5: Progression of DC over epochs on the validation set

For the quantitative comparison, the mean Dice score coefficient (DC) was
chosen. This score is defined as

$DC=\frac{2TP}{2TP+FP+FN}$

where TP, FP and FN are the number of true positives, false positives and
false negatives respectively.

Table 1: Comparison of quantitative results | DC | HD | BL (%)
---|---|---|---
MwoS | MwS | MwoS | MwS | MwoS | MwS
Val. | 0.907 | 0.931 | 0.303 | 0.291 | 14.0 | 12.9
Test | 0.853 | 0.880 | - | - | - | -

For further validation, we also compared two additional metrics - Hausdorff
distance (HD) and the difference in boundary length of prediction and ground
truth (BL). The obtained values (for both MwoS and MwS) for these are
presented for the validation, and test sets in table 1. After training for 200
epochs, a peak DC of 0.931 and 0.907 were achieved with MwS and MwoS,
respectively, on the validation set. Ideally, higher values (close to 1) are
good for DC and lower values are good for HD and BL. It is evident from table
1 that MwS performs better than MwoS as per all 3 metrics. The test set of 90
volumes is released by challenge organizers without annotations. Evaluation
results (only DC) on this set can be obtained by submitting prediction masks
on the challenge website. The DC value drops by roughly the same amount. The
first ranked performance on the KiTS challenge leaderboard achieves 0.974 DC
for kidney segmentation by employing 6-fold data augmentation and training for
1000 epochs [15]. In comparison, we have achieved the reported result without
any data augmentation and by training for $\frac{1}{5}^{th}$ number of epochs.
This further reinforces the fast convergence of our proposed method.

Fig. 5 shows the progression of DC values over epochs for the MwS and MwoS for
the validation set. For clarity of visualization, the plots are shown after a
moving average operation. Relative to MwoS, MwS can be seen to yield higher DC
values even in the first few epochs and converge much faster, with a higher DC
value at the end of the 200 epochs.

Fig. 4 shows the segmentation results of MwS, overlaid on the volume and
ground truth. The segmentation results of MwoS overlaid on the volume, are
compared on the right side. It can be seen that with MwS, the segmentation
captures the fine structures as well without over-segmenting. In the case of
MwoS, voxels towards the edges of the kidneys are seen to be misclassified as
background while in some locations some over-segmentation can be observed.

##  4 Conclusion

We demonstrated the application of self-supervised learning for segmentation
in medical imaging. A novel proxy task was used to perform self-supervised
learning for segmentation. Self-supervision for segmentation is promising as
it leads to faster convergence and a boost in results without using any
additional data or data augmentation. In the future, generalized criteria for
the selection of a good proxy task may be formulated. The success of self-
supervised learning in a more generic manner would lead towards reducing the
need for large datasets for transfer learning.

## References

  * [1] Menegola et al.,  “Knowledge transfer for melanoma screening with deep learning,”  in 2017 IEEE 14th International Symposium on Biomedical Imaging (ISBI 2017). IEEE, 2017, pp. 297–300.
  * [2] Agrawal et al.,  “Learning to see by moving,”  in Proceedings of the IEEE International Conference on Computer Vision, 2015, pp. 37–45.
  * [3] Doersch et al.,  “Multi-task self-supervised visual learning,”  in Proceedings of the IEEE International Conference on Computer Vision, 2017, pp. 2051–2060.
  * [4] Everingham et al.,  “The pascal visual object classes (voc) challenge,”  International journal of computer vision, vol. 88, no. 2, pp. 303–338, 2010.
  * [5] Vondrick et al.,  “Tracking emerges by colorizing videos,”  in Proceedings of the European Conference on Computer Vision (ECCV), 2018, pp. 391–408.
  * [6] Jamaludin et al.,  “Self-supervised learning for spinal MRIs,”  in Deep Learning in Medical Image Analysis and Multimodal Learning for Clinical Decision Support, pp. 294–302. Springer, 2017.
  * [7] Li et al.,  “Non-rigid image registration using self-supervised fully convolutional networks without training data,”  in 2018 IEEE 15th International Symposium on Biomedical Imaging (ISBI 2018). IEEE, 2018, pp. 1075–1078.
  * [8] J Frimann-Dahl,  “Normal variations of the left kidney: an anatomical and radiologic study,”  Acta radiologica, , no. 3, pp. 207–216, 1961.
  * [9] Bromley et al.,  “Signature verification using a” siamese” time delay neural network,”  in Advances in neural information processing systems, 1994, pp. 737–744.
  * [10] Milletari et al.,  “V-net: Fully convolutional neural networks for volumetric medical image segmentation,”  in 2016 Fourth International Conference on 3D Vision (3DV). IEEE, 2016, pp. 565–571.
  * [11] Huang et al.,  “Densely connected convolutional networks,”  in Proceedings of the IEEE conference on computer vision and pattern recognition, 2017, pp. 4700–4708.
  * [12] Sabour et al.,  “Dynamic routing between capsules,”  in Advances in neural information processing systems, 2017, pp. 3856–3866.
  * [13] Wu et al.,  “Group normalization,”  in Proceedings of the European Conference on Computer Vision (ECCV), 2018, pp. 3–19.
  * [14] Heller et al.,  “The KiTS19 challenge data: 300 kidney tumor cases with clinical context, ct semantic segmentations, and surgical outcomes,” 2019.
  * [15] Fabian et al.,  “An attempt at beating the 3D U-Net,”  arXiv preprint arXiv:1908.02182, 2019.


# A Further Look at the Bayes Blind Spot

Mark Shattuck  Department of Mathematics, University of Tennessee, 37996
Knoxville, TN, USA<EMAIL_ADDRESS>and  Carl Wagner  Department of
Mathematics, University of Tennessee, 37996 Knoxville, TN, USA
<EMAIL_ADDRESS>
###### Abstract.

Gyenis and Rédei have demonstrated that any prior $p$ on a finite algebra,
however chosen, severely restricts the set of posteriors accessible from $p$
by Jeffrey conditioning on a nontrivial partition. Their demonstration
involves showing that the set of posteriors not accessible from $p$ in this
way (which they call the _Bayes blind spot of_ $p$) is large with respect to
three common measures of size, namely, having cardinality $c$, (normalized)
Lebesgue measure $1$, and Baire second category with respect to a natural
topology. In the present paper, we establish analogous results for probability
measures defined on any infinite sigma algebra of subsets of a denumerably
infinite set. However, we have needed to employ distinctly different
approaches to determine the cardinality, and especially, the topological and
measure-theoretic sizes of the Bayes blind spot in the infinite case.
Interestingly, all of the results that we establish for a single prior $p$
continue to hold for the intersection of the Bayes blind spots of countably
many priors. This leads us to conjecture that Bayesian learning itself might
be just as culpable as the limitations imposed by priors in enabling the
existence of large Bayes blind spots.

_Keywords:_ Bayes blind spot, Jeffrey conditioning, Baire category, Bayesian
learning

##  1\. Introduction

In a pair of groundbreaking papers, Gyenis and Rédei (2017, 2021) initiated a
study of the following important epistemological question: Given a prior
probability measure $p$, and a family $R$ of possible methods for revising $p$
in response to new evidence, how large is the set of probability measures $q$
that are _inaccessible_ from $p$ using any of the stipulated revision methods?
The authors call the set of probability measures that are inaccessible from
$p$ its _Bayes blind spot_ , denoted by $BS(p)$. In their foundational 2017
paper, Gyenis and Rédei (henceforth, G&R) situate the accessibility problem in
the broadly comprehensive setting in which revision is conceptualized in terms
of conditional expectation. In their 2021 paper, they explore in detail the
case in which probability measures are defined on a _finite_ algebra $\bf{A}$
of subsets of an arbitrary set $Y$,1 and $R$ consists of all possible ways of
revising $p$ by Jeffrey conditioning on any (necessarily finite) partition of
$Y$ consisting of members of $\bf{A}$, at least one of which has at least two
elements. They prove in this case that the Bayes blind spot is large with
respect to three common measures of size, namely, having cardinality $c$,
(normalized) Lebesgue measure $1$, and Baire second category with respect to a
natural topology. They observe that their result emphasizes the heretofore
insufficiently recognized crucial role of priors in Bayesian learning, and
demonstrates that any prior, however chosen, severely restricts the set of
posteriors that are in principle derivable from that prior by a single
application of Jeffrey conditioning.2

In the present paper, we establish analogous results for probability measures
defined on any infinite sigma algebra of subsets of a denumerably infinite set
$Y$.3 However, we have needed to employ distinctly different approaches to
determine the cardinality, and the topological and measure-theoretic sizes of
the Bayes blind spot in the infinite case. Interestingly, all of the results
that we establish for a single prior $p$ continue to hold for the intersection
of the Bayes blind spots of countably many priors. This leads us to conjecture
that Bayesian learning itself might be just as culpable as the limitations
imposed by priors in enabling the existence of large Bayes blind spots. Before
presenting our results, however, we offer a brief review of some essential
features of Jeffrey conditioning.

##  2\. A primer on Jeffrey conditioning

Suppose $X$ is _any_ set of possible states of the world, ${\bf A}$ is a sigma
algebra of subsets of $X$, and ${\bf E}=\\{E_{i}\\}$ is a countable partition
of $X$ with each $E_{i}\in{\bf A}$. It will be convenient throughout this
paper to assume that every prior $p$ is a _strictly coherent_ probability
measure on ${\bf A}$, in the sense that $p(A)>0$ for all nonempty
$A\in\bf{A}$.4 A probability measure $q$ on $\bf{A}$ is said to come from $p$
by _Jeffrey conditioning_ (henceforth, JC) on E if

(2.1) $q(A)=\sum_{i}q(E_{i})p(A|E_{i}),\quad\text{for all }A\in\bf{A}.$

######  Theorem 2.1.

Formula (2.1) is equivalent to the rigidity condition

(2.2) $q(A|E_{i})=p(A|E_{i}),\text{ for each }i\text{ such that
}q(E_{i})>0\text{ and all }A\in{\bf A},$

and, if $X$ is countable and ${\bf A}=2^{X}$, to the following identities:

(2.3) $\text{For each }i\text{ and all }x,x^{\prime}\in
E_{i},~{}\frac{q(x)}{p(x)}=\frac{q(x^{\prime})}{p(x^{\prime})},$

where we slightly abuse notation by writing, for example, $q(x)$, instead of
$q(\\{x\\})$.

###### Proof.

See Jeffrey (1992, pp. 117–118).

The partition $\\{\\{x\\}:x\in X\\}$, in which each block is a singleton, is
called the _trivial partition of_ $X$. All other partitions are _nontrivial_.
It is easy to see that if $X$ is countable and $p$ is strictly coherent on
$2^{X}$, then _every_ probability measure $q$ on $2^{X}$ comes from $p$ by JC
on the trivial partition of $X$. In this case, JC amounts to _total
reassessment_ , which is why we restrict consideration to JC on nontrivial
partitions.

The following theorem is an immediate consequence of Theorem 2.1.

######  Theorem 2.2.

If $X$ is countable, $p$ and $q$ are probability measures on $2^{X}$, and $p$
is strictly coherent, then $q$ comes from $p$ by JC on a nontrivial partition
$\bf{E}$ of $X$ if and only if there exist distinct $x$ and $x^{\prime}$ in
$X$ such that $q(x)/p(x)=q(x^{\prime})/p(x^{\prime})$.

###### Proof.

Necessity. Since at least one block of a nontrivial partition contains at
least two elements of $X$, this follows from (2.3) above. Sufficiency. By
assumption, the map $x\mapsto q(x)/p(x)$ from $X$ to $[0,\infty)$ is not
injective. For each member $r$ in the range $R$ of this map, let
$E_{r}:=\\{x\in X:q(x)/p(x)=r\\}$. Clearly, ${\bf E}:=\\{E_{r}:r\in R\\}$ is a
countable, nontrivial partition of $X$, and for each $r\in R$, if
$x,x^{\prime}\in E_{r}$, then $q(x)/p(x)=q(x^{\prime})/p(x^{\prime})=r$. So by
Theorem 2.1, $q$ comes from $p$ by JC.

_Remark:_ The partition ${\bf E}$ introduced above in the proof of sufficiency
is the _coarsest partition_ 5 of $X$ on which $q$ comes from $p$ by JC. See
Diaconis and Zabell (1982, 824) and van Fraassen (1980).

From Theorem 2.2, the following characterization of $BS(p)$ is immediate.

######  Corollary 2.3.

The probability measure $q$ belongs to $BS(p)$ if and only if the ratios
$q(x)/p(x)$ are distinct for all $x\in X$.

In the following sections, the above corollary will be invoked repeatedly in
determining the size of $BS(p)$ in terms of its cardinality, its Baire
category and an appropriately constructed measure.

##  3\. The cardinality of $BS(p)$

Let $X=\\{x_{i}\\}_{i\geq 1}$ be a countable set with $|X|\geq 2$. Since all
probability measures $q$ on $2^{X}$ are completely determined by the values
$q_{i}:=q(x_{i})$, we may focus attention on the associated probability
distributions (i.e., mass functions), denoted by ${\bf q}=(q_{i})_{i\geq 1}$.
We prove here a more general result concerning the cardinality of what we now
denote by $BS(\bf{p})$. Given a nonempty set ${\bf P}$ of probability
distributions on $X$, let $BS({\bf P})$ be the Bayes blind spot of ${\bf P}$
defined as the intersection of $BS({\bf p})$ for all ${\bf p}\in{\bf P}$.

######  Theorem 3.1.

Let ${\bf P}$ be a nonempty countable set of probability distributions on a
countable set $X$ with $|X|\geq 2$, where all probabilities in each
distribution are positive. Then the cardinality of $BS({\bf P})$ is equal to
$c$.

###### Proof.

We treat only the case when $X$ is denumerably infinite, as the simple
modifications required to accommodate the finite case of $X$ will be apparent.
We show first that $BS({\bf P})$ is nonempty. Let ${\bf P}=\\{{\bf
p}^{(1)},{\bf p}^{(2)},\ldots\\}$, where ${\bf p}^{(k)}=(p_{i}^{(k)})_{i\geq
1}$ with $\sum_{i\geq 1}p_{i}^{(k)}=1$ and $p_{i}^{(k)}>0$ for all $i$, for
each $k\geq 1$. Let $m_{1}$ be any positive real number. If $i\geq 2$, and the
positive real numbers $m_{1},\ldots,m_{i-1}$ have been chosen, choose $m_{i}$
such that (i) $0<m_{i}<2^{-i}$ and (ii)
$\frac{m_{i}}{p_{i}^{(k)}}\neq\frac{m_{j}}{p_{j}^{(k)}}$ for all $1\leq j\leq
i-1$ and $k\geq 1$. Note that such a sequence $(m_{i})_{i\geq 1}$ can be
constructed, since at each step, only a countable number of possible values
for $m_{i}$ are being excluded from the (uncountable) set $(0,2^{-i})$.
Further, the series $\sum_{i\geq 1}m_{i}$ is convergent, with, say,
$\sum_{i\geq 1}m_{i}=m$. Define ${\bf q}=(q_{i})_{i\geq 1}$ by setting
$q_{i}=\frac{m_{i}}{m}$. Then the sequence $\frac{q_{i}}{p_{i}^{(k)}}$ for
$i\geq 1$ has all its terms distinct for each $k\geq 1$, by construction, with
$\sum_{i\geq 1}q_{i}=1$, so ${\bf q}\in BS({\bf P})$.

Let $\epsilon=\min\\{1-q_{1},q_{2}\\}$. Given $0<\delta<\epsilon$, let
$s_{\delta}^{(k)}$ be the sequence defined by
$\frac{q_{1}+\delta}{p_{1}^{(k)}},\frac{q_{2}-\delta}{p_{2}^{(k)}},\frac{q_{3}}{p_{3}^{(k)}},\frac{q_{4}}{p_{4}^{(k)}},\ldots$
for each $k\geq 1$. Note that ${\bf q}\in BS({\bf P})$ implies that there are
only a countable number of values of $\delta$ for which $s_{\delta}^{(j)}$
fails to have all its terms distinct for some $j\geq 1$. Then for all other
values of $\delta$ in the interval $(0,\epsilon)$, we have
$q_{\delta}=(q_{1}+\delta,q_{2}-\delta,q_{3},q_{4},\ldots)\in BS({\bf P})$.
Hence, $|BS({\bf p})|\geq c$, as it contains an uncountable subset. On the
other hand, $BS({\bf p})\subset[0,1]^{\mathbb{P}}$, where
$\mathbb{P}=\\{1,2,\ldots\\}$, and so $|BS({\bf
p})|\leq|[0,1]^{\mathbb{P}}|=c$, whence $|BS({\bf p})|=c$.

##  4\. A topological analysis of $BS(p)$

In this section and the next, $X$ represents a denumerably infinite set. Given
a real sequence ${\bf s}=(s_{i})_{i\geq 1}$, the $\ell_{1}$-norm of ${\bf s}$
is defined as $\sum_{i\geq 1}|s_{i}|$, with the distance between $\bf{s}$ and
${\bf t}=(t_{i})_{i\geq 1}$ given by $|{\bf s}-{\bf t}|=\sum_{i\geq
1}|s_{i}-t_{i}|$. A fundamental result from analysis states that the subset of
the real sequences for which the $\ell^{1}$-norm is finite (i.e., $\sum_{i\geq
1}|s_{i}|$ converges) is a complete metric space (with metric $d$ defined as
$d({\bf s},{\bf t})=|\bf{s}-\bf{t}|$); see, e.g., Friedman (1982, Theorem
3.2.3). Let $S$ denote the set of sequences summing to $1$ with nonnegative
entries, which are synonymous with the probability distributions on $X$. Then
$S$ itself is complete (in the topology induced by the $\ell^{1}$-norm), being
a closed subset of a complete metric space.

In this (and the following) section, let ${\bf p}=(p_{i})_{i\geq 1}$ denote a
member of $S$ where $p_{i}>0$ for all $i$. We shall make use of the
characterization of $BS({\bf p})$ supplied by Corollary 2.3 as consisting of
those ${\bf q}=(q_{i})_{i\geq 1}$ in $S$ such that the $\frac{q_{i}}{p_{i}}$
values for $i\geq 1$ are all distinct from one another.

Then $BS({\bf p})$ satisfies the following topological properties.

######  Theorem 4.1.

Let ${\bf p}=(p_{i})_{i\geq 1}\in S$, with $p_{i}>0$ for all $i$. Then
$BS({\bf p})$ in the $\ell^{1}$-norm topology on $S$ (i) is of second
category, (ii) is dense in $S$, and (iii) has an empty interior.

###### Proof.

(i) Given $1\leq i<j$, let $S_{i,j}$ denote the subset consisting of those
members ${\bf q}=(q_{i})_{i\geq 1}$ of $S$ such that
$\frac{q_{i}}{p_{i}}=\frac{q_{j}}{p_{j}}$. Note that $S_{i,j}$ is a closed
subset of $S$ having an empty interior, as one can clearly find ${\bf v}\in
S-S_{i,j}$ with $|{\bf v}-{\bf q}|<\epsilon$ for any given $\epsilon>0$ and
${\bf q}$ (upon perturbing the $q_{i}$ and $q_{j}$ entries of ${\bf q}$
slightly). Since the closure of $S_{i,j}$ has empty interior, it is nowhere
dense for each $i$ and $j$ and thus

$BS({\bf p})^{c}=\bigcup_{1\leq i<j}S_{i,j}$

is a countable union of nowhere dense sets. By definition, $BS({\bf p})^{c}$
is of first category. Now $S$ complete implies it is of the second category,
by the Baire category theorem, see Friedman (1982, p. 106). But then
$S=BS({\bf p})\cup BS({\bf p})^{c}$ implies $BS({\bf p})$ must be of the
second category, for otherwise $BS({\bf p})\cup BS({\bf p})^{c}$ would be of
first category being a countable union of nowhere dense sets.

(ii) Let ${\bf q}\in S$. We will find a member of $BS({\bf p})$ whose
$\ell_{1}$-distance from ${\bf q}$ is arbitrarily small. Let
$0<\epsilon<\frac{1}{2}$ be given. Define a sequence ${\bf
r}=(r_{1},r_{2},\ldots)$ of nonnegative real numbers as follows. Let $r_{1}$
be given and for $n>1$, let $r_{n}\geq 0$ be such that $\frac{r_{n}}{p_{n}}$
is distinct from all values of $\frac{r_{i}}{p_{i}}$ for $1\leq i\leq n-1$.
Further, we may assume $|r_{n}-q_{n}|<\frac{\epsilon}{2^{n}}$ for all $n\geq
1$. Then ${\bf r}$ is a convergent series with

$|{\bf r}-{\bf q}|<\sum_{n\geq 1}\frac{\epsilon}{2^{n}}=\epsilon,$

and hence $||{\bf q}|-|{\bf r}||<\epsilon$ implies
$\frac{1}{2}<1-\epsilon<|{\bf r}|<1+\epsilon$. Let
$r_{n}^{\prime}=\frac{r_{n}}{|{\bf r}|}$ and ${\bf
r^{\prime}}=(r_{n}^{\prime})_{n\geq 1}$. Then ${\bf r^{\prime}}\in BS({\bf
p})$ and

$\displaystyle|{\bf r^{\prime}}-{\bf q}|$ $\displaystyle=\frac{1}{|{\bf
r}|}|{\bf r}-|{\bf r}|{\bf q}|\leq 2|{\bf r}-|\bf{r}|{\bf q}|$
$\displaystyle\leq 2\left(|{\bf r}-{\bf q}|+|1-|\bf{r}||\cdot|{\bf
q}|\right)<2(\epsilon+\epsilon\cdot 1)=4\epsilon,$

whence $BS({\bf p})$ is dense in $S$.

(iii) Let ${\bf q}=(q_{i})_{i\geq 1}\in BS({\bf p})$. We will find ${\bf
q^{\prime}}=(q_{i}^{\prime})_{i\geq 1}\in S-BS({\bf p})$ such that $|{\bf
q}-{\bf q^{\prime}}|$ is arbitrarily small. Let $0<\epsilon<q_{2}$, where we
assume for now $q_{2}>0$. Let $n\geq 1$ be large enough so that
$\max\\{q_{i},\frac{p_{i}}{p_{1}}\\}<\epsilon$ for all $i\geq n$. First assume
$\frac{q_{n}}{p_{n}}-\frac{q_{1}}{p_{1}}>0$ and let
$\delta=q_{n}-\frac{p_{n}q_{1}}{p_{1}}>0$. Let the $(q_{i}^{\prime})_{i\geq
1}$ be given by $q_{i}^{\prime}=q_{i}$ if $i\neq n,n+1$, with
$q_{n}^{\prime}=\frac{q_{1}p_{n}}{p_{1}}$ and
$q_{n+1}^{\prime}=q_{n+1}+\delta$. One may verify ${\bf q^{\prime}}\in
S-BS({\bf p})$. Then $\frac{p_{n}q_{1}}{p_{1}}\leq\epsilon q_{1}<\epsilon$
implies $|q_{n}-q_{n}^{\prime}|<\epsilon$, being the difference of two
nonnegative real numbers less than $\epsilon$, and further
$|q_{n+1}-q_{n+1}^{\prime}|=\delta<\epsilon$. Thus, we get $|{\bf q}-{\bf
q^{\prime}}|=|q_{n}-q_{n}^{\prime}|+|q_{n+1}-q_{n+1}^{\prime}|<2\epsilon$.

Now assume $\frac{q_{n}}{p_{n}}-\frac{q_{1}}{p_{1}}<0$. Let
$\rho=\frac{p_{n}q_{1}}{p_{1}}-q_{n}>0$ and note
$\rho\leq\frac{p_{n}q_{1}}{p_{1}}\leq\epsilon q_{1}<q_{2}$. Define the
$q_{i}^{\prime}$ in this case by $q_{i}^{\prime}=q_{i}$ if $i\neq 2,n$, with
$q_{2}^{\prime}=q_{2}-\rho$ and $q_{n}^{\prime}=\frac{p_{n}q_{1}}{p_{1}}$
(note $n>2$ since $\epsilon<q_{2}$, by assumption). Then we have
$|q_{2}-q_{2}^{\prime}|,|q_{n}-q_{n}^{\prime}|<\epsilon$ and thus $|{\bf
q}-{\bf q^{\prime}}|<2\epsilon$. On the other hand, if $q_{2}=0$, then ${\bf
q}\in BS({\bf p})$ implies $q_{3}>0$, and one may proceed similarly as before
with $q_{3}$ in place of $q_{2}$, upon requiring $0<\epsilon<q_{3}$. This
implies in all cases that ${\bf q}$ cannot be an interior point of $BS({\bf
p})$, whence $BS({\bf p})$ has an empty interior.

The preceding result may be extended as follows.

######  Theorem 4.2.

The blind spot $BS({\bf P})$, where ${\bf P}$ is a nonempty countable subset
of $S$, satisfies the same conditions (i)–(iii) given above with respect to
the $\ell^{1}$-norm as $BS({\bf p})$.

###### Proof.

We make simple modifications to the preceding proof where required. For
property (i) concerning $BS({\bf P})$, note that the countable intersection of
second category sets each of whose complement is of first category is of
second category. For (ii), observe that the same proof applies when there are
a countable number of ${\bf p}$ since for each $n>1$, one has an interval of
potential values for $r_{n}$ wherein at most a denumerably infinite number of
values are excluded as possibilities. Property (iii) follows from the fact
that the interior operator respects subset inclusions.

_Discussion:_ Due to (iii), Theorem 4.1 demonstrates that the infinite case of
$X$ differs fundamentally from the earlier comparable result from G&R (2021)
when $X$ was finite where it was shown that $BS({\bf p})$ is an open dense
subset in the set of all probability distributions on $X$. For a more basic
example of a subset $T$ in a metric space $Y$ such that $T$ is dense in $Y$
and of second category, but yet has an empty interior, consider the subset of
irrationals within the set of reals. Further, it is seen that $BS({\bf p})$ is
neither open nor closed in the infinite case, as both $BS({\bf p})$ and
$BS({\bf p})^{c}$ have empty interior.

Recall that the $\ell^{p}$-norm for each $p\geq 1$ is defined for ${\bf
s}=(s_{i})_{i\geq 1}$ as $\left(\sum_{i\geq
1}|s_{i}|^{p}\right)^{\frac{1}{p}}$, with the limiting case as
$p\rightarrow\infty$ denoted by $\ell^{\infty}$ given by $\max\\{|s_{i}|:i\geq
1\\}$. Since the topology of the $\ell^{p}$-norm strictly refines that of the
$\ell^{q}$-norm for $1\leq p<q\leq\infty$, with the $\ell^{p}$-metric complete
for each $p\geq 1$, the prior two theorems apply also to all of the
$\ell^{p}$-norms on $S$. Further, the results are seen to apply to the
complete metric $d$ defined by $d({\bf u},{\bf v})=\frac{|{\bf u}-{\bf
v}|}{1+|{\bf u}-{\bf v}|}$ for ${\bf u},{\bf v}\in S$, where $|\cdots|$
denotes an $\ell^{p}$-norm. Note that since the topology on $S$ induced by $d$
is bounded, it is seen not to be equivalent to the topology induced by any of
the $\ell^{p}$-norms. It would be interesting to prove analogues of Theorems
4.1 and 4.2 for other kinds of topologies on $S$ by considering a greater
variety of metrics on $S$.

Finally, the result in (iii) from Theorem 4.1 can be generalized in another
way as follows. Consider the subset $S_{\ell}$ of $S$ where $\ell\geq 1$
consisting of those distributions ${\bf q}=(q_{i})_{i\geq 1}$ in which there
are at least $\ell$ pairs $(n,m)$ where $1\leq n<m$ such that
$\frac{q_{n}}{p_{n}}=\frac{q_{m}}{p_{m}}$. Then the set $S_{\ell}$ is dense in
$S$ for each $\ell\geq 1$, the details of the proof we leave to the reader.
Note that the $\ell=1$ case corresponds to (iii).

_Open question:_ To what degree can the results in Theorems 4.1 and 4.2 be
extended beyond the class of topologies on $S$ corresponding to the
$\ell^{p}$-norms? In particular, is it possible to find general sufficient
(more desirably, necessary and sufficient) conditions for a complete metric
$d$ on $S$ which would ensure $BS(\bf{p})$ is of second category in $(S,d)$
for all ${\bf p}$?

##  5\. A measure-theoretic analysis of $BS(p)$

Let $S$ denote the set of all probability distributions on a countably
infinite set $X$ whose associated mass functions are represented sequentially.

######  Theorem 5.1.

There exists a probability measure on $S$ such that $BS({\bf p})$ has
probability $1$ with respect to this measure for each ${\bf p}=(p_{i})_{i\geq
1}\in S$ having all positive components.

###### Proof.

Let $X_{1}$ be a uniform random variable on the interval $[0,1)$. Define the
random variables $X_{i}$ for $i>1$ recursively by letting $X_{i}$ be uniform
on the interval $[0,1-\sum_{j=1}^{i-1}x_{j})$, where $X_{j}=x_{j}$ for $1\leq
j\leq i-1$. Then $X_{1}+X_{2}+\cdots\rightarrow 1$ almost surely, and we
consider the set of possible outcomes $(x_{1},x_{2},\ldots)$ wherein
$X_{i}=x_{i}$ for all $i$, which are synonymous with the members of the set
$S$. Note that outcomes $(x_{1},x_{2},\ldots)$ which are finitely nonzero have
probability zero of occurring and hence the subset comprising the
corresponding members of $S$ has measure zero.

Given integers $1\leq a<b$, consider $P(X_{a}=cX_{b})$, where $c=c_{a,b}$ is
the constant defined by $c=\frac{p_{a}}{p_{b}}$. Then we have

$\displaystyle P(X_{a}=cX_{b})=\int\int$ $\displaystyle
P(X_{a}=ct|\sum_{i=1}^{a-1}X_{i}=s\text{ and }X_{b}=t)$
$\displaystyle\quad\times(\text{Joint density of }\sum_{i=1}^{a-1}X_{i}\text{
and }X_{b}\text{ evaluated at }s\text{ and }t)dsdt.$

Note that the probability $P(X_{a}=ct|...)$ in the prior integral is zero
since for each $s$ and $t$, it can be shown that $X_{a}$ has a (conditional)
density, whence $P(X_{a}=cX_{b})=0$. Considering all pairs $(a,b)$ where $a<b$
(a countable number of possibilities) implies the probability that some
$(x_{1},x_{2},\ldots)$ is not in $BS({\bf p})$ is zero. Thus, $BS({\bf p})$
has probability $1$ in this measure on $S$.

_Discussion:_ Since $BS({\bf p})^{c}$ has probability $0$ with respect to the
measure $M$ defined on $S$ in the preceding proof, then so does $BS({\bf
P})^{c}$, where ${\bf P}$ consists of a countable number of distributions
$\bf{p}$. Hence, $BS({\bf P})$ has probability $1$ with respect to $M$.

Note that there is apparently not a straightforward extension to the infinite
case of the argument from G&R (2021) in the finite case of $X$ which
demonstrated that the complement of $BS({\bf p})$ has measure zero with
respect to the Lebesgue measure on $\mathbb{R}^{n-1}$, where $n=|X|$. This is
due in part to the fact that there is apparently not an analogous measure on
$\mathbb{R}^{\infty}$ that permits a comparable analysis. It should be
remarked that the proof of Theorem 5.1 can be extended to the finite case of
$X$ by setting $X_{n}=1-\sum_{i=1}^{n-1}x_{i}$ and terminating the recursive
procedure of defining random variables. Note however that the resulting
measure for the probability distributions on $X$ differs from that supplied by
(normalized) Lebesgue measure on $\mathbb{R}^{n-1}$.

Further, different measures on $S$ for which $BS(\bf{p})$ has probability $1$
can be obtained by allowing the $X_{i}$ to assume other continuous
distributions on a finite interval. For example, one could let $X_{1}$ be a
$\beta$ distribution on $[0,1)$ and then subsequently define the $X_{i}$ for
$i>1$ as appropriate scaled versions of the $X_{1}$ distribution on intervals
of decreasing length. Note that it is not a requirement that the $X_{i}$ all
have the same kind of distribution, provided they are continuous and
$\sum_{i\geq 1}X_{i}$ converges to $1$ almost surely.

_Open Question:_ Is it possible to find a general criterion for a probability
measure on $S$ to ensure that $BS(\bf{p})$ has probability $1$ for all
possible $\bf{p}$?

##  6\. Summary and conclusion

We initiated this project to determine if Jeffrey conditioning on partitions
of a denumerably infinite set might offer a degree of flexibility sufficient
to ameliorate the limitative results established by G&R in the finite case. On
a superficial level, the answer turns out to be negative. For the Bayes blind
spot of a prior distribution on a denumerably infinite set $X$ has cardinality
$c$, is of second Baire category in the set $S$ of all probability
distributions on $X$ for a natural topology on $S$, and has measure 1 for an
appropriately defined measure on $S$. Our proof that $|BS({\bf p})|=c$ holds
for all countable $X$ and, in the case when $X$ is finite, differs from,
though it is in the same spirit as, the proof given by G&R. Further, the proof
of G&R may be extended to all countable $X$, and applies also to show
$|BS({\bf P})|=c$ in the finite, though not the infinite, case of ${\bf P}$.
However, our approach to the measure-theoretic and topological sizes of
$BS(\bf{p})$ when $X$ is denumerably infinite differs more dramatically from
that employed by G&R in the finite case, as detailed below.

The difference is particularly striking in the topological analysis, where we
show that $BS(\bf{p})$ is (i) of second category, (ii) is dense in $S$, and
(iii) _has an empty interior_ , with the topology on $S$ derived from its
$\ell^{1}$-norm (or any $\ell^{p}$-norm, where $1\leq p\leq\infty$). Indeed,
it is precisely the flexibility afforded by probabilities defined on a
denumerably infinite set that allows us, for any ${\bf q}\in BS({\bf p})$, to
find a member of $S-BS({\bf p})$ whose $\ell^{1}$-distance from ${\bf q}$ is
arbitrarily small. So this topology does not reduce to the topology employed
by G&R in the finite case, where $BS({\bf p})$ was shown to be an _open_ ,
dense subset of $S$. If, as we suspect, separate treatments of the finite and
denumerably infinite cases are unavoidable here, this represents a somewhat
unusual situation in mathematics, where one typically expects a unified
treatment under the rubric of countability of the relevant underlying sets.

In our measure-theoretic analysis, we constructed a measure $M$ on the set of
probability distributions on the denumerably infinite set $X$ such that
$M(BS({\bf p}))=1$. Moreover, a slight variation on this construction when $X$
is finite produces a measure $m$ such that $m(BS({\bf p}))=1$. So we have here
a more or less unified treatment for all countable sets $X$. This measure $m$
differs, however, from the (normalized) Lebesgue measure employed by G&R in
their approach to the finite case.

It is particularly noteworthy that if ${\bf P}$ is any countable family of
priors on a denumerably infinite set and $BS({\bf P}):=\bigcap_{{\bf p}\in{\bf
P}}BS({\bf p})$, then $BS({\bf P})$ continues to have the topological and
measure-theoretic sizes of the Bayes blind spot of a single prior, as
described in the preceding two paragraphs, and has the same cardinality as
well. In particular, (1) $|BS({\bf P})|=c$, (2) $BS({\bf P})$ is of second
category, is dense in $S$, and has empty interior for the topology on $S$
derived from its $\ell^{1}$-norm, and (3) $M(BS({\bf P}))=1$. This raises an
intriguing question, which we have been unable to resolve, and with which we
conclude this paper:

Do these results strengthen the observation of G&R that any prior, however
chosen, puts severe limitations on Bayesian learning originating in that prior
(after all, the intersection of the Bayes blind spots of countably many
Bayesian agents, each with a different prior, and each employing Jeffrey
conditioning, remains topologically and measure-theoretically large)? Or do
they suggest that _Bayesian learning itself_ might be equally culpable in
allowing the existence of large Bayes blind spots (even with access to
countably many priors, the union of all the probabilities accessible from
these priors by Jeffrey conditioning remains topologically and measure-
theoretically small)?

##  7\. Notes

1\. Actually, Gyenis and Rédei treat the case in which this algebra is
$2^{X}$, where $X$ is a finite set. This involves no loss of generality, since
any finite algebra $\bf{A}$ of subsets of an arbitrary set is isomorphic to
$2^{X}$ for some finite set $X$, namely, the set of atoms of $\bf{A}$ (Rényi,
1970, Theorem 1.6.1). It may be worth recalling here that a finite algebra of
subsets of an arbitrary set is _ipso facto_ a sigma algebra, since every
countable union of such subsets is equal to some finite union of those
subsets. Similarly, every finitely additive probability measure on a finite
algebra is countably additive since, in every infinite sequence of pairwise
disjoint sets from that algebra, all but finitely many are equal to the empty
set.

2\. Gyenis and Rédei (2021, section 5.1) also analyze the Bayes blind spot
from the perspective of repeated applications of Jeffrey conditioning.

3\. In fact, we are able, with no loss of generality, to restrict
consideration to probability measures defined on $2^{Y}$, where $Y$ is
denumerably infinite. See Rényi (1970, Theorem 1.6.2).

4\. This simplifying assumption is made to avoid continually having to specify
in various definitions and theorems that certain probabilities are nonzero.
Recall that if probabilities are construed as the (linear utility) prices one
is willing to pay for certain bets, then confirming your probabilities with
the axioms of finitely additive probability (so-called _coherent_
probabilities) protects you against accepting a finite sequence of bets on
which you are sure to sustain a net loss. It is commonly believed that
probability measures must be strictly coherent in order to avoid accepting
bets on which a net gain is impossible, but a net loss is possible (a so-
called _weak Dutch book_). But see Wagner (2007) regarding a slightly modified
conception of subjective probability in which mere additivity suffices not
only to protect against sure loss, but also against vulnerability to a weak
Dutch book.

5\. If ${\bf E}$ and ${\bf F}$ are partitions of a set $X$, then ${\bf F}$ is
_coarser than_ ${\bf E}$ if, for every $E\in{\bf E}$, there exists an
$F\in{\bf F}$ such that $E\subseteq F$ (equivalently, if every $F\in{\bf F}$
is a union of members of ${\bf E}$).

## References

  * [1] P. Diaconis and S. Zabell (1982), Updating subjective probability, _Journal of the American Statistical Association_ 77 (380), 822–830.
  * [2] A. Friedman (1982), _Foundations of Modern Analysis_ , Dover.
  * [3] Z. Gyenis and M. Rédei (2017), General properties of Bayesian learning as statistical inference determined by conditional expectations, _The Review of Symbolic Logic_ 10 (4), 719–737.
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