5656institutetext:  Department of Surgical Sciences, section of
Neuroradiology, Uppsala University, Sweden 22institutetext: Department of
Radiology, University of California San Diego, CA, USA 33institutetext:
Department of Electical and Computer Engineering , Cornell University and
Cornell Tech, New York, NY, USA 44institutetext: Department of Radiology,
Weill Cornell Medicine 55institutetext: Department of Pathology and Laboratory
Medicine, Indiana Univeristy, IN, USA 66institutetext: Department of
Neuroradiology, Heidelberg University, Germany 77institutetext: University of
Laval, Quebec City, QC, Canada 88institutetext: Department of Radiation
Oncology, Duke University, Durham, NC, USA 99institutetext: Radiology
Department, University of Missouri, Columbia, MO, USA 1010institutetext:
Department of Electrical and Computer Engineering, Duke University, Durham,
NC, USA 1111institutetext: Department of Radiology, Nassau Medical Center,
East Meadow, NY, USA 1212institutetext: Department of Radiology, Stanford
University, CA, USA 1313institutetext: Neuroimaging Unit, ASST Ovest Milanese
- Legnano (Milan) - Italy 1414institutetext: Ross University School of
Medicine, Bridgetown, Barbados 1515institutetext: Department of Radiology,
University of California Los Angeles, CA, USA 1616institutetext: C.M.H. Lahore
Medical College & Institute of Dentistry, Lahore, Pakistan 1717institutetext:
Department of Imaging Sciences, University of Rochester Medical Center,
Rochester, NY, USA 1818institutetext: Department of Radiology, Department of
Radiology, Northwestern University, Chicago, IL, USA 1919institutetext:
Department of Radiology, VA San Diego Medical Center 2020institutetext: School
of Medicine, Iran University of Medical Sciences, Tehran, Iran
2121institutetext: Neuroradiology Department, Hospital Garcia de Orta -
Unidade Local de Saúde de Almada Seixal, EPE; Almada, Portugal
2222institutetext: Imaging Department, Hospital da Luz Lisboa, Lisbon,
Portugal 2323institutetext: Department of Radiation Oncology, University of
California Irvine, CA, USA 2424institutetext: Department of Diagnostic and
Interventional Radiology, University Hospital Ulm, Ulm, Germany
2525institutetext: Department of Diagnostic and Interventional Neuroradiology,
School of Medicine, Klinikum rechts der Isar, Technical University of Munich,
Munich, Germany 2626institutetext: Department of Radiology, Scripps Clinic
Medical Group, CA, USA 2727institutetext: Medical Artificial Intelligence
Laboratory (MAI Lab), Lagos, Nigeria 2828institutetext: Sage Bionetworks
2929institutetext: Montreal Neurological Institute (MNI), McGill University,
Montreal, QC, Canada 3030institutetext: Department of Radiology, Mayo Clinic,
Rochester, MN, USA 3131institutetext: National Institute of Health, USA
3232institutetext: Division of Neurosurgery, The Children’s Hospital of
Philadelphia, PA, USA 3333institutetext: Department of Neurosurgery,
University of Pennsylvania, Philadelphia, PA, USA 3434institutetext: Athinoula
A Martinos Center for Biomedical Imaging, Massachusetts General Hospital,
Boston, MA, USA 3535institutetext: Helmholtz AI, Helmholtz Munich, Germany
3636institutetext: Children’s National Hospital, Washington DC, USA
3737institutetext: George Washington University, Washington DC, USA
3838institutetext: University of Zurich, Switzerland 3939institutetext:
Department of Radiology, Mercy Catholic Medical Center, Darby, PA, USA
4040institutetext: Department of Radiology, Northwestern University, Chicago
IL, USA 4141institutetext: Klinikum rechts der Isar, Technical University of
Munich, Germany 4242institutetext: Department of Radiology , University of
Michigan, MI, USA 4343institutetext: Department of Radiology and Biomedical
Imaging, University of California San Francisco, CA, USA 4444institutetext:
Pathology Department, University of Missouri, Columbia, MO, USA
4545institutetext: Department of Radiation Oncology, University of California
San Diego, CA, USA 4646institutetext: Department of Radiology, Duke
University, Durham, NC, USA 4747institutetext: Cortechs.ai 4848institutetext:
School of Medicine, University of California San Diego, CA, USA
4949institutetext: Department of Neurosurgery, Indiana Univeristy, IN, USA
5050institutetext: Department of Radiology, The Children’s Hospital of
Philadelphia, PA, USA 5151institutetext: Departments of Radiation Medicine and
Applied Sciences, Radiology, and Bioengineering, University of California San
Diego, CA, USA 5252institutetext: Department of Radiology and Imaging
Sciences, Emory University, Atlanta, GA, USA 5353institutetext: Stanford
University, CA, USA 5454institutetext: Department of Radiology, Brigham and
Women’s Hospital, Boston, MA, USA 5555institutetext: Harvard Medical School,
Boston, MA, USA 5656institutetext: Division for Computational Radiology &
Clinical AI, University Hospital Bonn
* Equal first authors.
+ Senior author.
† 2024 BraTS Post-Treatment Challenge Organizer.
‡ 2024 BraTS Organizer.
§ Annotator.
¶ Data Contributor.
** Corresponding author: 5656email<EMAIL_ADDRESS>
# The 2024 Brain Tumor Segmentation (BraTS) Challenge: Glioma Segmentation on
Post-treatment MRI

Maria Correia de Verdier Rachit Saluja Louis Gagnon Dominic LaBella Ujjwall
Baid Nourel Hoda Tahon Martha Foltyn-Dumitru Jikai Zhang Maram Alafif Saif
Baig Ken Chang Gennaro D’Anna Lisa Deptula Diviya Gupta Muhammad Ammar Haider
Ali Hussain Michael Iv Marinos Kontzialis Paul Manning Farzan Moodi Teresa
Nunes Aaron Simon Nico Sollmann David Vu Maruf Adewole Jake Albrecht Udunna
Anazodo Rongrong Chai Verena Chung Shahriar Faghani Keyvan Farahani Anahita
Fathi Kazerooni Eugenio Iglesias Florian Kofler Hongwei Li Marius George
Linguraru Bjoern Menze Ahmed W. Moawad Yury Velichko Benedikt Wiestler Talissa
Altes Patil Basavasagar Martin Bendszus Gianluca Brugnara Jaeyoung Cho Yaseen
Dhemesh Brandon K.K. Fields Filip Garrett Jaime Gass Lubomir Hadjiiski Jona
Hattangadi-Gluth Christopher Hess Jessica L. Houk Edvin Isufi Lester J.
Layfield George Mastorakos John Mongan Pierre Nedelec Uyen Nguyen Sebastian
Oliva Matthew W. Pease Aditya Rastogi Jason Sinclair Robert X. Smith Leo P.
Sugrue Jonathan Thacker Igor Vidic Javier Villanueva-Meyer Nathan S. White
Mariam Aboian Gian Marco Conte Anders Dale Mert R. Sabuncu Tyler M. Seibert
Brent Weinberg Aly Abayazeed Raymond Huang Sevcan Turk Andreas M. Rauschecker
Nikdokht Farid Philipp Vollmuth Ayman Nada Spyridon Bakas Evan Calabrese
Jeffrey D. Rudie
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+ +††‡‡§§¶¶

###### Abstract

Gliomas are the most common malignant primary brain tumors in adults and one
of the deadliest types of cancer. There are many challenges in treatment and
monitoring due to the genetic diversity and high intrinsic heterogeneity in
appearance, shape, histology, and treatment response. Treatments include
surgery, radiation, and systemic therapies, with magnetic resonance imaging
(MRI) playing a key role in treatment planning and post-treatment longitudinal
assessment. The 2024 Brain Tumor Segmentation (BraTS) challenge on post-
treatment glioma MRI will provide a community standard and benchmark for
state-of-the-art automated segmentation models based on the largest expert-
annotated post-treatment glioma MRI dataset. Challenge competitors will
develop automated segmentation models to predict four distinct tumor sub-
regions consisting of enhancing tissue (ET), surrounding non-enhancing
T2/fluid-attenuated inversion recovery (FLAIR) hyperintensity (SNFH), non-
enhancing tumor core (NETC), and resection cavity (RC). Models will be
evaluated on separate validation and test datasets using standardized
performance metrics utilized across the BraTS 2024 cluster of challenges,
including lesion-wise Dice Similarity Coefficient and Hausdorff Distance.
Models developed during this challenge will advance the field of automated MRI
segmentation and contribute to their integration into clinical practice,
ultimately enhancing patient care.

###### Keywords:

Brain Tumors, Post-treatment, Segmentation, Cancer, Challenge, Glioma, Diffuse
Glioma, Glioblastoma, MICCAI, BraTS, Machine Learning, Artificial
Intelligence, AI, Data-centric Machine Learning

##  1 Introduction

Gliomas are among the deadliest types of cancer and constitute the most
prevalent malignant primary brain tumors in adults. According to the Central
Brain Tumor Registry of the United States, gliomas as a group represent
approximately 25% of all primary brain tumors and 80% of malignant primary
brain and central nervous system tumors, with diffuse gliomas being the most
common malignant subtype [1]. Diffuse gliomas are a genetically diverse group
of disorders, comprising astrocytoma, oligodendroglioma, and glioblastoma,
encompassing both low- and high-grade gliomas (World Health Organization [WHO]
grades 2-4) [2]. They are characterized by their infiltrative growth pattern
within the central nervous system, presenting substantial challenges for
treatment and monitoring due to their variability in biological behavior,
prognosis, and response to therapy.

Treatment for diffuse gliomas involves a multi-modal approach tailored to the
tumor’s characteristics and the patient’s health and includes surgery,
radiation therapy, and systemic therapies (i.e. chemotherapy, immunotherapy,
and targeted therapy). Years of extensive research to improve diagnosis,
characterization, and treatment have decreased mortality rates, but gliomas
remain one of the deadliest types of malignancies. The prognosis varies
depending on tumor grade and molecular type, the patient’s age and health, and
how they respond to treatment. Median survival is lowest for glioblastoma
(isocitrate dehydrogenase [IDH]-wildtype, WHO grade 4), with overall survival
of 8 months, extending to 14 months with standard-of-care treatment, while it
is highest for oligodendroglioma at approximately 17 years [1, 3]. Magnetic
resonance imaging (MRI) remains the gold standard for post-treatment imaging
across the spectrum of diffuse gliomas. It provides crucial information on
tumor size, location, and morphological changes over time. Post-treatment
imaging of diffuse gliomas is a fundamental part of patient management that
significantly influences clinical decision making and outcomes.

Most prior research on brain tumor segmentation has been performed in the pre-
treatment setting, with a recent survey finding that 98.3% of published glioma
segmentation studies had been performed on pre-operative imaging [4]. These
studies typically utilized pre-operative data from The Cancer Imaging Archives
(TCIA) [5] or prior Brain Tumor Segmentation (BraTS) challenges [6, 7, 8],
which first began in 2012.

The combination of treatment-related changes, including resection cavities,
blood products, post-radiation inflammation, and gliosis, combined with the
already naturally ill-defined tumor borders seen in infiltrative diffuse
gliomas makes segmentation a challenging task, which is not addressed by
segmentation models trained on untreated tumors. While several recent studies
involving glioma segmentation have been performed in the post-treatment
setting [9, 10, 11, 12, 13, 14, 15, 16, 17], relatively little post-treatment
glioma data with voxelwise annotations exists publicly [17, 18].

The 2024 BraTS challenge is the first to focus on post-treatment gliomas. The
goal is to create a public database of annotated post-treatment glioma MRIs
and a benchmarking environment for the development and evaluation of deep
learning segmentation algorithms. The 2024 BraTS post-treatment glioma
challenge utilizes a new dataset consisting only of post-treatment MRIs of
low- and high-grade diffuse gliomas. It includes the resection cavity (RC) as
a new tissue subregion. Algorithms developed from this challenge could be used
as an objective measure for assessing residual tumor volume, which can guide
treatment. Moreover, the dataset can serve as a starting point for future
studies aimed at distinguishing treatment changes from residual/recurrent
tumor, predicting outcomes, and evaluating response to treatment.

##  2 Methods

###  2.1 Data

Retrospective multi-institutional cohorts of patients, diagnosed with diffuse
gliomas having already undergone treatment, which may include surgery,
radiation, and/or systemic therapy. Data was contributed from seven different
academic medical centers (Figure 1, Table 1): Duke University, University of
California San Francisco (UCSF; which consists of the previously described
UCSF Longitudinal Post-Treatment Diffuse Glioma [UCSF-LPTDG]) [17], University
of Missouri Columbia, University of California San Diego, Heidelberg
University Hospital, University of Michigan and Indiana University.

Contributing Sites | Number of Cases (approximate)
---|---
Duke University | 680
University of California San Francisco | 600
University of Missouri Columbia | 400
University of California San Diego | 350
Heidelberg University Hospital | 300
University of Michigan | 100
Indiana University | 70
Total | 2200
Table 1: Number of cases contributed by each site.

The patients have been clinically scanned with multiparametric MRI (mpMRI)
acquisition protocols including the following MRI sequences:

  1. 1.

Pre-contrast T1-weighted (T1)

  2. 2.

Contrast-enhanced T1-weighted (T1-Gd)

  3. 3.

T2-weighted (T2)

  4. 4.

T2-weighted fluid-attenuated inversion recovery (FLAIR)

Figure 1: Map showing institutions from USA and Germany contributing data to
the 2024 Brain Tumor Segmentation (BraTS) post-treatment glioma challenge and
their relative sample size.

###  2.2 Data Preprocessing

The preprocessing pipeline (Figure 2) applied to all the data considered in
the 2024 BraTS post-treatment glioma challenge was similar to the one
evaluated and followed by the BraTS 2017-2023 challenges [6, 7, 8, 19]. Raw
MRI scans were first carefully reviewed by radiologists from individual
institutions. The T1, T1-Gd, T2, and FLAIR sequences were then extracted and
named according to the standard BraTS naming convention. The dcm2niix software
was applied to the four sequences to convert the raw scans from their original
Digital Imaging and Communications in Medicine (DICOM) file format to the
Neuroimaging Informatics Technology Initiative (NIfTI) file format [20].
Following the conversion to NIfTI files, we performed brain extraction to
remove any apparent non-brain tissue (e.g., neck fat, skull, eyeballs) using
HD-BET111https://github.com/CCI-Bonn/HD-BET [21]. The brain-extracted T1,
T1-Gd, T2, and FLAIR sequences were registered to the Linear Symmetrical MNI
Atlas using affine registration via the CapTK/Greedy software [22]. Several
institutions that contributed to the dataset processed their data in a similar
in-house pipeline or the FeTS 2.0 platform222https://fets-ai.github.io/FL-
PoST/.

The four sequences, or the image volume data, were then segmented using five
different pre-segmentation approaches, all of which used the nnU-Net [23].

  1. 1.

3D U-Net trained by nnU-Net V1 on post-treatment glioma data acquired by Duke
and UCSF using five-fold cross-validation, which is the Federated Learning for
Postoperative Segmentation of Treated glioblastoma (FL-PoST) pre-segmentation
method.

  2. 2.

3D U-Net trained by nnU-Net V2 on post-treatment glioma data acquired by Duke
and UCSF using five-fold cross-validation.

  3. 3.

3D SegResNet[24] trained using MONAI333MONAI SegResNet Block(Medical Open
Network for AI) and nn-UNet V2 on post-treatment glioma data acquired by Duke
and UCSF using five-fold cross-validation.

  4. 4.

A 3D proprietary nnU-Net trained by Cortechs.ai (Neuroquant GliomaTM) on pre-
and post-treatment glioma data from various datasets from the cancer imaging
archives (TCIA)[16].

  5. 5.

A 3D nnU-Net trained on the UCSF-LPTDG dataset and various TCIA post-treatment
glioma datasets annotated by Cortechs.ai.

The five segmentations produced by these networks were combined using the
Simultaneous Truth and Performance Level Estimation (STAPLE) fusion algorithm
[25]. In addition, digital subtraction images were created between the T1-Gd
and T1 sequences to facilitate the annotation process for radiologists in
cases of more subtle enhancement or confounding areas of T1 intrinsic
hyperintensity. This data was handed off to radiologists to make the
annotation process easier and more efficient.

Figure 2: Data Processing and Annotation Workflow for creating the 2024 BraTS
post-treatment glioma challenge dataset.

###  2.3 Tumor Annotation Protocol

The data considered in the 2024 BraTS post-treatment glioma challenge was
similar to the paradigm of the BraTS 2021-2023 challenge data [6, 7, 8, 19],
though with modifications specific to the post-treatment setting. The
annotation (Figure 3) of these data followed a pre-defined clinically approved
annotation protocol (defined by expert neuroradiologists and radiation
oncologists). This annotation protocol was provided to all clinical
annotators, describing in detail instructions on what the segmentations of
each tumor sub-region should include with numerous examples of more
challenging cases444Tumor Annotation Protocol Document.

Summary of specific instructions:

  1. 1.

Enhancing tissue (ET): This delineates the hyperintense signal on T1-Gd, after
excluding the vessels. Any areas of thick or nodular enhancement are included
in the ET class, though typical treatment-related thin linear enhancement
along and within resection cavities and along the dura was not included in the
ET class. A T1-Gd - T1 subtraction image was provided to help identify subtle
areas of enhancement and to distinguish areas of intrinsic T1 hyperintensity
from enhancement.

  2. 2.

Non-enhancing tumor core (NETC): This outlines regions appearing dark on both
T1 and T1-Gd images (denoting necrosis/cysts), and dark regions on T1-Gd that
appear brighter on T1 and are not otherwise clearly represented by a prior RC.

  3. 3.

Surrounding non-enhancing FLAIR hyperintensity (SNFH): This tissue typically
includes edema and infiltrating tumor. Given the post-treatment nature of the
scans, any T2/FLAIR signal abnormalities, including radiation-related
hyperintensity, gliosis, edema, and non-enhancing tumor, were included in the
SNFH label. Symmetric or patchy white matter hyperintensities clearly related
to chronic microvascular ischemic disease or periventricular capping are not
included.

  4. 4.

Resection cavity (RC): The RC class consists of both recent and chronic
resection cavities. Chronic resection cavities, which are typically older than
3-6 months, were considered those with signal intensity isointense to
cerebrospinal fluid on both T1 and T2/FLAIR images. More recent resection
cavities often contained air, blood, and/or proteinaceous materials,
exhibiting variable signal characteristics.

The annotators were provided with the four mpMRI sequences as well as a T1
contrast subtraction (T1-Gd - T1) image and given the flexibility to use their
tool of preference for making the segmentations, following a hybrid approach
where the presegmentations were refined manually. Once the tumor segmentations
were refined by the annotators, the approver reviewed the segmentations.
Annotators spanned across various experience levels and clinical/academic
ranks, while the approvers were board-certified neuroradiologists on the
organizing committee with prior annotation experience. The approver would
review the tumor segmentations in tandem with the corresponding MRI scans,
and, if the segmentations were not of satisfactory quality, would send them
back to the annotators for further refinement. This iterative approach was
followed for all cases until their respective segmentations reached
satisfactory quality for public availability and were noted as the final
ground truth segmentation labels for these scans. The test data were annotated
and approved by two sets of annotators/approvers to assess inter-rater
reliability.

Figure 3: Tumor sub-regions considered in the 2024 BraTS post-treatment glioma
challenge. Image panels with the tumor sub-regions annotated in the different
mpMRI scans and combined segmentations on mpMRI. The enhancing tissue (blue)
visible on a T1-Gd scan, the non-enhancing tumor core (red) visible on a T1-Gd
scan, the surrounding non-enhancing FLAIR hyperintensity (green) visible on a
FLAIR scan and the resection cavity (yellow) visible on a T2 scan. The
combined segmentations generating the final tumor sub-region labels visible on
mpMRI, as provided to the challenge participants: enhancing tissue (blue), the
surrounding non-enhancing FLAIR hyperintensity (green), the non-enhancing
tumor core (red), and the resection cavity (yellow).

Common errors of automated segmentations (Figure 4): Based on observations, we
have identified some common errors in automated segmentations. The most
typical errors in the current challenge were:

  1. 1.

The segmentation of vessels and choroid plexus as ET.

  2. 2.

The segmentation of non-enhancing lesions that have intrinsic T1
hyperintensity as ET.

  3. 3.

Undersegmentation of SNFH or ET when subtle.

  4. 4.

The segmentation of white matter changes from microvascular disease unrelated
to treatment as SNFH.

  5. 5.

The segmentation of RC as NETC or vice versa.

  6. 6.

Extension of RC into ventricles or extra-axial spaces.

Figure 4: Common errors in automated segmentations in the 2024 BraTS post-
treatment glioma challenge. The top row shows typical segmentation errors, and
the bottom row shows manually corrected labels. Color codes: blue for
enhancing tissue (ET), red for non-enhancing tumor core (NETC), green for
surrounding non-enhancing FLAIR hyperintensity (SNFH), and yellow for
resection cavity (RC). Additional Detailed examples are available in the Tumor
Annotation Protocol Document.

##  3 Performance Evaluation

The challenge is hosted on the Synapse Platform (Sage Bionetworks). Following
the paradigm of algorithmic evaluation in machine learning, the data included
in the 2024 BraTS post-treatment glioma challenge were divided into training
(70%), validation (10%), and testing datasets (20%). Challenge participants
receive ground truth labels exclusively for the training dataset, while the
validation dataset is provided without any associated ground truth, and the
testing dataset remains completely hidden. The evaluation metrics used in this
challenge are the same metrics from previous BraTS 2023 challenges [26, 27,
28, 29, 30, 31].

In terms of evaluation metrics, we use:

  1. 1.

Lesion-wise Dice Similarity Coefficient (DSC), which measures voxelwise
segmentation overlap between predicted and ground truth segmentations,
ignoring true negative voxels.

  2. 2.

Lesion-wise 95% Hausdorff distance (HD95), which measures the distance between
the center of the predicted and ground truth segmentations.

The lesion-wise DSC555https://github.com/rachitsaluja/BraTS-2023-Metrics and
lesion-wise HD95 metrics were designed to assess model performance at the
level of individual lesions rather than across the entire image. This
methodology prevents our evaluation from favoring models that only detect
larger lesions, a limitation often seen with standard DSC. By evaluating
models on a lesion-by-lesion basis, we can better understand their ability to
segment multi-focal and multi-centric disease.

In terms of the assessed and evaluated tumor sub-regions:

  1. 1.

ET describes the regions of active tumor as well as nodular areas of
enhancement.

  2. 2.

NETC denotes necrosis and cysts within the tumor.

  3. 3.

SNFH includes edema, infiltrating tumor, and post-treatment changes.

  4. 4.

RC consists of both recent and chronic resection cavities and typically
contains fluid, blood, air, and/or proteinaceous materials.

  5. 5.

Tumor core (ET plus NETC) describes what is typically resected during a
surgical procedure.

  6. 6.

Whole tumor (ET plus SNFH plus NETC) defines the whole extent of the tumor,
including the tumor core, infiltrating tumor, peritumoral edema and treatment-
related changes.

For ranking of multidimensional outcomes (or metrics), for each team, we will
compute the summation of their ranks across the average of the metrics
described above as a univariate overall summary measure. This measure will
decide the overall ranking for each specific team. To visualize the results in
an intuitive fashion, we propose to visualize the outcome via an augmented
version of radar plot [32].

Similar to BraTS 2017-2023 challenges, uncertainties in rankings will be
assessed using permutational analysis [7]. Performance for the segmentation
task will be assessed based on the relative performance of each team on each
tumor sub-region and for each segmentation measure. These will be combined by
averaging ranks for the measures, and statistical significance will be
evaluated only for the segmentation performance measures and will be
quantified by permutation the relative ranks for each segmentation measure and
sub-region per subject of the testing data.

##  4 Discussion

Assessment of diffuse glioma treatment response is challenging given the
complex appearance on MRI after treatment. Automated segmentation of post-
treatment diffuse glioma has significant potential to increase workflow
efficiency with more rapid and accurate volumetric assessments for treatment
planning [11, 15]. The 2024 BraTS post-treatment glioma challenge differs from
previous BraTS glioma challenges as it consists of an entirely new dataset of
exclusively post-treatment diffuse gliomas and includes a novel tissue class,
the RC. The aim of the 2024 BraTS post-treatment glioma challenge will be to
identify current state-of-the-art segmentation algorithms for post-treatment
gliomas, and hence target a more clinically relevant question of monitoring
tumor progression.

Prior studies in post-treatment glioma have shown slightly lower performance
than for pre-treatment glioma regarding segmentations of large areas of
abnormal FLAIR signal and areas of ET [9, 10, 11, 12, 13, 14]. However,
smaller areas of contrast enhancement still remain challenging, and many of
these studies have not included the RC as a separate label. Providing labeled
segmentations of the RC is important for correct therapy planning, especially
for patients receiving adjuvant radiotherapy after surgical resection [33,
34].

Automated segmentation of post-treatment diffuse glioma tumor subregions has
significant potential to increase workflow efficiency for radiologists, neuro-
oncologists, radiation oncologists, and neurosurgeons[9, 15, 35]. Given the
significant effort required to annotate multilabel tissue classes in the post-
treatment setting, the release of this dataset, which includes expert-
voxelwise segmentations of tumor subregions, will support future studies
seeking to validate automated segmentation tools for post-treatment diffuse
gliomas.

Although efforts were made during challenge design and dataset preparation
there are limitations to this study that should be mentioned. The annotator-
approval model for manual segmentation corrections, while consistent with
prior BraTS challenges [8], does not address inter-observer variability. To
mitigate this variability, annotators received written instructions and an
expert reviewer evaluated all segmentations prior to their inclusion in the
dataset. To quantify this variability in this dataset, the test set was
annotated and approved by two sets of annotators/approvers.

There are several ways to improve and expand future BraTS glioma challenges.
The annotation protocol combines abnormal signals related to
residual/recurrent tumor and post-treatment changes due to overlapping imaging
features. Confidently distinguishing these entities is a challenging clinical
task, which becomes even more difficult in the context of this dataset without
relevant treatment history, prior imaging, or advanced imaging (diffusion and
perfusion). Thus, future challenges would need to incorporate longitudinal and
multimodal data in order to tackle the crucial issue of distinguishing between
residual/recurrent enhancing and infiltrative tumor from edema, gliosis,
granulation tissue and pseudoprogression. Such a model could significantly aid
in developing optimal patient management plans.

Another challenge lies in identifying specific imaging features that can
accurately anticipate responses to different treatments. Given its non-
invasive nature, MRI is widely used in neuro-oncology practice for response
evaluation, typically by measuring tumor size before and after treatment.
However, because response patterns can be complex and heterogeneous, this
approach does not always lead to an accurate assessment of the underlying
biological response. Creating a model tailored for this task would require
adding more clinical information to the dataset. By developing such a model,
clinicians could potentially gain valuable insights into the disease’s
progression and tailor treatment strategies accordingly.

##  5 Conclusion

The objective of the 2024 BraTS post-treatment glioma challenge is to
establish a benchmark and define a community standard for automated
segmentation on post-treatment MRI, utilizing the largest, publicly available,
expert-annotated post-treatment glioma MRI dataset. The developed state-of-
the-art models will provide a crucial tool for objectively assessing residual
tumor volume for follow-up examinations and treatment planning that has the
potential to improve patient management and outcomes. Additionally, they will
lay the foundation for future studies aimed at identifying tumor subtypes,
assessing aggressiveness, and predicting recurrence risk based solely on MRI
findings.

## Acknowledgments

The success of any challenge in the medical domain depends on the development
of a large and well-curated dataset. We express our gratitude to all data
contributors, annotators, and approvers for their dedication and effort.

## References

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# Asymptotic Behavior of Free Energy When Optimal Probability Distribution Is
Not Unique

Shuya Nagayasu and Sumio Watanabe
Department of Mathematical and Computing Science
Tokyo Institute of Technology, Japan

###### Abstract

Bayesian inference is a widely used statistical method. The free energy and
generalization loss, which are used to estimate the accuracy of Bayesian
inference, are known to be small in singular models that do not have a unique
optimal parameter. However, their characteristics are not yet known when there
are multiple optimal probability distributions. In this paper, we
theoretically derive the asymptotic behaviors of the generalization loss and
free energy in the case that the optimal probability distributions are not
unique and show that they contain asymptotically different terms from those of
the conventional asymptotic analysis.

##  1 Introduction

In statistical learning theory, a probability distribution which generates a
sample is called a true distribution and one with a parameter is called a
statistical model or a learning machine. An probability distribution is
estimated by applying a training algorithm to a statistical model. Then, the
difference between the true distribution and the estimated one is defined by
some measure, for example, the Kullback-Leibler (KL) divergence. In practical
applications, the true distribution is unknown, hence the free energy and the
generalization loss, which give the relative difference of KL divergence, are
used to evaluate the estimated one.

The theoretical values of the free energy and the generalization loss strongly
depend on the geometrical situations of the true distribution and a
statistical model. A statistical model is called regular if the parameter
which minimizes the KL divergence of a true distribution and the statistical
model is unique and Hessian matrix of the KL divergence at the minimum point
is regular. For the regular case, the asymptotic behavior of the
generalization loss was revealed by Akaike[1], while that of the free energy
was revealed by Schwarz[8]. These results have been applied to statistical
model selection criteria, i.e. Akaike(AIC), Bayesian(BIC), Deviance(DIC)[9],
and Adjusted Bayesian(ABIC)[2].

If a statistical model is not regular, then it is called singular. There are
two different singular cases. One is that multiple optimal parameters exist,
but the probability distributions of the optimal parameters are the same and
unique. This case occurs when a statistical model such as a neural network or
a normal mixture is redundant for to the true distribution. From here on, we
will refer to this case as the multiple parameters and unique distribution
case.

The other case is when there are multiple optimal parameters that give
different multiple probability distributions. This case occurs when there
exist different optimal statistical models have the same KL divergence from
the true distribution. We will call this case the multiple parameters and
multiple distribution case.

In the former singular case, the asymptotic behavior of the free energy was
revealed by Watanabe[10], where it was applied to the statistical model
criteria WAIC[12] and WBIC[14]. In contrast, asymptotic behavior in the latter
case remains unknown. There, we decide to study the multiple parameters and
multiple distribution case, and clarify the asymptotic behavior of the free
energy. In particular we find that there is a new term that is not included in
the previous statistical learning theory.

##  2 Main Result

In this section, we explain the framework of Bayesian inference and show the
main result of this paper.

We assume that, from a true distribution $q(x)$, a set of $n$ independent
random variables $X^{n}=(X_{1},X_{2},\cdots,X_{n})$ and $X$ are generated,
where $X_{i}\in\mathbb{R}^{N}(0\leq i\leq N)$ and $X^{n}$ and $X$ are
independent. Let $\mathbb{E}_{X}[\;\;]$ and $\mathbb{E}[\;\;]$ denote
expectation values over $X$ and $X^{n}$. A statistical model and a prior are
represented by probability density functions $p(x|w)$ and $\varphi(w)$, $w\in
W\subset R^{d}$, respectively. In Bayesian inference, for a given inverse
temperature $\beta>0$, the marginal likelihood $Z_{n}(\beta)$, the posterior
distribution $p(w|X^{n})$, and the predictive distribution $p(x|X^{n})$ are
defined as

$\displaystyle Z_{n}(\beta)$
$\displaystyle=\int\prod_{j=1}^{n}p(X_{j}|w)^{\beta}\varphi(w)dw,$ (1)
$\displaystyle p(w|X^{n})$
$\displaystyle=\frac{1}{Z_{n}}\prod_{j=1}^{n}p(X_{j}|w)^{\beta}\varphi(w),$
(2) $\displaystyle p(x|X^{n})$ $\displaystyle=\int p(x|w)p(w|X^{n})dw.$ (3)

Note that $\beta=1$ results in the conventional Bayesian estimation. The free
energy $F_{n}(\beta)$ and the generalization loss $G_{n}(\beta)$ are defined
as

$\displaystyle F_{n}(\beta)$ $\displaystyle=-\frac{1}{\beta}\log
Z_{n}(\beta),$ (4) $\displaystyle G_{n}(\beta)$
$\displaystyle=-\mathbb{E}_{X}[\log p(X|X^{n})].$ (5)

The set of optimal parameters $W_{0}$ is defined as the set of all parameters
that minimize the KL divergence of $q(x)$ and $p(x|w)$,

$\displaystyle W_{0}=\\{w\in W\;;\;\int q(x)\log\frac{q(x)}{p(x|w)}dx\mbox{ is
minimized. }.\\}$ (6)

The log density ratio function for $w_{0}\in W_{0}$ and $w\in W$ is defined as

$\displaystyle f(x,w_{0},w)=\log\frac{p(x|w_{0})}{p(x|w)}.$ (7)

Then, by defining

$L(w)=-\mathbb{E}_{X}[\log p(X|w)],$

it follows that

$\mathbb{E}_{X}[f(X,w_{0},w)]=L(w)-L(w_{0}).$

It is said that the log density ratio function has a relatively finite
variance if the following condition is satisfied,

$\displaystyle\exists c_{0}>0,\forall w\in W,\forall w_{0}\in
W_{0},\;\;\mathbb{E}_{X}[f(X,w_{0},w)]\geq
c_{0}\mathbb{E}_{X}[f(X,w_{0},w)^{2}],$ (8)

It was proved in [10] that, if the log density ratio function has a relatively
finite variance, then the free energy and the generalization loss have
asymptotic expansions,

$\displaystyle\mathbb{E}[F_{n}(1)]$ $\displaystyle=nL(w_{0})+\lambda\log
n-(m-1)\log\log n+O(1),$ (9) $\displaystyle\mathbb{E}[G_{n}(1)]$
$\displaystyle=L(w_{0})+\frac{\lambda}{n}-\frac{m-1}{n\log
n}+o\left(\frac{1}{n\log n}\right),$ (10)

even if the Fisher information matrix is singular, where $\lambda>0$ is a
rational number called the real log canonical threshold (RLCT), and $m\geq 1$
is a natural number called the multiplicity. It was revealed that the
assumption (8) is satisfied if a parameter $w_{0}$ exists that satisfies
$p(x|w_{0})=q(x)$, or if the Fisher information matrix is positive definite at
a unique $w_{0}$.

However, in general, the assumption (8) is not always satisfied. For example,
it is known that, if the optimal probability distribution is not unique, that
is to say,

$\displaystyle\exists w_{01},w_{02}\in W_{0},\;\;p(x|w_{01})\neq p(x|w_{02}),$
(11)

then assumption (8) does not hold (See Lemma 1). Hence neither eq. (10) nor
eq. (11) does not holds in general. Previous research [13] treated the case
the assumption (8) does not hold, but the optimal probability distribution is
unique.

In this paper, we study the case in which the optimal probability distribution
is not unique, as shown in eq.(11), and show that the asymptotic behaviors of
the free energy and the generalization loss are given by

$\displaystyle\mathbb{E}[F_{n}(1)]$
$\displaystyle=nL(w_{0})-\mu\sqrt{n}+\hat{\lambda}\log n-(\hat{m}-1)\log\log
n+O(1),$ (13) $\displaystyle\mathbb{E}[G_{n}(1)]$
$\displaystyle=L(w_{0})-\frac{\mu}{2\sqrt{n}}+o\left(\frac{1}{\sqrt{n}}\right),$
(14)

where $\mu,\hat{\lambda},\hat{m}$ are real numbers satisfying $\mu\geq 0$,
$\hat{\lambda}>0$, and $\hat{m}\geq 1$. Since $\mu>0$, these results show that
both the free energy and the generalization loss are made smaller in Bayesian
estimation if the optimal probability distribution is not unique.

##  3 Definitions and Notation

In this section, we summarize several definitions and notation. We estimate a
probability distribution $q(x)$ by Bayesian inference using a statistical
model $p(x|w)$ and prior distribution $\varphi(w)$. It is assumed that the set
of parameters $W$ is compact and that $p(x|w)$ is continuous for $w$.
$W_{0}\subset W$ is set of all optimal parameters that minimizes $L(w)$. In
this case, the log density ratio function does not have a relatively finite
variance. In fact, if eq. (8) holds, there exists a positive real number
$c_{0}>0$ such that for any $w_{01},w_{02}\in W_{0}$,

$\displaystyle 0=L(w_{01})-L(w_{02})=\mathbb{E}_{X}[f(X,w_{01},w_{02})]\geq
c_{0}E_{X}[f(X,w_{01},w_{02})^{2}],$

and it follows that $f(x,w_{01},w_{02})=0$ for all $x$, which means
$p(x|w_{01})=p(x|w_{02})$.

Problem treated in this paper. We study the case,

$\displaystyle\exists w_{01},w_{02}\in W_{0}\;\;p(x|w_{01})\neq p(x|w_{02}),$

resulting that the assumption, eq.(8), does not hold. We assume that the set
$W_{0}$ can represented by a disjoint union of $W_{0i}(i\in I)$ which
satisfies

$\displaystyle\cup_{i\in I}W_{0i}$ $\displaystyle=W_{0},$ $\displaystyle
W_{0i}\cap W_{0j}$ $\displaystyle=\varnothing(i\neq j),$

and in each subset $W_{0i}$, the assumption holds. Hence, for an arbitrary
$w_{0i}\in W_{0i}$, $p(x|w_{0i})$ is the same probability distribution.

The empirical log loss function is defined by

$\displaystyle L_{n}(w)$ $\displaystyle=-\frac{1}{n}\sum_{j=1}^{n}\log
p(X_{j}|w).$ (15)

Accordingly, $L(w_{0i})$ does not depend on $i\in I$, and the random variable
depends on $i$:

$L_{ni}=L_{n}(w_{0i}).$

The log loss function $L(w)$ and the average error function $K(w)$ are defined
by

$\displaystyle L(w)$ $\displaystyle=-\mathbb{E}_{X}[\log p(X|w)],$ (16)
$\displaystyle K(w)$
$\displaystyle=\mathbb{E}_{X}[f(X,w_{0i},w)]=L(w)-L(w_{0i}).$ (17)

Accordingly $K(w)$ does not depend on $w_{0i}$ and K(w) satisfies

$\displaystyle K(w_{0i})=0,K(w)\geq 0.$ (18)

The empirical error function $K_{ni}(w)$ is defined by

$\displaystyle K_{ni}(w)=\frac{1}{n}\sum_{j=1}^{n}f(X_{j},w_{0i},w).$ (19)

It follows that

$L_{n}(w)=K_{ni}(w)+L_{ni}.$

##  4 Asymptotic Properties of Free Energy

In this section, we discuss the case in which the number of optimal
probability distributions is finite. If the number of optimal probability
distributions is a finite natural number m, the elements of the index set $I$
are natural numbers $(1,2,\cdots m)$. We will show that

$\displaystyle\mathbb{E}[F_{n}(\beta)]=nL(w_{0})-\sqrt{n}\mu+\frac{1}{\beta}(\hat{\lambda}\log
n-(\hat{m}-1)\log\log n)+O(1),$ (20)

holds. If $\mathbb{E}[G_{n}(1)]$ has an asymptotic expansion, then

$\displaystyle\mathbb{E}[G_{n}(1)]=L(w_{0})-\frac{\mu}{2\sqrt{n}}+o\left(\frac{1}{\sqrt{n}}\right)$
(21)

holds.

Example. The following is an example for the case that the optimal probability
distribution is not unique. We suppose supervised learning of $q(y|x)$ using a
statistical model $p(y|x,a,b)$. The variables $a,b$ are parameters. We suppose
true distribution $q(y|x),q(x)$ is

$\displaystyle q(y|x)$
$\displaystyle=\frac{1}{\sqrt{2\pi}}\exp\left(-\frac{(y-f(x))^{2}}{2}\right),$
$\displaystyle q(x)$
$\displaystyle=\left\\{\begin{array}[]{cc}\frac{1}{4}&(-2\leq x\leq 2)\\\
0&(\mbox{otherwise})\end{array}\right.,$ $\displaystyle f(x)$
$\displaystyle=\left\\{\begin{array}[]{cc}x+2&(-2\leq x<-1)\\\ 1&(-1\leq
x<1)\\\ -x+2&(1\leq x\leq 2)\end{array}\right..$

We also suppose a statistical model $p(y|x,a,b)$,

$\displaystyle p(y|x,a,b)$
$\displaystyle=\frac{1}{\sqrt{2\pi}}\exp\left(-\frac{(y-\sigma(ax+b))^{2}}{2}\right)$
$\displaystyle\sigma(x)$ $\displaystyle=\frac{1}{1+\exp(-x)}.$

In this situation, the KL divergence between $q(y|x)$ and $p(y|x,a,b)$ can be
calculated as

$\displaystyle KL(q(y,x)|p(y,x|a,b))$ $\displaystyle=\int
q(y,x)\log\frac{q(y|x)}{p(y|x,a,b)}=\int
q(y|x)q(x)\log\frac{q(y|x)}{p(y|x,a,b)},$ $\displaystyle=\frac{1}{2}\int
q(y|x)q(x)(f(x)-\sigma(ax+b))^{2}.$

Note that $q(x)$ and $f(x)$ are even functions. In addition, $\sigma(ax+b)$
and $\sigma(-ax+b)$ are line symmetric on $x=0$. Therefore,
$KL(q(y|x)|p(y|x,a,b))=KL(q(y|x)|p(y|x,-a,b))$ holds. Thus, we can find two
optimal parameters as $(a_{0},b_{0})$ and $(-a_{0},b_{0})$. These two points
satisfy $p(y,x|a_{0},b_{0})\neq p(y,x|-a_{0},b_{0})$. A numerical calculation
shows that the optimal parameters in this case are $(5.13,7.71)$,
$(-5.13,7.71)$.

To show eq.(20), let us represent the set of parameters as

$W=\cup_{i=1}^{m}W_{i},$

where $W_{0i}\subset W_{i}$. The marginal likelihood of each domain
$Z_{n}^{(i)}$ is also defined by

$\displaystyle Z_{n}^{(i)}=\int_{w\in W_{i}}\exp(-n\beta
L_{n}(w))\varphi(w)dw.$ (22)

By using $L_{n}(w)=K_{ni}(w)+L_{ni}$, we have

$\displaystyle Z_{n}^{(i)}=\exp(-n\beta L_{ni})\int_{w\in W_{i}}\exp(-n\beta
K_{ni}(w))\varphi(w)dw.$ (23)

The normalized marginal likelihood of each domain $Z_{n}^{0i}$ is defined by

$\displaystyle Z_{n}^{0i}=\int_{w\in W_{i}}\exp(-n\beta
K_{ni}(w))\varphi(w)dw.$ (24)

In accordance with these definitions, the marginal likelihood is given by

$\displaystyle Z_{n}=\int_{w\in W}\exp(-n\beta
L_{n}(w))\varphi(w)dw=\sum_{i=1}^{m}\exp(-n\beta L_{ni})Z_{n}^{0i}.$ (25)

The normalized marginal likelihood $Z_{n}^{0i}$ can be divided into
$Z_{n}^{1i}$ and $Z_{n}^{2i}$ as

$\displaystyle Z_{n}^{0i}=Z_{n}^{1i}+Z_{n}^{2i},$ (22) $\displaystyle
Z_{n}^{1i}=\int_{w\in W_{i},K(w)<\epsilon}\exp(-nK_{ni}(w))\varphi(w)dw,$ (26)
$\displaystyle Z_{n}^{2i}=\int_{w\in
W_{i},K(w)\geq\epsilon}\exp(-nK_{ni}(w))\varphi(w)dw.$ (27)

By taking $\epsilon>0$ to be a monotonically decreasing function of n which
satisfies

$\displaystyle\lim_{n\rightarrow\infty}\epsilon=0,$
$\displaystyle\lim_{n\rightarrow\infty}\sqrt{n}\epsilon=\infty,$

it can be proved that $Z_{n}^{2i}=O_{p}(\exp(-\sqrt{n}))$[10]. The optimal
parameter set in each $W_{i}$ is $W_{0i}$; therefore, in this set, the optimal
probability distribution is only $p(x|w_{0i})$. By assumption,

$\displaystyle\exists c_{0}\forall w\in W_{i}\forall
w_{0i}\;\;E_{X}[f(x,w_{0i},w_{i})]\geq c_{0}E_{X}[f(x,w_{0i},w_{i})^{2}].$

Then, applying the results of [10], by using the resolution theorem on each
$W_{i}$, there exists a measure $du_{i}^{*}$ such that

$\displaystyle Z_{n}^{(1i)}=\frac{(\log n)^{m_{i}-1}}{n^{\lambda_{i}}}\int
du^{*}_{i}\int t^{\lambda_{i}-1}\exp(-\beta
t+\beta\sqrt{t}\xi_{ni}(u_{i}))dt+o_{p}\left(\frac{(\log
n)^{m_{i}-1}}{n^{\lambda_{i}}}\right).$ (28)

In this equation, $\lambda_{i}$ and $m_{i}$ are respectively the real log
canonical threshold and the multiplicity of zeta functions, and
$\xi_{ni}(u_{i})$ is an empirical process that converges in distribution to a
Gaussian process.

Thus, the marginal likelihood is

$\displaystyle Z_{n}$ $\displaystyle=\sum_{i=1}^{m}\exp(-n\beta
L_{ni})Z_{n}^{0i}$ $\displaystyle=\sum_{i=1}^{m}\exp(-n\beta
L_{ni})\frac{(\log n)^{m_{i}-1}}{n^{\lambda_{i}}}\left(\int du^{*}_{i}\int
t^{\lambda_{i}-1}\exp(-\beta
t+\beta\sqrt{t}\xi_{ni}(u_{i}))dt+o_{p}\left(1\right)\right).$ (29)

From (4), the free energy is given by

$\displaystyle F_{n}(\beta)$ $\displaystyle=-\frac{1}{\beta}\log Z_{n}(\beta)$
$\displaystyle=f_{1}+f_{2}+f_{3}+o_{p}(1),$ (30)

where

$\displaystyle f_{1}$
$\displaystyle=-\frac{1}{\beta}\log\left(\sum_{k=1}^{m}e^{(-n\beta
L_{ni})}\right),$ $\displaystyle f_{2}$
$\displaystyle=-\frac{1}{\beta}\log\left(\frac{\sum_{i=1}^{m}e^{(-n\beta
L_{ni}-\lambda_{i}\log n+(m_{i}-1)\log\log n)}}{\sum_{k=1}^{m}e^{(-n\beta
L_{ni})}}\right),$ $\displaystyle f_{3}$
$\displaystyle=-\frac{1}{\beta}\log\left(\frac{\sum_{i=1}^{m}e^{(-\Theta(\beta,\xi_{ni})-n\beta
L_{ni}-\lambda_{i}\log n+(m_{i}-1)\log\log n)}}{\sum_{i=1}^{m}e^{(-n\beta
L_{ni}-\lambda_{i}\log n+(m_{i}-1)\log\log n)}}\right).$

In the above equations equation, we have used the notation,

$\displaystyle\Theta(\beta,\xi_{ni})$ $\displaystyle\qquad=-\log\left(\int
du_{i}^{*}\int_{0}^{\infty}dt\,t^{\lambda_{i}-1}\exp(-\beta
t+\beta\sqrt{t}\xi_{ni}(u_{i}))\right).$ (31)

In the following, we examine the asymptotic behaviors of the three terms
eq.(30). First, to study $f_{1}$, we define $i_{max}$ and $Y$ by

$\displaystyle i_{max}$
$\displaystyle=\underset{i}{\mbox{argmax}}\;(-L_{ni}),$ $\displaystyle Y$
$\displaystyle=-n\beta L_{ni_{max}}.$

From the definition,

$\displaystyle\log\left(\sum_{i=1}^{m}e^{(-n\beta
L_{ni})}\right)-Y=\log\left(1+\sum_{i\neq i_{max}}^{m}e^{(-n\beta
L_{ni}-Y)}\right).$ (32)

Since $Y+n\beta L_{ni}\geq 0$,

$\displaystyle 0<\log\left(\sum_{i=1}^{m}e^{(-n\beta L_{ni})}\right)-Y\leq
m\log 2.$

Therefore,

$f_{1}=Y/\beta+O_{p}(1)=-nL_{ni_{max}}+O_{p}(1).$

Note that the average of $L_{ni}$ is $L(w_{0i})$, which does not depend on
$i$. Let us define a random variable,

$\displaystyle\mathscr{L}_{n}(w_{0i})\equiv\sqrt{n}(-L_{n}(w_{0i})+L(w_{0i})).$
(33)

By using the central limit theorem, $\mathscr{L}_{n}(w_{0i})$ $(i=1,2,...,m)$
converges in distribution to an m-dimensional Gaussian random variable
$\mathscr{L}(w_{0i})$ on $w_{0i}\in W_{0}$ whose average is zero and variance-
covariance matrix $V=(V_{ij})$ is

$\displaystyle V_{ij}=\mathbb{E}_{X}[(\log p(X|w_{0i})+L(w_{0}))(\log
p(X|w_{0j})+L(w_{0}))].$ (34)

Then we obtain

$f_{1}=nL(w_{0})-\sqrt{n}\underset{w_{0}\in
W_{0}}{\mbox{max}}\mathscr{L}_{n}(w_{0})+O_{p}(1).$

Using $\mathscr{L}(w_{0})$, the asymptotic behavior of its average is given by

$\mathbb{E}[f_{1}]=nL(w_{0})-\mathbb{E}[\sqrt{n}\underset{w_{0}\in
W_{0}}{\mbox{max}}\mathscr{L}(w_{0})]+O(1).$

Now, let us examine the second term $f_{2}$ in eq.(30). If there exist
multiple $i_{max}$, we define $i_{max}$ as the $i$ whose RLCT is smallest, and
if the RLCTs are the same, we define $i_{max}$ as the $i$ whose multiplicity
is biggest. Using $i_{max}$ so defined, the asymptotic behavior of the second
term $f_{2}$ is given by

$\displaystyle-\beta f_{2}$
$\displaystyle=\log\left(\frac{\sum_{i=1}^{m}e^{(-n\beta
L_{ni}-\lambda_{i}\log n+(m_{i}-1)\log\log n)}}{\sum_{i=1}^{m}e^{(-n\beta
L_{ni})})}\right)$ $\displaystyle\qquad=-\lambda_{i_{max}}\log
n+(m_{k_{max}}-1)\log\log n$
$\displaystyle\qquad+\log\left(\frac{1+\sum_{k\neq i_{max}}^{m}e^{(-n\beta
L_{nk}-Y-(\Delta\lambda_{k})\log n+(\Delta m_{k})\log\log n)}}{1+\sum_{k\neq
i_{max}}^{m}e^{(-n\beta L_{ni}-Y)})}\right)$
$\displaystyle\qquad=-\lambda_{i_{max}}\log n+(m_{i_{max}}-1)\log\log
n+\log\left(\frac{1+o_{p}(e^{-n\beta a})}{1+o_{p}(e^{-n\beta a})}\right)$
$\displaystyle\qquad=-\lambda_{i_{max}}\log n+(m_{i_{max}}-1)\log\log
n+o_{p}(e^{-n\beta a}),$ (37)

where $\Delta\lambda_{i}$ and $\Delta m_{i}$ are
$\lambda_{i}-\lambda_{i_{max}}$ and $m_{i}-m_{i_{max}}$ respectively, and
$a\geq 0$ is the difference between $Y$ and the second biggest $-nL_{ni}$.

$\displaystyle\mathbb{E}[f_{2}]=\frac{1}{\beta}\left(\lambda_{i_{max}}\log
n-(m_{i_{max}}-1)\log\log n\right)+O(1).$ (38)

Next, let us study study the third term $f_{3}$ in eq.(30). We define a random
variable $a_{i}$ as follows.

$\displaystyle a_{i}=\frac{e^{-n\beta L_{ni}-\lambda_{i}\log
n+(m_{i}-1)\log\log n}}{\sum_{i=1}^{m}e^{(-n\beta L_{ni}-\lambda_{i}\log
n+(m_{i}-1)\log\log n)}}.$ (39)

The sum of $a_{i}$ over $i$ is 1. Using $a_{i}$ is 1. We can describe the
third term using $a_{i}$, so we can describe the free energy as follows.

$\displaystyle F_{n}(\beta)=nL(w_{0})$
$\displaystyle-\sqrt{n}\underset{w_{0}\in
W_{0}}{\mbox{max}}\mathscr{L}_{n}(w_{0})+\frac{\lambda_{i_{max}}}{\beta}\log
n-\frac{(m_{i_{max}}-1)}{\beta}\log\log n$
$\displaystyle-\log\left(\sum_{i=1}^{m}a_{i}e^{-\Theta(\beta,\xi_{ni})}\right)+O_{p}(1).$
(40)

Lastly, in order to derive the asymptotic behavior of the average
$\mathbb{E}[F_{n}]$, we show that the average of the random variable

$f_{4}\equiv\log\left(\sum_{i=1}^{m}a_{i}e^{-\Theta(\beta,\xi_{ni})}\right)$

is finite. By the Cauchy-Schwarz inequality,

$\displaystyle-\frac{t+\mbox{sup}_{u}|\xi_{ni}(u_{i})|^{2}}{2}\leq\sqrt{t}\xi_{ni}(u_{i})\leq\frac{t+\mbox{sup}_{u}|\xi_{ni}(u_{i})|^{2}}{2}$
(41)

holds. In the integral range of $u_{i}$,$[0,1]^{d}$, we have

$\displaystyle-\log\int du_{i}^{*}-\log\int
dt\,t^{\lambda_{i}-1}\exp(-\frac{\beta}{2}t)-\frac{\beta}{2}\underset{u_{i}\in[0,1]^{d}}{\sup}|\xi_{ni}(u_{i})|^{2}$
$\displaystyle\leq\Theta(\beta,\xi_{ni}(u_{i}))$ $\displaystyle\leq-\log\int
du_{i}^{*}-\log\int
dt\,t^{\lambda_{i}-1}\exp(-\frac{3\beta}{2}t)+\frac{\beta}{2}\underset{u_{i}\in[0,1]^{d}}{\sup}|\xi_{ni}(u_{i})|^{2}.$
(42)

Using Jensen’s inequality,

$\displaystyle f_{4}$
$\displaystyle\qquad\geq\sum_{i=1}^{m}a_{i}(-\Theta(\beta,\xi_{ni}(u)))$
$\displaystyle\qquad\geq\sum_{i=1}^{m}a_{i}\left(\log\int du_{i}^{*}+\log\int
dt\,t^{\lambda_{i}-1}\exp^{-\frac{3\beta}{2}t}-\frac{\beta}{2}\underset{u_{i}\in[0,1]^{d}}{\sup}|\xi_{ni}(u_{i})|^{2}\right)$
(43)

holds. From the nature of emprical processes, $\xi_{ni}(u_{i})$ converge in
low to Gaussian processes $\xi_{i}(u_{i})$ and

$\displaystyle\lim_{n\rightarrow\infty}E[\underset{u_{i}\in[0,1]^{d}}{\sup}|\xi_{ni}(u_{i})|^{2}]=\lim_{n\rightarrow\infty}E[\underset{u_{i}\in[0,1]^{d}}{\sup}|\xi_{i}(u_{i})|^{2}]$

holds. In regard to the lower bound of
$\log\left(\sum_{i=1}^{m}a_{i}e^{-\Theta(\beta,\xi_{ni}(u))}\right)$, the only
term that may diverge as a random variable is $\xi_{ni}(u_{i})$, so
$E[\log\left(\sum_{i=1}^{m}a_{i}e^{-\Theta(\beta,\xi_{ni}(u))}\right)]$ can be
shown to be bounded below. In addition, because
$\underset{u_{i}\in[0,1]^{d}}{\sup}|\xi_{ni}(u_{i})|^{2}\geq 0$ holds, we have

$\displaystyle f4$ $\displaystyle\quad\leq\log\left(\max_{i}\int
dt\,t^{\lambda_{i}-1}e^{-\frac{\beta}{2}t}\int
du_{i}^{*}\right)\left(me^{\frac{\beta}{2}\sum_{i=1}^{m}\mbox{sup}_{u}|\xi_{ni}(u_{i})|^{2}}\right)$
$\displaystyle\quad=\log\left(\max_{i}\int
dt\,t^{\lambda_{i}-1}e^{-\frac{\beta}{2}t}\int du_{i}^{*}\right)+\log
m+\frac{\beta}{2}\sum_{i=1}^{m}\mbox{sup}_{u}|\xi_{nk}(u_{i})|^{2}.$ (44)

Hence as we did for the lower bound it can be shown that
$\mathbb{E}[\log\left(\sum_{i=1}^{m}a_{i}e^{-\Theta(\beta,\xi_{ni}(u))}\right)]$
is bounded from above. By summing up the above equations, the asymptotic
behavior of $\mathbb{E}[F_{n}]$ can be described as

$\displaystyle\mathbb{E}[F_{n}(\beta)]$
$\displaystyle=nL(w_{0})-\sqrt{n}\mathbb{E}[\underset{w_{0}\in
W_{0}}{\mbox{max}}\mathscr{L}(w_{0})]$
$\displaystyle+\frac{1}{\beta}\sum_{k=1}^{m}\alpha_{i}(\lambda_{i}\log
n-(m_{i}-1)\log\log n)+O(1),$ (45)

where $\alpha_{i}$ is the probability that $i=i_{max}$. By putting
$\hat{\lambda}=\sum_{k=1}^{m}\alpha_{i}\lambda_{i}$ and
$\hat{m}=\sum_{k=1}^{m}\alpha_{i}m_{i}$, we obtain

$\displaystyle\mathbb{E}[F_{n}(\beta)]$
$\displaystyle=nL(w_{0})-\sqrt{n}\mathbb{E}[\underset{w_{0}\in
W_{0}}{\mbox{max}}\mathscr{L}(w_{0})]+\frac{\hat{\lambda}}{\beta}\log
n-\frac{\hat{m}-1}{\beta}\log\log n+O(1)$ (46)

holds.
When $\beta=1$ holds, we have [3, 5]

$\displaystyle\mathbb{E}[G_{n}(1)]$
$\displaystyle=\mathbb{E}[F_{n+1}(1)]-\mathbb{E}[F_{n}(1)].$ (47)

Using this equation, and assuming that $\mathbb{E}[G_{n}(1)]$ has an
asymptotic expansion, we find that

$\displaystyle\mathbb{E}[G_{n}(1)]=L(w_{0})-\frac{1}{2\sqrt{n}}\mathbb{E}[\underset{w_{0}\in
W_{0}}{\mbox{max}}\mathscr{L}(w_{0})]+o\left(\frac{1}{\sqrt{n}}\right).$ (48)

##  5 Experiment

In this section, we show the results of an experiment for the case when the
optimal probability distribution is not unique.We set the true distribution as

$\displaystyle
q(y|x)=\frac{1}{\sqrt{2\pi(0.2)^{2}}}\exp\left(-\frac{(y-f(x))^{2}}{2(0.2)^{2}}\right).$
$\displaystyle f(x)=\begin{cases}x+2&(-2\leq x<-1)\\\ 1&(-1\leq x<1)\\\
-x+2&(1\leq x\leq 2)\end{cases}$ $\displaystyle
q(x)=\begin{cases}\frac{1}{4}&(-2\leq x\leq 2)\\\
0&\mbox{otherwise}.\end{cases}$

We use following statistical model and prior distributions.

$\displaystyle
p(y|x,a,b)=\frac{1}{\sqrt{2\pi(0.2)^{2}}}\exp\left(-\frac{(y-\sigma(ax+b))^{2}}{2(0.2)^{2}}\right).$
$\displaystyle\sigma(x)=\frac{1}{1+\exp(-x)}.$
$\displaystyle\varphi(a)=\begin{cases}\frac{1}{20}&(0\leq x\leq 20)\\\
0&\mbox{otherwise}\end{cases}$
$\displaystyle\varphi(b)=\begin{cases}\frac{1}{40}&(-20\leq x\leq 20)\\\
0&\mbox{otherwise}.\end{cases}$

This statistical model has two optimal parameters

$\displaystyle w_{01}=(5.13,7.71),w_{02}=(-5.13,7.71).$

At these points,

$\displaystyle p(x|w_{01})\neq p(x|w_{02})$

holds. In this case, eq. (45) gives the theoretical asymptotic behavior of the
free energy versus inverse temperature for $\beta=1$. Note that the KL-
divergence between q(y,x) and p(y,x—a, b) in each neighborhood of the optimal
parameter is regular, so $\lambda=1andm=1$. The expectation of the maximum
value of a 2-dimensional Gaussian distribution follows a 1-dimensional
Gaussian distribution (see the appendix). We will show that the theoretical
behavior of free energy obeys

$\displaystyle\mathbb{E}[F_{n}(1)]$
$\displaystyle=nL(w_{0})-\sqrt{n}\sqrt{\frac{\mathbb{V}[\log(p(y|x,a_{0},b_{0}))-\log(p(y|x,-a_{0},b_{0}))]}{2\pi}}+\log
n+O(1).$ (49)

In eq. (49), $L(w_{0})$ and the coefficient of $\sqrt{n}$ can be calculated by
numerical integration. We used the average of $F_{n}$ calculated from the true
distribution $q(y|x),q(x)$ as the experimental value of $\mathbb{E}[F_{n}]$.
The prior distribution $p(a)$, $p(b)$ does not have an effect on the
asymptotic behavior. For this reason, we used equally spaced fixed values for
integration. We compared this experimental values and theoretical values,
except for the $O(1)$ term.

Figure 1: Experimental value and theoretical free energy depending on the
sample size. The error bar is the SE of the average of free energy. The
theoretical value of log likelihood$(nL(w_{0}))$ is subtracted from each
value. Figure 2: Difference between experimental value and theoretical value
in Figure1. The error bar is the same as in Figure1.

We calculated the experimental values of $\mathbb{E}[F_{n}]$ whose sample
sizes were $n=100$ to $600$ every $100$. The number of samples ranged from
$10000$ to $60000$ in steps of $10000$ for each sample sizes. Figure1 compares
the experimental and theoretical values. The experimental behavior of the free
energy depending on the sample size is similar to the theoretical behavior.
Figure2 shows the difference between the theoretical value and experimental
value. This difference corresponds to $O(1)$ term. This difference is remains
on this order regardless of the sample size. The experimental results support
the theoretical formula, eq.(45).

##  6 Discussion

We found that if the optimal probability distribution is not unique, the
apparent bias or the variance gets smaller for a finite sample number $n$
corresponding to a Gaussian process determined by the log loss of the optimal
parameter set and the reduction converges to 0 asymptotically.This behavior
can be explained qualitatively as follows: when there are two or more optimal
probability distributions, the posterior distribution can be selected to be
the nearest optimal probability distribution by bias of data, and this makes
the generalization loss smaller than the average generation of data. As the
number of samples and the bias increase, data generates averagely and
generalization loss gets larger.
In this paper, we showed that the asymptotic behavior of the free energy and
generalization loss are determined by $n^{\frac{1}{2}}$ and $n^{-\frac{1}{2}}$
order. Previous research[13] provides concrete example in which there is a
unique optimal probability distribution but assumption(8) does not hold. In
that paper, the asymptotic behavior of the free energy and the generalization
loss are determined by $n^{\frac{1}{3}}$ and $n^{-\frac{2}{3}}$ order, so we
predict that the lowest order determining the asymptotic behavior of the free
energy and generalization loss are $n^{\frac{1}{2}}$ and $n^{-\frac{1}{2}}$.
We showed that the asymptotic behaviors of the free energy and generalization
loss are determined by the maximum value of a Gauusian process. The
probability distribution of the maximum value of a Gaussian process or
multivariate normal distribution can not be calculated analytically, but an
approximate calculation, called the “tube method”[4] exists. There is also a a
method for calculating the upper and lower bounds of the expectation of the
maximum value of a Gauusian process, called “chaining”[6].This maximum value
is what determines the free energy and generalization loss in this paper.

##  7 Conclusion

We examined the case of when an important assumption in singular learning
theory about the log density ration function is loosened. In this case there
is a new term that is determined by a Gaussian process, whereby the
generalization loss asymptotically increases as the size of the dataset
increase. In the future, we should examine the asymptotic behavior of the
generalization loss as a random variable, in particular the asymptotic
equivalence of WAIC [12] and WBIC [14] in this case, and in the case in which
the assumption is completely removed.

## Appendix

We will derive the following equation.

$\displaystyle\mathbb{E}[{\mbox{max}}\mathscr{L}(w_{0})]=\sqrt{\frac{\mathbb{V}[\log(p(x|w_{01}))-\log(p(x|w_{02}))]}{2\pi}}$
(A1)

In this equation, $\mathscr{L}(w_{0})$ is a 2-dimensional gaussian
distribution which average is $0$ and variance-covariance matrix is

$\displaystyle V_{ij}=\mathbb{E}[(\log p(x|w_{0i})+L(w_{0}))(\log
p(x|w_{0j})+L(w_{0}))]$ (A2)

We define two random variables as

$\displaystyle z_{1}=\mathscr{L}(w_{01})-\mathscr{L}(w_{02}).$ $\displaystyle
z_{2}=\mathscr{L}(w_{01})+\mathscr{L}(w_{02})$

The random variables $(z_{1},z_{2})$ are also from 2-dimensional Gaussian
distribution whose average is $0$ and variance-covariance matrix is

$\displaystyle\left(\begin{array}[]{ccc}1&-1\\\
1&1\end{array}\right)\left(\begin{array}[]{ccc}V_{11}&V_{12}\\\
V_{21}&V_{22}\end{array}\right)$
$\displaystyle\left(\begin{array}[]{ccc}1&1\\\ -1&1\end{array}\right)$
$\displaystyle=\left(\begin{array}[]{ccc}V_{11}+V_{22}-V_{12}-V_{21}&V_{11}-V_{22}+V_{12}-V_{21}\\\
V_{11}-V_{22}-V_{12}+V_{21}&V_{11}+V_{22}+V_{12}+V_{21}\end{array}\right).$

The marginal distribution about $z_{1}$ is a 1-dimensional Gaussian
distribution whose average is $0$ and the variance is

$\displaystyle V_{11}+V_{22}-V_{12}-V_{21}.$

According to eq.(16) and eq.(34), we have

$\displaystyle V_{11}$ $\displaystyle+V_{22}-V_{12}-V_{21}$
$\displaystyle=\mathbb{E}[(\log p(x|w_{01})(\log
p(x|w_{01})]-L(w_{0})^{2}+\mathbb{E}[(\log p(x|w_{02})(\log p(x|w_{02})]$
$\displaystyle\qquad\qquad-L(w_{0})^{2}-2(\mathbb{E}[(\log p(x|w_{01})(\log
p(x|w_{02})]-L(w_{0})^{2})$ $\displaystyle=\mathbb{E}[(\log p(x|w_{01})-\log
p(x|w_{02}))^{2}]$ $\displaystyle=\mathbb{V}[(\log p(x|w_{01})-\log
p(x|w_{02}))]$ (A3)

We define a random variable $z_{3}$

$\displaystyle z_{3}=\begin{cases}z_{1}&z_{1}\geq 0\\\ 0&z_{1}<0\end{cases}$

By using $z_{3}$ we can describe the maximum value of $\mathscr{L}(w_{0})$ in
the following way,

$\displaystyle{\mbox{max}}\mathscr{L}(w_{0})=\mathscr{L}(w_{02})+z_{3}.$ (A4)

Considering the average of $\mathscr{L}(w_{02})$ is $0$, we find that

$\displaystyle\mathbb{E}[{\mbox{max}}\mathscr{L}(w_{0})]=\mathbb{E}[z_{3}].$
(A5)

$\mathbb{E}[z_{3}]$ is the expectation of a positive value in a Gaussian
distribution. This integration of a Gaussian whose variance is $\sigma^{2}$
can be calculated as

$\displaystyle\int_{0}^{\infty}\frac{x}{\sqrt{2\pi}}\exp\left(-\frac{x^{2}}{2\sigma^{2}}\right)$
$\displaystyle=\left[-\frac{\sigma}{\sqrt{2\pi}}\exp\left(-\frac{x^{2}}{2\sigma^{2}}\right)\right]_{0}^{\infty}$
$\displaystyle=\frac{\sigma}{\sqrt{2\pi}}.$ (A6)

From (A3), (A5), and(A6), we have

$\displaystyle\mathbb{E}[{\mbox{max}}\mathscr{L}(w_{0})]=\mathbb{E}[z_{3}]=\sqrt{\frac{\mathbb{V}[\log(p(x|w_{01}))-\log(p(x|w_{02}))]}{2\pi}}.$

Therefore (A1) holds.

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