# UQ for Credit Risk Management: A deep evidence regression approach

Ashish Dhiman
Department of Industrial and Systems Engineering
Georgia Institute of Technology
Atlanta, GA 30323
<EMAIL_ADDRESS>
###### Abstract

Machine Learning has invariantly found its way into various Credit Risk
applications. Due to the intrinsic nature of Credit Risk, quantifying the
uncertainty of the predicted risk metrics is essential, and applying
uncertainty-aware deep learning models to credit risk settings can be very
helpful. In this work, we have explored the application of a scalable UQ-aware
deep learning technique, Deep Evidence Regression and applied it to predicting
Loss Given Default. We contribute to the literature by extending the Deep
Evidence Regression methodology to learning target variables generated by a
Weibull process and provide the relevant learning framework. We demonstrate
the application of our approach to both simulated and real-world peer to peer
lending data.

##  1 Introduction

###  1.1 Credit Risk Management

Credit risk management is assessing and managing the potential losses that may
arise from the failure of borrowers or counterparties to fulfil their
financial obligations. In other words, it identifies, measures, and mitigates
the risks associated with lending money or extending credit to individuals,
businesses, or other organizations.

Credit risk’s anticipated loss (EL) comprises three components: Probability of
Default (PD), Loss Given Default (LGD), and Exposure at Default (EAD). PD is
the likelihood that a borrower will fail to fulfill their financial
commitments in the future. LGD refers to the proportion of the outstanding
amount that is lost in the event of default. Lastly, EAD refers to the
outstanding amount at the time of default.[8]

LGD prediction is important as accurate prediction of LGD not only supports a
healthier and risk-less allocation of capital, but is also vital for pricing
the security properly. [8] & [14]. There is a large body of literature using
advanced statistical and machine learning methods for prediction of LGD [8].
However the machine learning literature on LGD has yet to address an essential
aspect, which is the uncertainty surrounding the estimates and
predictions.[4].

UQ techniques like Bayesian Neural Network, Monte Carlo Dropout and ensemble
methods as outlined in [1] present a natural first step towards quantifying
uncertainty. However, almost all these methods are computationally and memory
intensive, and require sampling on test data after fitting the network, making
them difficult to adapt for complex neural network architectures that involve
a large number of parameters.

###  1.2 Deep Evidence Regression

The primary inspiration of this work is taken from the work done by Amini et
al in [2]. The paper develops a unique approach, Deep Evidence Regression as a
scalable and accurate UQ aware deep learning technique for regression
problems. This approach predicts the types of uncertainty directly within the
neural network structure, by learning prior distributions over the parameters
of the target distribution, referred to as evidential distributions. Thus this
method is able to quantify uncertainty without extra computations after
training, since the estimated parameters of the evidential distribution can be
plugged into analytical formulas for epistemic and aleatoric uncertainty, and
target predictions.

The setup of the problem is to assume that the observations from the target
variable, $y_{i}$ are drawn i.i.d. from a Normal distribution with unknown
mean and variance parameters $\theta={\mu,\sigma^{2}}$. With this we can write
the log likelihood of the observation as:

$Lik(\mu,\sigma^{2})=log(p(y_{i}|\mu,\sigma^{2})=-\frac{1}{2}\log(2\pi\sigma^{2})-\frac{(y_{i}-\mu)^{2}}{2\sigma^{2}}$

Learning $\theta$ that maximises the above likelihood successfully models the
uncertainty in the data, also known as the aleatoric uncertainty. However,
this model is oblivious to its predictive epistemic uncertainty. [2].
Epistemic uncertainty, is incorporated by placing higher-order prior
distributions over the parameters $\theta$. In particular a Gaussian prior is
placed on the unknown mean and an Inverse-Gamma prior on the unknown variance.

$\mu\sim\mathcal{N}(\gamma,\sigma^{2}\nu^{-1})\quad\sigma^{2}\sim\Gamma^{-1}(\alpha,\beta)$

Following from above the posterior $p(\mu,\sigma^{2}|\gamma,\nu,\alpha,\beta)$
can be approximated as $p(\mu|\gamma,\nu)*p(\sigma^{2}|\alpha,\beta)$. Hence:

$p(\mu,\sigma^{2}|\gamma,\nu,\alpha,\beta)=\frac{\beta^{\alpha}\sqrt{\nu}}{\Gamma(\alpha)\sqrt{2\pi\sigma^{2}}}(1/\sigma^{2})^{\alpha+1}\exp\big{(}{-\frac{2\beta+\nu(\gamma-\mu)^{2}}{2\sigma^{2}}}\big{)}$

Amini et al [2] thus find the likelihood of target variable given evidential
parameters, as:

$p(y_{i}|\gamma,\nu,\alpha,\beta)=\int_{\theta}p(y_{i}|\theta)p(\theta|\gamma,\nu,\alpha,\beta)$

where $\theta=\\{\mu,\sigma^{2}\\}$. Then a Neural Network is trained t infer,
the parameters $m=\\{\gamma,\nu,\alpha,\beta\\}$, of this higher-order,
evidential distribution.

###  1.3 Weibull distribution

The Weibull distribution is a continuous probability distribution commonly
used in reliability analysis to model the failure time of a system or
component.[7] The probability density function (PDF) of the Weibull
distribution is given by:

$f(x;\lambda,k)=\begin{cases}\frac{k}{\lambda}\left(\frac{x}{\lambda}\right)^{k-1}e^{-(x/\lambda)^{k}}&\text{if
}x\geq 0,\\\ 0&\text{if }x<0,\end{cases}$ (1)

where $\lambda>0$ is the scale parameter and $k>0$ is the shape parameter. The
scale parameter determines the location of the distribution, while the shape
parameter controls the rate at which the failure rate changes over time. There
is a body of lietrature that explores the application of the weibull
distribution to various credit risk applications. [9] [11].

The work by [12] assumes a normal distribution on LGD values. While this
assumption might be true in a lot of settings, however it does not follow in
the context of Loss Given Default. While normal distribution is symmetric and
has a support over entire real line, however the LGD values are restricted to
a range of $[0,1]$ and might not necessarily be symmetric.

Hence in the section below we provide a novel theoretical framework to learn
target variables which follow Weibull distribution. We provide the following
theoretical results, in the setting of target variables following a Weibull
dataset.

  * •

Log Likelihood

  * •

Mean Prediction

  * •

Prediction Uncertainty

We also provide results testing our approach on both simulated and real world
dataset.

##  2 Deep Evidence Regression for Weibull Data

###  2.1 Problem setup

We consider the problem where the observed targets, $y_{i}$, are drawn iid
from a Weibull distribution, with a known shape or rate parameter k and an
unknown scale $\lambda$. Although ideally we would want to keep both the
parameters unknown, however with both $\lambda$ and $k$ there are no priors
with which likelihood can be computed analytically [3]. Hence we have decided
to simplify the problem setup by assuming known shape $k$.

$\displaystyle y_{i}\sim Weibull(k,\lambda)$ (2) $\displaystyle\text{where
}k\in\mathrm{R}^{+},\lambda\in\mathrm{R}^{+}$ (3) Hence pdf of $y_{i}$ is (4)
$\displaystyle\implies
p(y_{i};\lambda,k)=\begin{cases}\frac{k}{\lambda}\big{(}\frac{y_{i}}{\lambda}\big{)}^{k-1}e^{-(y_{i}/\lambda)^{k}},&\text{if
$y_{i}\geq 0$}\\\ 0,&\text{otherwise}\end{cases}$ (5)

For the above setting we want to place priors on the unknown parameter,
$\lambda$, such that we are able to get solve for the likelihood of $y_{i}$
given the parameters of the prior distribution. Hence similar to work in [17]
and [5], we define the following prior.

$\displaystyle\theta=\lambda^{k}$ (6) Hence the pdf of $y_{i}$ becomes: (7)
$\displaystyle
p(y_{i}|\theta,k)=\frac{k}{\theta}{y_{i}}^{k-1}\exp{(-y_{i}^{k}/\theta)}$ (8)
$\displaystyle\text{And we place a Inverse Gamma Prior on }\theta$ (9)
$\displaystyle\theta\sim\Gamma(\alpha,\beta)\quad(\alpha>2)$ (10) Hence pdf of
$\theta$ is (11) $\displaystyle
p(\theta|\alpha,\beta)=\frac{\beta^{\alpha}}{\Gamma(\alpha)}\frac{1}{\theta^{\alpha+1}}\exp{(-\frac{\beta}{\theta})}$
(12)

###  2.2 Learning Log-Likelihood

Hence we can define likelihood of $y_{i}$ given the higher order evidential
parameters $\alpha,\beta$ can be defined as :

$\displaystyle
Lik=p(y_{i}|\alpha,\beta)=\int_{\theta}p(y_{i}|\theta,k)p(\theta|\alpha,\beta)d\theta$
(13) $\displaystyle\text{Now given }\lambda,k>0\implies\theta>0$ (14)
$\displaystyle
p(y_{i}|\alpha,\beta)=\int_{\theta=0}^{\infty}\big{(}\frac{k}{\theta}{y_{i}}^{k-1}\exp{(-y_{i}^{k}/\theta)}\big{)}\big{(}\frac{\beta^{\alpha}}{\Gamma(\alpha)}\frac{1}{\theta^{\alpha+1}}\exp{(-\frac{\beta}{\theta})}\big{)}d\theta$
(15)
$\displaystyle=ky_{i}^{k-1}\frac{\beta^{\alpha}}{\Gamma(\alpha)}\int_{\theta=0}^{\infty}\frac{1}{\theta^{\alpha+2}}\exp(-\frac{y_{i}^{k}+\beta}{\theta})$
(16)
$\displaystyle=ky_{i}^{k-1}\frac{\beta^{\alpha}}{\Gamma(\alpha)}\frac{\Gamma(1+\alpha)}{{(y_{i}^{k}+\beta)}^{1+\alpha}}$
(17) $\displaystyle=\frac{\alpha
ky_{i}^{k-1}\beta^{\alpha}}{(y_{i}^{k}+\beta)^{\alpha+1}}$ (18)

Hence the log-likelihood for i’th observation is defined as:

$\displaystyle Log-Lik_{i}=L^{lik}_{i}=\log\alpha_{i}+\log k+(k-1)\log
y_{i}+\alpha_{i}\log\beta_{i}-(\alpha_{i}+1)({y_{i}}^{k}+\beta_{i})$ (19)

We set up our neural network to minimise the negative Log-Likelihood plus some
regularisation cost, discussed in section below.

###  2.3 Mean Prediction and UQ of prediction

Given the main advanatge of Deep Evidence Regression over other UQ aware deep
learning methods like Bayesian NN, esembling etc, is due to existence of
analytical solution for both predictions and unceratinty from NN output,
without the need for sampling. Hence this section details the derivation of
mean prediction and total prediction uncertainty.

####  2.3.1 Mean Prediction

$\displaystyle\text{We define the mean prediction as }E[Z|\alpha,\beta]$ (20)
$\displaystyle\text{where }Z=E[y_{i}]$ (21) Now given $y_{i}\sim
Weibull(k,\lambda)$ (22) $\displaystyle
E[Z]=E[\lambda*\Gamma(1+\frac{1}{k})]=E(\lambda)*\Gamma(1+\frac{1}{k})\quad\text{(k
is known)}$ (23) $\displaystyle E[\lambda]=\int_{\lambda}\lambda
p(\lambda)d\lambda$ (24)

Hence to solve for mean prediction we need to find pdf $p(\lambda)$. Because
we know $\theta=\lambda^{k}\sim\Gamma^{-1}(\alpha,\beta)$, we can use change
of variable to find pdf of $\lambda$ [18].

$\displaystyle\implies
E[\lambda|\alpha,\beta]=\frac{k\beta^{\alpha}}{\Gamma(\alpha)}*\Gamma(\frac{k\alpha-1}{k})*\frac{1}{k}*\frac{1}{\beta^{\frac{k\alpha-1}{k}}}$
(26)

The mean prediction can thus be simplified as:

$\displaystyle E[Z|\alpha,\beta]=E(\lambda)*\Gamma(1+\frac{1}{k})$ (27)
$\displaystyle=\frac{k\beta^{\alpha}}{\Gamma(\alpha)}*\Gamma(\frac{k\alpha-1}{k})*\frac{1}{k}*\frac{1}{\beta^{\frac{k\alpha-1}{k}}}*\Gamma(1+\frac{1}{k})$
(28)
$\displaystyle=\Gamma(1+\frac{1}{k})\frac{1}{\Gamma(\alpha)}\Gamma(\alpha-\frac{1}{k})*\beta^{1/k}$
(29)

####  2.3.2 Prediction Uncertainty

We quantify the total uncertainty as $Var(Z)$ with defined as above, i.e.
$Z=E[y_{i}]$

$\displaystyle Var(Z|\alpha,\beta)=Var(\lambda)*\Gamma^{2}(1+\frac{1}{k})$
(30) $\displaystyle=(E[\lambda^{2}]-E[\lambda])*\Gamma^{2}(1+\frac{1}{k})$
(31) With $E(\lambda)$ defined as in 26, we only need $E(\lambda^{2})$ (32)
$\displaystyle E[\lambda^{2}]=\int_{\lambda}\lambda^{2}p(\lambda)d\lambda$
(33)

Similar to approach outlined in 2.3.1, we get:

$\displaystyle
E[\lambda^{2}|\alpha,\beta]=\Gamma(\frac{k\alpha-2}{k})\frac{\beta^{2/k}}{\Gamma(\alpha)}$
(35)

Hence we can write

$\displaystyle
Var(Z)=\Gamma^{2}(1+\frac{1}{k})*\big{[}\Gamma(\frac{k\alpha-2}{k})\frac{\beta^{2/k}}{\Gamma(\alpha)}-\big{(}\Gamma(\frac{k\alpha-1}{k})\frac{\beta^{1/k}}{\Gamma(\alpha)}\big{)}^{2}\big{]}$
(36)

or

$Var(Z)\propto\frac{\beta^{2/k}}{\Gamma(\alpha)^{2}}\big{[}\Gamma(\alpha)\Gamma(\frac{k\alpha-2}{k})-\Gamma^{2}(\frac{k\alpha-1}{k})\big{]}$

###  2.4 Regularisation Cost

In this section, we outline the process of regularization during training by
implementing a regularisation penalty, which involves assigning a high
uncertainty cost. The purpose of this penalty is to inflate the uncertainty
associated with incorrect predictions, thereby improving the overall efficacy
of the model. As followed in [2], the intuition behind the regularisation cost
is to increases the variance of prediction in cases where it’s unsure. This
utility of this approach has been demonstrated in classification setting by
[13] and in regression setting by [2].

Hence we define the Regularisation cost for the i’th observation as

$L_{i}^{reg}=|error_{i}|*(\frac{\alpha_{i}}{\beta_{i}})\\\ $

where $error_{i}=y_{i}-Z_{i}$ and Z is defined in 27.

Note The regularization cost mentioned earlier has been determined as the most
effective through experimentation. However, in order to precisely determine
the coefficients of $\alpha$ and $\beta$ in the regularization cost, further
theoretical analysis is required. By conducting a deeper theoretical
investigation, we can establish the optimal values for these coefficients,
which will enhance the regularization process and improve the overall
performance of the model.

###  2.5 Neural Network training schematic

The learning process is then set up with a deep neural network with two output
neurons to predict the parameters of the prior/evidential distribution,
$\alpha$ and $\beta$. The neural network is trained using the cost function:

$L^{NN}_{i}=-L^{lik}_{i}+c*L^{reg}_{i}$

Figure 1: NN training schematic

where c is the hyper-parameter governing the strength of regularisation.

##  3 Results and Experiments

In this section, we present the results of experiments conducted on both
simulated and real data. The aim was to evaluate the performance of our
proposed method and compare it with existing methods. The simulated data was
generated based on example data given in [2], while the real dataset was
obtained from a peer to peer lending company.

###  3.1 Simulated Data

Here generate a target variable following a Weibull distribution. The target
variable is generated as:

$y_{i}=x_{i}^{2}+\epsilon,\epsilon\sim Weibull(k=1.6,\lambda)$

The train set is comprised of uniformly spaced $x\in[-4,4]$ while test set is
$x\in[-5,5]$. The value of $\lambda$ is varied between $[0.2,0.4]$ to test the
effect of noise magnitude on the approach.

Figure 2: Synthetic data generated for varying $\lambda$

Next we fit both the original deep evidence regression and proposed weibull
version of deep evidence regression. Since our approach assumes known $k$, k
is estimated from the training set.

Comparing the two versions qualitatively, we observe that the original model’s
predictions exhibit consistent uncertainty regardless of whether the data is
within or outside the distribution. In contrast, the proposed version
demonstrates improved capability in capturing prediction uncertainty. The
proposed model’s prediction interval gradually expands beyond the training
data range $|x|>4$, indicating its ability to account for uncertainty in Out-
of-distribution data. On comparison, for the benchmark version of the model
displays a slightly narrower prediction interval at the edges of the training
window, contrary to expectations of interval widening.

Figure 3: Deep evidence regression (left) vs Weibull evidence Regression
(right). We see that uncertainty is much better captured by proposed version.

Quantitatively to assess the performance of our proposed method, we compare it
to the benchmark model by evaluating key metrics such as mean squared error
(MSE) and negative log-likelihood (NLL).

We can see that the proposed version exhibits significantly lower negative log
likelihood compared to the original Deep Evidence Regression model. This
indicates that the proposed model better aligns with the actual distribution
of the data, capturing the uncertainty more accurately. However, despite this
improvement, the original model outperforms in capturing the underlying signal
beyond the training window, as evidenced by its lower mean squared error (MSE)
values.

Table 1: Deep Evidence regression original vs proposed results on simulated data for varying $\lambda$. We see MSE (or mean squared error) is similar for both, while NLL (or Negative log likelihood) values are much better captured by proposed version. $\lambda$ | MSE(test) | NLL(test)
---|---|---
benchmark | proposed | benchmark | proposed
0.2 | 0.099303 | 0.571519 | 70.64365 | 7.416122
0.25 | 0.119299 | 0.504875 | 41.71667 | 6.667958
0.3 | 0.142722 | 3.871369 | 36.16713 | 6.202275
0.35 | 0.143117 | 3.202328 | 57.02918 | 5.773156
0.4 | 0.172697 | 8.981477 | 42.53032 | 5.471559

###  3.2 Real Data: Loss Given Default for peer to peer lending

In this subsection, we showcase the utility of our proposed learning approach
by extending it to the intricate and complex domain of credit risk management
in the context of peer to peer lending. Peer-to-peer lending, which is an
emerging form of credit aimed at funding borrowers from small lenders and
individuals seeking to earn interest on their investments. Through an online
platform, borrowers can apply for personal loans, which are typically
unsecured and funded by one or more peer investors. The P2P lender acts as a
facilitator of the lending process and provides the platform, rather than
acting as an actual lender.

Credit risk management is crucial for peer-to-peer lending data as it helps
mitigate the potential default risks associated with borrowers, ensuring a
healthier loan portfolio and reducing financial losses. By effectively
analyzing and managing credit risk, P2P lending platforms can maintain
investor confidence, attract more participants, and sustain the long-term
viability of the lending ecosystem.

The dataset under consideration pertains to peer to peer mortgage lending data
during the period of 2007 to 2014 sourced from Kaggle [16]. However, the data
does not include the loss given default values. Instead, the recovery rate has
been used as a proxy, which is calculated as the ratio of recoveries made to
the origination amount. The dataset contains approximately 46 variables
denoted as ’x,’ which include features such as the time since the loan was
issued, debt-to-income ratio (DTI), joint applicant status, and delinquency
status, among others. In total, the dataset comprises around 23,000 rows.

Figure 4: Distribution and Weibull fit of the recovery rate (left). Histogram
of recovery rate for train/test split (right). It appears that Weibull
distribution might not be a good fit to this data.

As described in the approach the shape parameter was found as $k=1.254$ by
fitting a Weibull distribution on the train dataset. Also given the
sensitivity of both the approaches to regression cost, hyper parameter
optimisation was done to arrive at the best regularisation cost. After
arriving at the best regularisation cost 10 trials of neural network training
were conducted with this best regularization cost for benchmark and proposed
model separately.

Similar to the synthetic case, we see that the proposed model demonstrates
superior performance compared to the benchmark model in terms of mean squared
error (MSE) and negative log likelihood. Additionally, it exhibits the ability
to generate more accurate prediction intervals. The difference in performance
between the benchmark and proposed models is even more pronounced in this case
compared to the simulated data, and the benchmark fails even to retrieve the
underlying signal, let alone the prediction uncertainty. To reinforce this,
the benchmark model was also run with 0 regularisation cost and it was found
to not improve the MSE. This behaviour outlines the difficulty of tuning
regularisation parameter for benchmark model. It is also worth noting that the
benchmark model also predicts $\infty$ as uncertainty for a significant number
of observations.

Table 2: Results for original vs proposed model for recovery rate. proposed version does not only has both lower MSE and NLL | MSE | NLL
---|---|---
| benchmark | proposed | benchmark | proposed
test | 84.333 $\pm$ 0.352 | 40.498 $\pm$ 4.745 | 2.757 $\pm$ 0.012 | 2.311 $\pm$ 0.024
train | 84.320 $\pm$ 0.216 | 39.909 $\pm$ 4.911 | 2.766 $\pm$ 0.009 | 2.314 $\pm$ 0.0228
Figure 5: Predicted UQ for Benchmark Regression (left) vs proposed
model(right). Again we see that the updated model is much better able to
capture the UQ. With Uncertainty increasing after recovery rate increases
beyond 40, which is a less dense region and has much fewer observations 4

##  4 Related Work

This work is primarily inspired by the work done by Amini et al [2], which
proposes Deep Evidentail Learning approach. Maximillian et al [12] have used
the same approach and shown it’s utility to Loss given default for bonds from
Moody’s recovery database. Additionally there’s a huge body of prior work on
uncertainty estimation [10] [6] and the utilization of neural networks for
modeling probability distributions. In another line of work, Bayesian deep
learning utilizes priors on network weights estimated with variational
inference [1]. There also exists alternative techniques to estimate predictive
variance like as dropout and ensembling which require sampling. [1]

##  5 Conclusion, limitations and future work

We propose an improvement over Deep Evidence Regression, specifically targeted
to usecases where the target might follow weibull distribution. We then test
the proposed method both on simulated and real world dataset in the context of
Credit risk management. The proposed model exhibits enhanced suitability for
applications in which the target variable originates from a weibull
distribution, better capturing the uncertainty characteristics of such data.
Although we have specifically tested the model in the credit risk domain, this
method can be applicable to wide variety of safety critical regression tasks
where the target variable follows a weibull distribution. The proposed
approach thus serves as a valuable tool for capturing and quantifying
uncertainty in cases characterized by weibull distributions, thereby enhancing
the trustworthiness and explainability of model predictions, ultimately
leading to improved confidence in the modeling process and the corresponding
decision making.

However, we are not sure if the proposed approach would generalise to other
distributions apart from Weibull. Additionally, the proposed model has only
two outputs, which could limit its flexibility when compared to the benchmark
model, which had four outputs from the neural network. In consequence the
proposed model requires a deeper network architecture compared to the
benchmark model. Furthermore we find that both the models exhibit a high
sensitivity to regularization cost, which means that changes in the
regularization coefficient can significantly impact the model’s performance.
The cylical learning rate as outlined in the [15], which proposes varying the
learning rate between reasonable boundary values might be of help to mitigate
this issue. Overall, these points suggest that both models have their
strengths and weaknesses, and selecting the most appropriate model depends on
the specific task requirements and considerations.

Considering the wide application of beta distributions in the credit risk
domain, there may also be value in further extending the proposed technique to
target variables characterized by beta distributions, as it has the potential
to provide valuable insights and improved modeling capabilities in the context
of credit risk management.

## References

  * [1] Moloud Abdar, Farhad Pourpanah, Sadiq Hussain, Dana Rezazadegan, Li Liu, Mohammad Ghavamzadeh, Paul Fieguth, Xiaochun Cao, Abbas Khosravi, U. Rajendra Acharya, Vladimir Makarenkov, and Saeid Nahavandi.  A review of uncertainty quantification in deep learning: Techniques, applications and challenges.  Information Fusion, 76:243–297, 2021.
  * [2] Alexander Amini, Wilko Schwarting, Ava Soleimany, and Daniela Rus.  Deep evidential regression.  In H. Larochelle, M. Ranzato, R. Hadsell, M.F. Balcan, and H. Lin, editors, Advances in Neural Information Processing Systems, volume 33, pages 14927–14937. Curran Associates, Inc., 2020.
  * [3] Pasquale Erto and Massimiliano Giorgio.  A note on using bayes priors for weibull distribution, 2013.
  * [4] Paolo Gambetti, Geneviève Gauthier, and Frédéric Vrins.  Recovery rates: Uncertainty certainly matters.  Journal of Banking & Finance, 106:371–383, 2019.
  * [5] Pseudodifferential Operators (https://stats.stackexchange.com/users/320438/pseudodifferential operators).  Find a conjugate prior for the weibull distribution under reparametrization.  Cross Validated.  URL:https://stats.stackexchange.com/q/594050 (version: 2022-10-30).
  * [6] Alex Kendall and Yarin Gal.  What uncertainties do we need in bayesian deep learning for computer vision?  In I. Guyon, U. Von Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vishwanathan, and R. Garnett, editors, Advances in Neural Information Processing Systems, volume 30. Curran Associates, Inc., 2017.
  * [7] Chin-Diew Lai, D. Murthy, and Min Xie.  Weibull distributions and their applications.  Springer Handbook of Engineering Statistics, Chapter 3:63–78, 02 2006.
  * [8] Matthias Nagl, Maximilian Nagl, and Daniel Rösch.  Quantifying uncertainty of machine learning methods for loss given default.  Frontiers in Applied Mathematics and Statistics, 8, 2022.
  * [9] Ismail Noriszura, Jamil Jaber, and Siti Mohd Ramli.  Credit risk assessment using survival analysis for progressive right-censored data: a case study in jordan.  Journal of Internet Banking and Commerce, 22:1–18, 05 2017.
  * [10] Harris Papadopoulos and Haris Haralambous.  2011 special issue: Reliable prediction intervals with regression neural networks.  Neural Netw., 24(8):842–851, oct 2011.
  * [11] Rebeca Peláez, Ricardo Cao, and Juan M. Vilar.  Bootstrap Bandwidth Selection and Confidence Regions for Double Smoothed Default Probability Estimation.  Mathematics, 10(9):1–25, May 2022.
  * [12] Murat Sensoy, Lance Kaplan, and Melih Kandemir.  Evidential deep learning to quantify classification uncertainty.  In S. Bengio, H. Wallach, H. Larochelle, K. Grauman, N. Cesa-Bianchi, and R. Garnett, editors, Advances in Neural Information Processing Systems, volume 31. Curran Associates, Inc., 2018.
  * [13] Murat Sensoy, Lance Kaplan, and Melih Kandemir.  Evidential deep learning to quantify classification uncertainty.  In S. Bengio, H. Wallach, H. Larochelle, K. Grauman, N. Cesa-Bianchi, and R. Garnett, editors, Advances in Neural Information Processing Systems, volume 31. Curran Associates, Inc., 2018.
  * [14] Luc Severeijns.  Challenging lgd models with machine learning, 2018.
  * [15] Leslie N. Smith.  Cyclical learning rates for training neural networks, 2017.
  * [16] Shawn Sun.  Loan data for credit risk modeling.  https://www.kaggle.com/datasets/shawnysun/loan-data-for-credit-risk-modeling, 2021\.
  * [17] tomka (https://math.stackexchange.com/users/118706/tomka).  Conjugate prior for the weibull distribution.  Mathematics Stack Exchange.  URL:https://math.stackexchange.com/q/3529658 (version: 2020-02-04).
  * [18] Wikipedia contributors.  Change of variables — Wikipedia, the free encyclopedia, 2023.  [Online; accessed 1-May-2023].