Machine-Learning Methods for Insurance Applications

Machine-Learning Methods

for Insurance Applications

A Survey

January 2019

Machine-Learning Methods for Insurance

Applications

A Survey

Caveat and Disclaimer

The opinions expressed and conclusions reached by the authors are their own and do not represent any official position or opinion of the Society of

Actuaries or its members. The Society of Actuaries makes no representation or warranty to the accuracy of the information

ALEX DIANA

PhD Candidate in Statistics

University of Kent

SPONSOR

Society of Actuaries

JIM E. GRIFFIN

PhD, Professor of Statistics

University of Kent

PROJECT OVERSIGHT

GROUP

Syed Danish Ali

Mary Pat Campbell

Thomas Farmer

Valerie Gingras

Lie Jen Houng

Mervyn Kopinsky

Min Mercer

Michael Niemerg

Shisheng Qian

Steven Siegel

Ronora Stryker

Jim Wright

JAIDEEP OBEROI

PhD, Lecturer in Finance

University of Kent

JI YAO

PhD, FIA, CERA

Ernst & Young, China, and

Shanghai Jiao Tong University

CONTENTS

Executive Summary ................................................................................................................................................... 4

Introduction .............................................................................................................................................................. 5

Machine Learning ........................................................................................................................................................... 5

Data 6

Generalized Linear Model.......................................................................................................................................... 7

Issues With GLM in Insurance Modeling ....................................................................................................................... 7

Regularization ................................................................................................................................................................. 7

LASSO, Elastic Net and Ridge Regression ......................................................................................................... 7

Feature Selection: Subset Selection, Stepwise Regression and Bayesian Variable Selection ....................... 8

Nonlinear Classification and Regression Models ........................................................................................................ 9

Classification and Regression Trees (CART) .................................................................................................................. 9

Ensemble Methods .................................................................................................................................................. 11

Bagging .......................................................................................................................................................................... 11

Random Forests ............................................................................................................................................................ 12

Boosting and Gradient Boosting .................................................................................................................................. 14

Bayesian Additive Regression Trees (BART) ................................................................................................................ 17

Other Machine Learning Methods and Topics.......................................................................................................... 18

Multivariate Adaptive Regression Splines (MARS) ..................................................................................................... 18

Neural Nets and Deep Learning................................................................................................................................... 18

Unsupervised Learning................................................................................................................................................. 19

Cluster Analysis ................................................................................................................................................ 19

Additional Methods and Combinations ...................................................................................................................... 19

Existing Comparative Studies .................................................................................................................................. 20

Applications to Wider Problems for Actuaries ......................................................................................................... 20

Estimation Packages ................................................................................................................................................ 20

Challenges ............................................................................................................................................................... 21

Interpretation ............................................................................................................................................................... 21

Discovering Nonlinear Relationships, Dealing With Large Numbers of Variables and Sparse Data ........................ 21

Conclusion ............................................................................................................................................................... 21

References .............................................................................................................................................................. 23

About The Society of Actuaries ................................................................................................................................ 27

Machine-Learning Methods

for Insurance Applications

Executive Summary

This project has been awarded funding by the Society of Actuaries from the Research Expanding Boundaries (REX)

pool to contribute to actuarial practice expansion. The objective of the current report is to provide a literature survey

of methodological improvements from the field of machine learning to insurance claim modeling.

In this report, we describe and illustrate a range of machine-learning approaches that have been used in the insurance

literature or have the potential to be used. We emphasize variable selection methods, including those applicable to

the generalized linear model (GLM), the current workhorse for the industry. The methods covered include LASSO,

elastic net, ridge regression and Bayesian variable selection. The former three methods involve penalizing the

objective function as a means of shrinking certain parameter estimates toward zero. The latter approach assigns

weights to alternative models in order to combine them to select features that have predictive value.

We then discuss classification and regression trees (CART), a method that helps capture the nonlinearities that are

challenging for the linear approaches. However, it also suffers from high variance and lack of smoothness. CART can

in many cases be augmented by a range of ensemble methods, that combine more trees (estimated in parallel or

sequentially) to improve the trade-off between bias and variance. The methods we cover are bagging, random forests,

boosting and Bayesian additive regression trees (BART). In bagging and random forests, the idea is to average across

a number of trees fitted to subsets of the data, whereas boosting involves sequential estimation using the residuals

from the previous model fit. These methods are suited to trees but are not restricted to them. BART involves

combining simple trees, which uses a prior to discourage complicated tree structures.

We also include multivariate adaptive regression splines (MARS), a more traditional nonparametric regression method

that generates a nonlinear regression function by combining shorter line segments. In our illustrations, this method

is outperformed by the other methods considered.

This report is accompanied by two working files, prepared in the Jupyter software, that illustrate the implementation

of the models covered in the report using R. Each file uses a different data set. The first is the Group Long Term

Disability data set (see Kopinsky 2017), and the second is the Long Term Care Incidence data set (see Bodnar et al.

2015).

In addition to the regularization methods listed here, an important consideration for evaluating model predictions is

the loss function itself. We illustrate two alternatives, the mean-squared error (MSE, applied to the disability data)

and the receiver operating curve (ROC, applied to the long-term care data).

The evidence in the public domain about the performance of the various machine-learning methods in claims

prediction is still limited. Overall, the literature has found that different methods offer improvements for specific

applications, suggesting the need for more robust examinations of model performance. Although we compare the

models using simple out-of-sample predictive ability, we also find that there is potential for improving predictions

relative to linear methods. We recommend further detailed comparisons, which could be made possible by the

increased availability of standard public data sets for researchers to use.

Introduction

The actuarial profession strives to develop or adopt the best approaches to understanding claims data. Recent trends

in data collection and data science call for the attention of actuaries to examine whether existing approaches to

modeling can be improved and lead to better decisions. Data sets are getting larger as researchers collect and combine

information from a broad range of sources. This has necessitated the development of methods to select the most

relevant variables in predictive models and to expunge spurious relationships. Claims data tend to be sparse, volatile,

skewed and time variant. This makes them excellent candidates for analysis using machine-learning approaches,

particularly variable selection methods.

The industry has widely adopted generalized linear models (GLMs) as a standard approach to modeling claims.

Typically, claims are modeled either as a combination of two components (frequency and severity) or directly using a

distribution such as the Tweedie distribution. While this approach is more accurate than a multiway table approach,

its popularity is also based on the speed with which it can be implemented and its easy interpretability. However, the

standard models do have certain limitations, and there is a potential that some of these limitations may be addressed

by the methods in this survey.

This survey attempts to link the challenges of using GLMs with the motivation for using the methods covered here.

One such challenge is the time taken to build models when incorporating new information. This becomes more

important as data availability from a wide range of sources makes it possible to consider more complex relationships.

As a further challenge, the discovery of interaction effects is difficult, often forcing model builders to rely on summary

indicators (e.g., a credit score rather than the underlying census and credit information used to construct it). In

addition, missing data from external sources poses difficulties to the modeler, as does the issue of uncertainty in

sparse segments when estimating GLMs.

The machine-learning methods applied to insurance data covered here include tree-based methods and regularization

methods, such as the LASSO and Bayesian variable selection methods. While reviewing methods, we also note

evidence where available on other criteria, such as interpretability, resource requirements (difficulty, speed, scale),

stability and prediction performance.

Before proceeding to a review of the methods, we briefly introduce machine learning.

Machine Learning

Machine learning is a field that absorbs techniques from a wide range of disciplines with the objective of prediction

based on data. There are two broad categories of machine learning: supervised and unsupervised. In this review, we

report comparisons of methods classified as supervised learning, though we also document unsupervised methods

that have been applied to insurance problems.

There are now many textbooks that describe the range of machine-learning methods. Two references that are widely

used are Hastie et al. (2009) and Murphy (2012). The aim of machine-learning methods is to be able to produce

generalizable rules for prediction based on patterns identified in data. The methods offer natural tools for prediction

under a range of tasks (e.g., classification, clustering, regression). The discovered patterns can often be very complex.

Due to their complexity, it is also usually necessary to employ regularization to avoid overfitting that would lead to

poor prediction performance in unseen data. As a result, machine-learning methods tend to incorporate both

estimation and model selection within the same procedure. As the tasks and loss functions vary by context, the

development of machine-learning methods has been relatively more problem specific. Caruana and Niculescu-Mizil

(2006) provide comparisons of a range of methods based on multiple criteria, emphasizing the need to match the

method to the loss function.

Two articles that discuss the potential for machine-learning methods in econometrics are Varian (2014) and

Mullainathan and Spiess (2017). They provide a useful introduction to machine-learning methods applied to common

problems such as the estimation of treatment effects and the prediction of outcomes based on individual

characteristics.

Actuaries have recognized the potential of machine-learning methods as part of their focus on predictive analytics,

suggesting several applications of the methods. An early article summarizing a selection of machine-learning methods

with actuarial science applications was Shapiro (2000), in which he explained some optimization methods as well as

the use of neural networks and classification algorithms.

These resources provide both an overview of machine-learning methods and detailed reference on implementation.

In the sections that follow, we provide brief summaries of the most popular approaches related to GLMs and machine

learning, listing some of the existing findings in the literature.

We also present comparative estimates on two data sets that we will describe in the next subsection. Accompanying

this document are additional files produced in the Jupyter Notebook format that include the R code and results from

the prediction comparisons we have carried out. These files can be read in a browser, and the R code copied to

replicate the results. Jupyter Notebook is open-source software that integrates several languages. For those

interested in working within the Jupyter format to modify the R code directly and produce new results, we have also

made available the Jupyter files with a brief description and a link to the resource website for the software. The data

used for the two illustrations is provided in accompanying CSV files.

Data

The data sets used for the empirical illustrations are publicly available from the Society of Actuaries. The first is the

group long-term disability (GLTD) data kindly provided by Mervyn Kopinsky based on his report on using tree models

to predict recovery and mortality rates (Kopinsky 2017). The second is the Society of Actuaries Long-term Care

Intercompany Experience Study (Bodnar et al. 2015).

The GLTD recovery rate data consists of more than 500,000 observations of seven predictors that summarize up to

46 million records. Details of the data processing can be found in Kopinsky (2017). Summaries of the data are provided

in the accompanying Jupyter file.

The long-term care claim incidence (LTCI) data is composed of more than 10 predictors and more than 1 million

observations. We expand the pivot table provided and then collapse some of the categories with no actual claims.

Once again, details of the data and empirical work are provided in the accompanying Jupyter file.

An essential feature of building good data-driven models is the data itself. The data we use in this report is already

cleaned and prepared, requiring relatively minor adjustments. While we do not focus in this survey on the

preprocessing of data, it should be noted that each paper we report here spends a considerable amount of attention

describing the data and discussing its quality and relevance.

When comparing the models, we consider the mean-squared error (MSE) when estimating the GLTD data and the

area under the curve (AUC) when estimating the LTCI data. The AUC refers to the area under the receiver operating

characteristic curve, which is produced by plotting the true positive rate against the false positive rate at various points

(quantiles). When classification is perfect, the AUC equals 1.0 (the area of the square plot), whereas when it is random,

the AUC moves toward 0.5.

Performance criteria play an important role in machine-learning methods because they are directly aligned with the

objective function. As a result, model selection and estimation apply the same objective function.

Generalized Linear Model

GLMs are widely used for pricing in the insurance industry. The classic textbook on GLMs is McCullagh and Nelder

(1989). Applications of GLMs to insurance problems can be found in Brockman and Wright (1992), Anderson et al.

(2007), De Jong and Heller (2008), Ohlsson and Johansson (2010) and Dean (2014).

GLMs use a linear regression via a link function to predict variables that have potentially non-normal distributions, as

is often the case in actuarial science. In other words, a GLM is specified as:















 ,

where 







is a differentiable and strictly monotonic function,  is the dependent variable,  is a matrix of predictors

and  is the parameter vector. The data is assumed to be from the exponential family of distributions (including

common distributions such as the binomial, Poisson and Gaussian), and the model can be estimated using maximum

likelihood or Bayesian methods.

Issues With GLMs in Insurance Modeling

Some of the criticisms of GLMs include the following:

• Either zero or full credibility is given to the data, and there is no way to do blending.

• Prediction of a risk depends on data in other, potentially different segments.

• Model predictions depend on the mixture of rating factors in the data.

• Maximum-likelihood estimate of prediction is lower than the mean of the prediction distribution.

• Link function could bias the model prediction and significantly change the lower and upper bound of prediction.

• Model diagnostics are relevant only in the segments where the model is used.

In this report, we will consider two main broad approaches to this problem. The first approach assumes that the

structure of GLMs is correct but that there is a list of potential features. It is often ineffective to include all these

variables in the model, since parameter estimates can be poor with large standard errors leading to inaccurate

predictions. An effective method for choosing features that are useful for prediction can address these problems and

allow consideration of a large number of potential features. This can allow more accurate predictions than careful

preselection of features. This is known as feature selection. The second approach assumes that the relationship

between the features and the mean of the target is not appropriately modeled by a GLM. Specifically, these methods

allow for nonlinear relationships and feature selection rather than the linear relationships (after a transformation

through the link function) and feature selection in the first approach. The second approach is more flexible but can

more easily overfit the data, leading to poor predictions. The first approach offers models that are often easier to

interpret but can predict poorly if the linearity assumption is not appropriate for the data.

For the first approach, a number of methods are available under the category of regularization methods. Next we

discuss general regularization methods that are widely applied to most estimation methods. Other methods will be

introduced in later sections.

Regularization

The essential objective with regularization is to fit regression models with large numbers of variables while avoiding

overfitting within the training data. Two broad groups of approaches are described next.

LASSO, Elastic Net and Ridge Regression

Regularization methods have become increasingly popular following the seminal work of Tibshirani (1996), where the

LASSO estimator for linear regression models was developed. The idea is to replace the maximum-likelihood estimator

of the regression coefficients (and any other model parameters) with a penalized maximum-likelihood estimator.

Suppose we have regression coefficients (and potentially other model parameters ); then the penalized maximum

likelihood estimator of  and  is













 







,

where   is the log-likelihood function for a GLM and  is a pre-specified penalty function. The penalty

function penalizes 



, whose absolute value is unrealistically large and leads to estimates for which the overall

magnitude of  is smaller than the maximum-likelihood estimate. The following choices for  are popular:

• LASSO penalty function: 







 













where we have  features

• Elastic net penalty function: 







 



 







 



















• Ridge penalty function: 







 













An important property for feature selection is that the penalized maximum-likelihood estimates for the effect of some

features can be exactly zero. This allows some variables to be removed from the model. The ability to generate zeros

is controlled by the choice of . The LASSO and elastic net penalty functions are able to generate zeros, whereas

the ridge penalty cannot. The elastic net penalty reduces to the LASSO penalty if 



  and reduces to the ridge

penalty if 



 . Therefore, the elastic net is seen as a compromise between the LASSO penalty and the ridge penalty.

There has been a substantial amount of work on the properties of these procedures, which is reviewed in Hastie et

al. (2015). A succinct review for actuaries is provided in Niemerg (2016). One important practical consideration is the

way that the feature selection method behaves when some features are highly correlated. If a group of highly

correlated features contains important features, the LASSO penalty will tend to set the coefficient for only one feature

to be nonzero, whereas the elastic net will tend to set more of these coefficients to be nonzero.

A practical problem in the use of regularization methods is the choice of the penalty parameters:  for the LASSO and

ridge penalty functions, and 



and 



for the elastic net penalty. These are usually chosen using the idea of separating

the data into different sections for validation purposes. The basic approach is to have in-sample data or a training

sample to estimate the model and out-of-sample data or a test sample to evaluate performance. The estimation step

itself requires regularization though methods such as cross-validation. In cross-validation, the data is separated into a

number of folds, and then estimation takes place by sequentially leaving out one or more folds for evaluation of the

predictive fit of the model. The process allows one to tune parameters that may, for instance, determine the

complexity of the model. At the least, to avoid overfitting, we need to separate the data into a training section and a

test section. Recent reports on predictive modeling from the Society of Actuaries (see, e.g., Xu et al. 2015; Ewald and

Wang 2015) highlight the use of separate training and test data sets as an essential element of building a predictive

model.

For both data sets that we have analyzed in this report, we set aside 30% of the sample as a test sample and use 70%

for training. Our initial checks using regression and a GLM with the logit link highlight the degree of nonlinearity,

something that linear models would struggle with. We are using a data set with relatively fewer variables than one

might have access to when conducting a pricing exercise. As a result, there is little room for methods such as the

LASSO to improve the out-of-sample performance of the model by removing overfitting or shrinking the parameters

of redundant predictors.

Feature Selection: Subset Selection, Stepwise Regression and Bayesian Variable Selection

In regularization methods, we concentrate on estimating the coefficients of the features, the s, and feature selection

arises as a by-product of the regularization. Other feature selection methods concentrate on calculating a goodness-

of-fit measure for different possible subsets of the potential features. For example, subset selection methods calculate

a goodness-of-fit measure, such as AIC or BIC, for all possible subsets. The final model is chosen to be the model with

the best value for the goodness-of-fit measure, such as the smallest value of the AIC or the BIC. There are 



possible

models if there are  possible features, so this method is feasible only if  is relatively small (for example,  less than

30). In large-scale problems, iterative algorithms can be used to find an optimal model. For example, in stepwise

methods, suppose we have a chosen subset of features at one iteration; the following iteration considers the effect

on the goodness-of-fit measure of including any one of the features that are currently not used or removing one

feature that is currently used. The algorithm would then choose the change that leads to the largest improvement in

the measure of goodness of fit. Randazzo and Kinney (2015) use a logistic regression to identify likely frequent visitors

to the emergency room. While they do not specify the variable selection approach, they do highlight the top 20

predictors in order of their contribution to 



Many approaches to feature selection concentrate on finding a single model. Bayesian approaches offer the

opportunity to combine all possible models in the analysis. Predictions can be made by combining the predictions for

each model with weights determined by the fit of each model to the data. In the Bayesian approach, a prior

distribution is defined for all parameters, and we are able to use the decision to include or exclude a feature as a

parameter. We define 



to be 1 if the ith feature is included and 0 if it is not included, and   







 is a vector

for all possible features. Then, conditional on the included feature, a prior distribution can be specified for the

regression coefficients for the included features, 



 and other parameters, . Although the prior distributions can be

chosen using expert information, they are usually chosen to lead to a suitably simple model. A typical setup is 







are independent and 







 



 . This implies that the prior expected number of included features is , and this

can be used to choose a value of  that leads to a suitable prior mean for the data.

Once a prior distribution has been chosen, Markov chain Monte Carlo (MCMC) algorithms or variational Bayes

algorithms (Carbonetto et al. 2017) can be used to calculate the posterior distribution of sets of included features.

This allows a prediction to be calculated by averaging predictions from the different sets of included features weighted

by their respective posterior probabilities.

Nonlinear Classification and Regression Models

Generalized linear regression models assume that the relationship between features and the response are linear

(usually, after some transformation of the effect of the features through a link function). In particular, this restricts

the expected response to be either increasing or decreasing as a function of each feature. Generalized linear

regression models can be made more flexible by including powers of features or interactions between features.

However, if the number of features is large, this approach can lead to models with huge numbers of parameters,

which makes estimation challenging. These difficulties with generalized linear models have led to interest in nonlinear

classification and regression models, which allow more general relationships between the features and the expected

response, including interactions. In this section, we will first review the popular classification and regression tree

(CART) approach before considering development of this approach to so-called ensemble methods that include

random forests and a Bayesian alternative, Bayesian additive regression trees (BART).

Classification and Regression Trees (CART)

In GLMs, a single model is used for all values of the feature. In contrast, a popular approach to nonlinear models

assumes that different models are used for different combinations of the features. For example, if we have two

features, age and sex, we might be able to build separate models for men over 60, men under 60, women over 50

and women under 50. All observations will fall within one of these categories, and we say we have partitioned the

feature space into the different categories. Note that the male and female populations have been split at different

ages. Figure 1 illustrates the idea.

Figure 1: A Simple Tree Structure

To use the approach, we need a way to describe the partition of feature spaces (i.e., the different categories), the

model for the categories, and a way to infer the partitions and models from data. In CART, the partitions are described

by a tree. This structure has benefits for inferring the partitions which can be learned sequentially. In each category,

we usually assume a constant level for the response for continuous data and assume a constant probability of positive

response in a classification problem. If we use 







to represent the partitions, then, in the continuous case, the

regression function is





















  



,

where 



is the level for the kth partition.

The advantage of using a tree structure is that the structure of the tree can be learned sequentially. We start by

considering the partition 



 



  and 



 



 , where the variable to be split j and the split point s

need to be chosen. These parameters can be chosen by finding the values that minimize







 























 















where 



is the mean of the 



s in 



and 



is the mean of the 



s in 



The minimization over the split points can

be simplified by noticing that we only need to consider the splits defined by each observed value of the jth variable

once the first split variable and split point have been found. The algorithm then subdivides 



and 



by finding split

variables and split points in each region. This process can be continued by subdividing all current regions at further

steps of the algorithm. Typically, the tree is grown until the minimum number of observations in a region (such as 5)

is reached. This usually leads to a large tree that will tend to overfit the data and lead to poor out-of-sample predictive

performance. This is addressed by pruning the tree by removing some splits.

The method has been extensively applied in the machine-learning literature, but some limitations have been

identified:

1. Instability of the tree. The estimated tree structure can be sensitive to sample of data. For example, dividing

the data into two halves and estimating trees can lead to very different inferred trees. Therefore, CART is

often considered a method with a high variance. This is a particularly important consideration for actuarial

Male = True Male = False

Age > 60 Age > 50Age 60 Age 50

applications, where the dependent variable (e.g., claim incidence) can be very unevenly distributed in the

data. Cross-validation and ensembling help to address this problem. Ewald and Wang (2015) also suggest the

use of stratified sampling.

2. Lack of smoothness. By its very nature, the tree allows two adjacent groups to have significantly different

models. While this may be appropriate in some circumstances, it may also lead to difficulties. For instance,

actuaries often interpolate or extrapolate predictions to categories where the data are insufficient (e.g., a

class where there is no experience of claims in the data).

Parkes (2015) uses ensemble decision trees to incorporate numerical measurements from electronic medical records

to improve prediction of care costs. He highlights that one of the advantages of using the method is its flexibility in

the presence of high levels of dependence among the features and the overall nonlinearity of the relationships

modeled.

Tree-related models can be studied using the rpart library in R. A careful evaluation of the use of CART is provided in

Kopinsky (2017), from whom we have received the GLTD data. We confirm his finding that CART improves on GLMs

for the GLTD data. However, when applied to the LTCI data, we do not find that CART outperforms GLMs.

Ensemble Methods

The previous methods have only considered the use of one model for prediction. In contrast, ensemble methods use

a number of different models to make predictions. Intuitively, this is a case of “hedging our bets.” If the different

models provide different predictions, then sensibly combining the predictions from each model can lead to much

better overall predictive performance. There are many ensemble methods, and we will consider some of the most

popular ones. Before introducing the idea of a random forest, it is useful to describe the idea of bagging. We will then

discuss boosting and BART.

Bagging

Bagging of trees was introduced to address the problem of the high variance of trees. In this method, B new data sets

are created from the original data set. Each new data set has n observations, which are sampled with replacement

from the original data set. A regression tree is estimated from each new data set, and the predictions for each tree

are averaged. Therefore, the ensemble estimate of the regression function is





























where 



is the regression function from fitting a regression tree to the bth new data set. The intuition behind the

idea is that the variance of an average will be lower than the variance for each element in the average.

To implement bagging with CART, one could use the Rborist package in R. When implementing bagging, we need to

choose the number of trees, . In practice, the prediction error on test data, which is often called the “bagging error”,

tends to stabilize as the number of trees increases, so we only need to find the point at which stabilization begins.

Figure 2 shows the results for different numbers of trees with the LTCI data. We find that the error rate stabilized

beyond approximately 250 trees, with values in the range 0.0011615 to 0.0011625.

Figure 2: Mean-Squared Error for Different Numbers of Trees: Held-Out Testing Sample With GLTD Data Set

When we applied bagging in the context of CART, we found an improvement in the MSE for the GLTD data set but no

improvement for the LTCI data set. This may be, in part, due to the differences in the incidence rates in the two data

sets, as the LTCI data set has very low incidence rates, leading to significant variation across small subsamples.

Random Forests

Breiman (2001) introduced the random forest as a variation on the idea of bagging trees. Again, B new data sets are

created by sampling with replacement, but the way the regression tree is constructed is restricted. In regression trees,

we are allowed to use any variable to make the split at any point in the tree. In a random forest, at any point in the

tree, a candidate set of c features are selected at random (without replacement) from the list of features, and only

these features can be used to define a split. The value of c is much smaller than the number of predictors p and is

typically chosen to be  



. To understand why this method might work, consider the argument for bagging trees,

i.e., that the variance of an average is much smaller than the variance of elements of the average. This is clearly true

if the elements are independent (which is essentially the central limit theorem), but the reduction will be small if the

elements are highly correlated. Therefore, reduction in the variance from bagging depends on the predictions from

the trees estimated from each new data set not being highly correlated. The extra step in random forests encourages

greater diversity in trees than bagging, since some variables are excluded as split variables at each step of the tree-

fitting algorithm.

One of the early applications of tree ensembles that supports their potential for insurance applications was Derrig and

Francis (2008). They showed improvements by random forest and TreeNet (an alternative algorithm) over logistic

regression in predicting fraud in vehicle claims.

Shehadeh et al. (2016) applied random forests to data on 130,000 applications for life insurance. They found that

stratified sampling was essential to exploit the performance of the random-forest algorithm, as the data was heavily

unbalanced and claims represented less than 5% of the data set.

The Rborist package also works for random forests. The user must choose hyperparameters, such as the number of

trees, the number of predictors chosen at each split, and the minimum size of a node (pruning choice). We used 250

trees following the results using bagging (although this could again be chosen using the MSE on a test set) and

compared different minimum sizes of nodes and the number of predictors chosen at each split.

Figure 3 shows the MSE for the test data after varying these two hyperparameters for the GLTD data. The test errors

are reduced by allowing three variables to be tried at each split (gold line), rather than two variables (red line), but

increases as we use four, five, six or seven variables. This is typical of random forests with an optimal number of

variables to be allowed at each split, providing the lowest MSE. In contrast, bagging allows all variables to be

considered at each split, so random forests usually provide a lower MSE. The plot also show that imposing a minimum

size of nodes leads to improved MSE on the data test. If we restrict to three possible variables at each split, the MSE

decreases as the minimum size of the node increases to 150 but then increases for larger minimum size. In our

applications, we find that random forests perform best of all models for the GLTD data but do not offer much

improvement for the LTCI data over the CART model.

Figure 3: Mean-Squared Error (MSE) for Different Parameters of the Random Forest Algorithm: Held-Out

Testing Sample With GLTD Data Set

Note: The MSE is reported for minimum node sizes ranging from 75 to 200, with c representing the number of

predictors chosen at each split.

A challenge for random forests is their complexity, as the interpretation becomes much more complex than for the

underlying trees. One way of addressing this is to provide measures of relative variable importance. In fitting a CART

model, the split variable is chosen by looking at the improvement of the fit to the data. A natural way to measure the

relative importance of a particular variable is to sum the improvements due to that variable. In a random forest, we

further sum across all these measures across all trees in the ensemble and weight according to the contribution of

each tree to the overall fit to the data. Figure 4 shows the relative importance for the variables in the LTCI data set.

This identifies a few variables as very important to the fit. Unfortunately, in contrast to the fitted random forest, this

method cannot describe the interactions between variables.

Minimum node size

MSE

Figure 4: Relative Importance of Variables: Random Forest Algorithm With GLTD Data Set

Boosting and Gradient Boosting

Boosting is an alternative method for building an ensemble of models. Unlike bagging, which fits models to new data

sets created by resampling the original data set, boosting works sequentially and fits a model based on the output

from the previous models. The approach is most easily understood as an algorithm. We initialize the regression

function 









  and residuals 



 



for all observations. The algorithm then repeats the following steps for  

:

1. Fit a tree model to the features X and responses r. The regression function for this tree is 





2. Set 



















 









3. Calculate new residuals 



 



 













The final fitted model is 









 













. Intuitively, this approach allows the model to adjust the regression

function in several iterations rather than in one simple fit, as in the basic CART model. By fitting the residuals from

previous fits, the model is able to concentrate on fitting the data in areas where previous models have performed

more poorly. Many applications of these methods have shown that boosted methods outperform standard

approaches.

Gradient boosting is a generalization of boosting to allowing the estimation problem to be defined by a loss function.

For example, if we have a continuous response variable, then we would want to learn the regression function ,

which is the mean of the response variable for a particular value of the features  In this case, a suitable loss function

is  







    









, since it is minimized when 







 . However, we could also consider another loss

Variable names

Importance

function such as  







    







, which might be more robust to a heavy-tailed distribution of regression

errors. The gradient-boosting algorithm starts with an initial regression function 









  and then involves the

following steps for   :

1. Calculate pseudo-residuals 









 

































2. Fit a model to the features X and responses r. The fitted regression function is 





3. Find 



that minimizes



















 





















4. Set 



















 













The final fitted model is 











Using the gradient-boosting example, we highlight the role of another hyperparameter important for the pruning of

trees: the maximum tree depth. In addition, we try different values of the shrinkage parameter around the default.

To implement boosting in R, we use the xgboost package. Several hyperparameters need to be chosen, and we initially

concentrate on three: the number of trees, the maximum depth of each tree and the 



parameter in the learning

algorithm. The MSEs are shown in Figure 5. Varying 



for a fixed number of trees and maximum depth leads to a

similar shape, where the MSE is highest for small and large values with an optimum in between. The optimum value

of 



tends to get smaller as the maximum depth and the number of trees increase. We find that the optimum MSE

over these parameters is given by the choices of max depth equals 4, number of trees equals 250, and 



 .

Many other parameters can be adjusted in the gradient-boosting algorithm, and results of varying these

hyperparameters are given in the accompanying HTML document. We find that these parameters have a much smaller

effect on MSE for the problems that we consider.

Note: We varied λ and the number of trees, with cases of maximum tree depth of 3, 4 and 5.

Figure 5: MSEs for Different Choices of Hyperparameters in Gradient-Boosting

Algorithm With GLTD Data

The relative importance of variables in gradient boosting can be measured in a similar way to the random forest.

The importance of a variable within each tree can be measured by summing the improvements in overall fit due to

split in that variable. In gradient boosting, these measures are combined across different trees in the ensemble by

weighting the th tree by 



. Figure 6 shows the results for the LTCI data.

Alternative scaling functions for the errors also can be used in the boosting iterations. Lee et al. (2015) and Lee and

Lin (2018) have proposed a modified model to take account of the fact that actuarial adjustments are often

multiplicative. In this approach, in the non-Gaussian cases, the residuals to be fitted are ratios rather than

differences.

Bayesian Additive Regression Trees (BART)

Bayesian Additive Regression Trees (BART) are a Bayesian ensemble method. In the case of a continuous response,

the additive regression model is



















 







Figure 6: The Importance of Different Variables for the Gradient-Boosting Algorithm

where 



 









and 



















are B tree models with different parameters (split variables and split

positions). The intuition is that each 



 function can estimate different aspects of the relationship between x and

y. The depth of a tree is the maximum number of splits between the root node, partition with no splits and the leaves,

the partitions at the end of the tree. Boosting approaches to estimation with trees will limit the depth of each tree to

avoid overfitting the data (to create weak learners). In this Bayesian modeling approach, a prior is placed on the depth

of the tree to avoid each tree becoming “too complicated” and leading to an ensemble of weak learners. This prior

that a node at depth d is terminal has the simple form









   



  





where      and    Increasing the parameter  for fixed  leads to a lower probability of terminating at

depth d and so encourages a larger tree. Increasing the parameter  for fixed  leads to a higher probability of

terminating at depth d and so encourages a smaller tree. The authors suggest using the values    and   

as a default, which encourages trees with depth of 2 or 3. Inference can be made in the model using Bayesian

backfitting MCMC (Hastie and Tibshirani 2000).

Other Machine-Learning Methods and Topics

Multivariate Adaptive Regression Splines (MARS)

MARS was introduced by Friedman (1991) as a nonparametric regression technique that essentially splits the

regression line into shorter segments:









 



















,

where 









, known as basis functions, are of the form 







  



, 



  



 



or a product of these

expressions. These effectively introduce kinks in the regression function. The model is estimated as a sequence of

forward regressions that are likely to lead to overfitting. Then terms are dropped by carrying out a backward

regression. While this is easier to visualize in a univariate context, it should be highlighted that MARS is meant to be

particularly useful in the multivariate context, because it explores the set of interactions.

To implement MARS in R, we use the package earth. On both our data sets, we find that MARS improves on GLMs,

suggesting the considerable nonlinearity required of our model. However, it is outperformed by other methods.

Francis (2003) compared MARS with neural networks on a closed-claims database for personal-injury protection, a

database used to predict fraudulent claims or suspicion of fraudulent claims. Her results were not conclusive, due to

the relatively small sample size, but she argues for considering MARS because of its relative interpretability.

Neural Nets and Deep Learning

One of the most widely known methods from machine learning is that of neural networks. The earlier literature has

focused extensively on “neural nets,” and they are now a standard part of the estimation tool kit in many applications.

However, for insurance, they typically suffer from the criticism of being difficult to interpret. The neural net is also the

underlying component of currently popular classification approaches such as deep learning. However, the issue of

interpretability still remains a key one for insurance applications.

We tried a simple neural net for the GLTD data and found that it underperformed other methods, so we have

continued to leave it outside the scope of this review. Francis (2003) compares MARS against neural networks for

predicting fraud and finds that neural networks are marginally superior on a number of criteria.

Unsupervised Learning

Unsupervised learning is different from the approaches we cover in this review because the objective is to discover

patterns in the data without any guidance—in particular, no predicted or dependent variable or model. Francis (2014)

provides an overview of the potential for use of unsupervised learning in insurance. Many of the techniques used for

supervised learning may also be used for unsupervised learning. For instance, Francis (2016) uses PRIDIT (see Ai et al.

2009) and random forests to help identify suspicious automobile insurance claims. Unsupervised-learning approaches

have been used in conjunction with supervised learning, often as a data-processing step. The most popular use of

such approaches is to group data into clusters, which we briefly describe in this section.

Cluster Analysis

Clustering is the process of grouping a set of data objects into clusters so that data objects within a cluster have high

similarity in comparison to one another but are dissimilar to objects in other clusters. Usually a similarity measure is

defined, and the clustering procedure is to optimize this measure locally or globally.

Clustering analysis is widely used in the pre-modeling phase to group granular data into a more manageable number

of levels for modeling. It enables a better understanding of data, as features of data may become clearer and more

meaningful after the data are grouped into clusters. It also reduces the volatility of data and may help attain more

stable rates over time. The result is that the modeler may reduce the number of levels in a rating factor, thereby

making it more likely that a model converges and produces statistically significant results. Yao (2016) provides

illustrations of various clustering methods with the help of auto insurance claims data in the UK. He also proposes an

exposure adjusted hybrid (EAH) method that relies on clustering for modeling claim risk.

There are various clustering methods, including the following:

• Partitioning methods: k-means method, k-medoids method, and expectation maximization

• Hierarchical methods: agglomerative nesting (AGNES), Divisia analysis (DIANA), balanced iterative reducing and

clustering using hierarchies (BIRCH), clustering using representatives (CURE) and Chameleon

• Density-based methods: density-based spatial clustering of application with noise (DBSCAN), ordering points to

identify the clustering structure (OPTICS), and density-based clustering (Denclue).

• Grid-based methods

• Kernel and spectral methods

Additional Methods and Combinations

As predictive analytics develops, more and more studies rely on a combination of the core machine learning and

statistical methodologies to carry out their analyses. Apart from data processing or preprocessing, previously

discussed, other combinations are regularly attempted in insurance. For instance, Kolyshkina et al. (2005) use MARS

as a preprocessing step to speed up the estimation of a GLM on car claims data by a significant multiple. Niemerg

(2016) evaluates the random GLM on a range of data sets and finds that, although it offers greater interpretability, its

prediction ability is not superior to random forests. Kunce and Chatterjee (2017) also propose a “machine learning

approach to parameter estimation” that combines a sequence of techniques including k-nearest neighbors, kernel

regression and relevance vector machines.

Existing Comparative Studies

There are not many large-scale studies comparing a wide range of models for specific insurance applications. Dugas

et al. (2003) carried out a rate-making study for a nationwide U.S. automobile insurer and compared a range of models

with an emphasis on neural networks. They provide a detailed explanation of the effect of the advantages of neural

networks to capture the nonlinearities in insurance problems. They compare a number of models (including multiway

tables, a GLM, GLMs with regularization, MARS, neural networks and mixture models). They find that mixture models

perform the best overall on criteria such as fairness (reasonable pricing for each subcategory of risk class) and

predictive-error minimization.

Duncan et al. (2016) compare the linear model for health care claim cost prediction against a variety of alternatives,

including LASSO, MARS, random forests, gradient boosting machines (GBM) and M5 decision trees (a variant of CART).

They present comparisons on criteria including the R

, mean absolute error, quantile truncated mean absolute

deviation, and the correlation between actual and predicted values out of sample. They also show that, while for some

parameters or features the various models offer similar predictions, this is not the case for all parameters. Using the

example of age (among male non-claimants in the current year) as a predictor of next year’s claim costs, they find a

wide range of results. While the linear model is the most restricted, the nonlinear models produce curves that are

difficult to interpret, highlighting the trade-offs documented in the literature.

Applications to Wider Problems for Actuaries

This review has focused on studies related to pricing but has also touched upon the application of machine learning

to other areas of actuarial practice, such as the identification of fraudulent claims. There is a wide range of applications

for which machine learning may be useful, particularly related to health care. In addition, the business management

areas such as marketing also benefit from the use of machine learning in, for instance, identifying appropriate

products for consumers. Studies investigating such management applications have not been covered in this report.

Estimation Packages

Most of the methods described in this review are readily applied using packages available in the open-source software

R. Those used in the empirical study have been listed in the relevant sections, with the code to call them provided in

the reports accompanying this document. Other packages in R are listed at the Comprehensive R Archive Network

website, https://cran.r-project.org/web/views/MachineLearning.html.

Due to the computational intensiveness and memory required to apply machine learning to large data sets, code is

often developed in other languages, including Python. We are not aware of a collated source for such code, so one

would need to search for the appropriate code on a case-by-case basis. Most commercial providers of statistical

software (e.g., MATLAB and SAS) also have well-developed machine-learning toolboxes.

Challenges

Many comparative studies have highlighted the advantages of specific machine-learning methods, and few have

compared their performance in different dimensions. There is a plethora of methods available, and researchers are

increasingly using them in various combinations to address particular problems. Ali (2016) discusses a classification of

the methods and possible applications. Guo (2003) offers an example of a study where the performance of machine-

learning methods is underwhelming, partly because the data used in the study is artificial.

An important judgment to make with respect to machine-learning methods is when it is appropriate to use them. For

this, the goals of the study and an exploratory analysis of the data are essential. One of the first questions to consider

is how interpretable we want our model to be. In terms of data, the common considerations are nonlinearity,

interactions and sparse or missing data.

Interpretation

Brockett and Golden (2007) and Golden et al. (2016) highlight the importance of interpretation by discussing social,

psychological and biomedical characteristics as the basis behind the use of over 30 credit-score-related variables to

predict claims. While credit scores have been shown to predict both incidence and severity of claims, some states

have legislated not to allow their use in setting insurance premiums. This highlights the potential that, while machine

learning might identify useful predictive features in the data, the incorporation of new variables from the growing

sets of big data will need to be justified on economic and legal grounds.

Discovering Nonlinear Relationships, Dealing With Large Numbers of Variables and Sparse Data

Very often, the problem we face is feature selection. Approaches to feature selection go a long way toward addressing

the overfitting problems in most models, particularly the nonlinear ones. They also help with the standard GLMs

approach. However, actuaries have been practicing other ways of dealing with large numbers of variables, including

feature design, which is currently of great interest to the machine-learning community. For instance, the grouping of

various credit report variables to assign a credit-related score would count as feature design.

Conclusion

There is a large and growing literature on machine-learning methods in general, but only limited evidence of their use

in insurance problems. The potential for their use is significant, given the complex classification problems and

nonlinear regression relationships seen in insurance data. An increase in the availability of larger data sets also offers

opportunities to adopt machine-learning methods suited to dealing with differently shaped data (e.g., tall, fat, sparse).

We have reviewed the existing literature on the application of machine-learning methods to problems in insurance

claim prediction. We have also provided two illustrative exercises using data publicly available from the SOA. Although

the prediction problems from the two data sets are similar, we find that different approaches work better, reflecting

the degree of nonlinearity in the relationship among other factors. In both cases, however, GLMs are outperformed

by the other models within the limited interpretability of the illustrations. This is despite the fact that regularization

of a GLM offers some improvement in the case of the LTCI data. Thus, it is difficult to conclude that a particular method

is superior in general, more so given the limited transparency of some of the methods. Achievement of reliable model

comparisons will require more robustness analyses on a variety of data sets.

One of the observations that arise from our survey is that only a limited amount of data is available to researchers for

robust comparisons of the various methods applicable to insurance problems. Data sets with large numbers of

variables and large numbers of records are regularly used in the insurance industry. It would be beneficial to develop

some standard shared databases that can be used to test and compare model predictions.

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About The Society of Actuaries

The Society of Actuaries (SOA), formed in 1949, is one of the largest actuarial professional organizations in the

world, dedicated to serving 32,000 actuarial members and the public in the United States, Canada and worldwide. In

line with the SOA Vision Statement, actuaries act as business leaders who develop and use mathematical models to

measure and manage risk in support of financial security for individuals, organizations and the public.

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industry, policy makers and the public. This distinct perspective comes from the SOA as an association of actuaries,

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