1. Introduction
Risk classification for property and casualty (P&C) insurance rating has traditionally been done with oneway, or univariate, analysis techniques. In recent years, many insurers have begun using generalized linear models (GLMs), a multivariate predictive modeling technique that addresses many shortcomings of univariate approaches and is currently considered the gold standard in insurance risk classification. At the same time, machine learning (ML) techniques, such as deep neural networks, have gained popularity in many industries due to their superior predictive performance over linear models (LeCun, Bengio, and Hinton 2015). In fact, there is a fastgrowing body of literature on applying ML to P&C reserving (Kuo 2019; Wüthrich 2018; Gabrielli, Richman, and Wüthrich 2019; Gabrielli 2019). However, ML techniques, often considered to be completely “black box,” have been less successful in gaining adoption in pricing, which is a regulated discipline and requires a certain amount of transparency in models.
If insurers knew more about how ML models behave in risk classification contexts, it would increase their ability to reassure regulators and the public that accepted ratemaking principles are being met. Being able to charge more accurate premiums would, in turn, make the risk transfer system more efficient and contribute to the betterment of society. In this paper, we aim to take a step toward liberating actuaries from the confines of linear models in pricing projects, by proposing a framework for explaining ML models for ratemaking that regulators, practitioners, and researchers in actuarial science can build upon.
The rest of the paper is organized as follows: Section 2 provides an overview of P&C ratemaking, Section 3 discusses the importance of interpretation, and Section 4 discusses model interpretability in the context of ratemaking and proposes specific tasks for model explanation. Section 5 describes current model interpretation techniques and applies them to the tasks defined in the previous section, and Section 6 concludes.
2. Property and Casualty Ratemaking
2.1. History of Ratemaking
Early classification ratemaking procedures were typically univariate in nature. For example, Lange (1966) noted that (at that time) most major lines of insurance used univariate methods based on the same principle: distributing an overall indication to territorial relativities or classification relativities based on the extent to which they deviated from the average experience.
Bailey and Simon (1960) introduced minimum bias methods, which were expanded throughout the 1960s, '70s, and '80s. As computing power developed, minimum bias began to give way to GLMs, with papers such as those by Brown (1988) and Mildenhall (1999) bridging the gap between the methods.
Arguably, GLMs predate minimum bias procedures by a significant margin. The term was coined by Nelder and Wedderburn (1972), but generalizations of least squares linear regression date back at least to the 1930s. Like minimum bias methods, GLMs did not become mainstream in actuarial science for some time. For example, the Syllabus of Basic Education of the Casualty Actuarial Society (CAS) does not seem to include any mention of GLMs prior to Brown (1988) in the 1990 syllabus. From there, GLMs seem to have received only passing mention until 2006 with the introduction of Anderson et al. (2005) to the syllabus. Beginning in 2016, the CAS introduced Goldburd, Khare, and Tevet’s (2016) monograph to the syllabus, which offers a comprehensive guide to GLMs.
2.2. Machine Learning in Ratemaking
Paralleling the development of GLMs was the development of ML algorithms throughout the middle part of the 20th century. Detailed histories of ML may be found in sources such as Nilsson (2009) and Wang and Raj (2017). Consistent with GLMs, ML remained relatively unpopular in actuarial science until the last 10 years as computing power has become cheaper and more easily available and as ML software packages have obviated the need for developing analyses from scratch each time an analysis is performed. Due to the breadth of ML as a field, it is difficult to identify the first time it entered the CAS syllabus; however, cluster analysis (in the form of kmeans) seems to have been first included in 2011 with Robertson (2009). More recently, the CAS MASI and MASII exams introduced in 2018 have included ML explicitly.
In the area of ratemaking, ML is still in its infancy. A significant portion of ML applications to ratemaking has been in the context of automobile telematics, such as Gao, Meng, and Wüthrich (2018), Gao and Wüthrich (2018, 2019), Roel, Antonio, and Claeskens (2018), or Paolo, Fricker, and Padman (2015). Presumably this focus has been a result of the high dimensionality and complexity of telematics data, making it a field in which the unique abilities of ML techniques demonstrate a clear advantage over traditional approaches.
Outside of telematics, Yang, Qian, and Zou (2018) use a gradienttreeboosting approach to capture nonlinearities that would be a challenge for GLMs. Henckaerts et al. (2018) make use of generalized additive models to improve predictions of GLMs. Many researchers, presumably to demonstrate the range of possibilities and advantages of ML, have approached the topic by comparing many different ML algorithms within a single study, such as in Dugas et al. (2003), Noll, Salzmann, and Wuthrich (2018), and Spedicato, Dutang, and Petrini (2018). These studies make use of such varied techniques as regression trees, boosting machines, supportvector machines, and neural networks.
2.3. Ratemaking Process
Regardless of the method used to determine the risk of various classifications, the actual process of setting rate relativities typically involves some variation of the following steps:

Obtain relevant policylevel data.

Prepare the data for analysis.

Perform analysis on the data, employing the desired method(s) to estimate needed rates.

Select final rates based on rate indications.

Present the rates to the regulator, including an explanation of the steps followed to derive the rates.

Answer questions from regulators regarding the method employed.
The focus of this paper is on steps 5 and 6. In many states, rate filings that exceed certain thresholds for magnitude of rate changes or filings that make use of new or sophisticated predictive models may be subject to particular regulatory scrutiny. In such cases, one must be able to explain the results of the modeling process in a way that is understandable without sacrificing statistical rigor.
It should be noted that communicating results is not simply a method of passing regulatory muster. Generating interpretable modeling output is an important—even essential—facet of model checking. Actuaries are bound by relevant standards to be able to exercise appropriate judgment in selecting risk characteristics as part of a risk classification system per Actuarial Standard of Practice 12 (“Risk Classification”). Therefore, the techniques discussed in this paper may be viewed through the lens of providing useful information to regulators, but they should also be considered as part of a thorough vetting of any rating model.
Although our focus is on communication with regulators, it should be said that selecting final rates based on indications (step 4 in the list above) may pose a unique challenge for blackbox models. This, too, provides strong motivation for techniques that could add to the modeler’s—or any stakeholder’s—understanding of the model, such as the relative importance of variables or the shapes of response curves. Such techniques could be usefully employed in making decisions about how best to select rates.
Similarly, although our focus is on communication in a pricing context, the techniques we explore in the paper (and many of the concerns discussed) may also be relevant to other contexts, such as claimlevel reserving / analytics or other applications of ML to the insurance industry.
3. The Need to See Inside the Black Box
Within the actuarial profession, Actuarial Standard of Practice 41 (“Actuarial Communications”) notes that “another actuary qualified in the same practice area [should be able to] make an objective appraisal of the reasonableness of the actuary’s work as presented in the actuarial report” (“Actuarial Standard of Practice No. 41  Actuarial Communications” 2010). Underlying this requirement is an assumption that the hypothetical other actuary qualified in the same practice area is adequately familiar with the relevant techniques employed. Although the Syllabus of Basic Education is constantly changing, there has at times been an assumption that all techniques and assumptions that have ever been a part of the syllabus needn’t be explained from first principles in general actuarial communications, and that an actuary practicing in the same field should be able to make an objective appraisal of the results from the methods found in the syllabus. This is notable because, beginning with the introduction of the CAS MASI and MASII examinations in July of 2018, several ML models were formally included in the Syllabus of Basic Education. These exams cover a wide range of topics, such as splines, clustering algorithms, decision trees, boosting, and principle components analysis (Casualty Actuarial Society 2018).
Nevertheless, ML poses something of a special challenge for ASOP 41 for several reasons:

ML models can be very ad hoc compared with traditional statistical models.

Because many ML models do not assume an underlying probability distribution or stochastic process, they may not admit of standard metrics for model comparison (e.g., calculating the Akaike information criterion over a neural network is not straightforward).

ML methods are often combined into ensembles that may not be easily separated and that may, as a collection, cease to resemble a single standard version of a model.

ML models can be black boxes insofar as the final form of response curve cannot be easily predicted and may depend heavily on the available data (which may not, in turn, be available to the reviewer).
This last item raises a final interesting issue. GLMs and their ilk are often fitted using one of a handful of standard and wellunderstood approaches (e.g., maximum likelihood estimation). However, that is not possible in general with ML models, as ML algorithms often use loss surfaces that are so complex that it may not be feasible to calculate the global minimum of the surface. Certainly, closedform representations of the loss surfaces are not generally available. For this reason, the training phase of an ML model is, in many ways, just as important to one’s understanding as the model form and the data on which the model is fitted. Because the final model result is inseparable from those three components (training method, model form, and data), simply knowing the method employed is generally insufficient to make an objective appraisal of the reasonableness of the result. More information is necessary.
These issues also pose particular challenges with respect to other standards. For instance, as discussed previously, ASOP 12 requires actuaries to be able to exercise appropriate judgment about risk classification systems. The recent ASOP 56 (“Models”) speaks to more general concerns in all practice areas that might make use of models. ASOP 56 requires the actuary to “make reasonable efforts to confirm that the model structure, data, assumptions, governance and controls, and model testing and output validation are consistent with the intended purpose” (“Actuarial Standard of Practice No. 56  Modeling” 2019). All such efforts may be hampered if it is not possible to peer into the black box of the model.
It should also be noted that these comments apply only within the actuarial profession, and outside of that profession communication of results may be more challenging. A 2017 survey conducted by the Casualty Actuarial and Statistical Task Force of the National Association of Insurance Commissioners (NAIC) found that a plurality of responding regulators identified “filing complexity and/or a lack of resources or expertise” as a key challenge impeding their ability to review GLMs or other predictive models (National Association of Insurance Commissioners 2017). Given that ML algorithms are generally regarded as more complex than GLMs, this implies that the challenge of communicating ML model results is significant.
In response to the same survey, 33 state regulators noted that it would be helpful or very helpful for the NAIC to develop information and tools to assist in reviewing rate filings based on GLMs, and 34 noted that it would be helpful to develop similar items to assist in reviewing “other advanced modeling techniques.” One outgrowth of this need was the development of a white paper, “Regulatory Review of Predictive Models” (National Association of Insurance Commissioners 2019), on best practices for regulatory review of predictive models. The paper focuses on review of GLMs, particularly with respect to private passenger automobile and homeowner’s insurance. Some of the guidance offered in this regard is therefore not strictly applicable to the review of ML models. For example, as previously noted, pvalues are not a concept that translates well to deterministic ML algorithms. However, among the guidelines applicable to ML algorithms are the following:

Determine the extent to which the model causes premium disruption for individual policyholders and how the insurer will explain the disruption to individual consumers that inquire about it.

Determine that individual input characteristics to a predictive model are related to the expected loss or expense differences in risk. Each input characteristic should have an intuitive or demonstrable actual relationship to expected loss or expense.

Determine that individual outputs from a predictive model and their associated selected relativities are not unfairly discriminatory.
The last of these items is an entire topic unto itself. The methods and concepts introduced in this paper are useful for exploring the question of whether rates are appropriately related to risk of loss as defined by the variables used in the model, but there are many other aspects of discriminationfree that are outside the scope of this paper. The methods in this paper may help in understanding the model, which is a necessary precursor to addressing the question of unfair discrimination.
The items in this list are by no means exhaustive, but they pertain to the concept of model interpretability for ratemaking that we develop next.
4. Interpretability in the Ratemaking Context
In this section, we attempt to develop a working definition of interpretability for ratemaking applications. Although we do not provide a comprehensive survey of the prolific and fastevolving ML interpretability literature, we do draw from it as appropriate in setting the stage for our discussion. Even among researchers in the subject, there is no consensus on how to define interpretability; here are a few such definitions from frequently cited papers:

the ability to explain or to present in understandable terms to a human (DoshiVelez and Kim 2017);

the degree to which an observer can understand the cause of a decision (Biran and Cotton 2017); and

a method is interpretable if a user can correctly and efficiently predict the method’s results (Kim, Khanna, and Koyejo 2016).
We motivate our discussion by considering several aspects of interpretability. As we proceed through the following points, we aim to arrive at a more scoped and relevant definition of what it means for a pricing model to be interpretable. In the remainder of this section, we clarify a couple of concepts regarding interpretable classes of models and the computational transparency of ML models, outline frameworks for understanding the communication goals of interpretability, and then discuss a potential framework for implementing ML interpretability in practice.
4.1. Not All Linear Models Are Interpretable
In the actuarial science literature, the GLM is probably the most oftcited example of an easily interpretable model. Given a set of inputs, we can easily reason about what output the model will produce. As an illustrative example, consider a claim severity model with driver age, driver sex, and vehicle age as predictors; assuming a log link function and letting
denote the response, we have\[\begin{aligned} \log{(E\lbrack Y \rbrack )} &= \beta_0 + \beta_1 \cdot \text{age} \\ & \quad + \beta_2 \cdot \text{vehicle} \_ \text{age} + \beta_3 \cdot \text{sex}_{\text{male}}.\end{aligned}\tag{1}\]
Here, we can tell, for example, what the model would predict for the expected severity if we were to increase age by a certain amount, all else being equal, because the relationship between the predictor and the response is simply multiplication by the coefficient
and applying the inverse link function.Another commonly cited example of an interpretable model is a decision tree. An illustrative example is shown in Figure 1. Here, the prediction is arrived at by following a sequence of ifelse decisions.
Now, it is worth pointing out that, when declaring that GLMs or decision trees are interpretable models, we are implicitly assuming that we are considering only a handful of predictors. In fact, the ease with which we can reason about a model declines as the number of predictors, transformations of them, and interactions increase, as in the following (somewhat pathological) example:
\[\begin{align} \log{(E \lbrack Y \rbrack )} &= \beta_0 + \beta_1 \cdot \text{age} + \beta_2 \cdot \text{vehicle} \_ \text{age} \\ &\quad + \beta_3 \cdot \text{vehicle} \_ \text{age}^2 \\ &\quad + \beta_4 \cdot \text{age} \cdot \text{vehicle} \_ \text{age} \\ &\quad + \beta_5 \cdot \text{sex}_{\text{male}}+\beta_6 \cdot \text{sex}_{\text{male}} \cdot \text{age.} \end{align} \tag{2}\]
Similarly, one can see that in Figure 2, larger trees are tough to reason about. In other words, even when working within the framework of an “interpretable” class of models, we may still end up with something that many would consider a black box.
4.2. The Machinery Is Not a Secret
Another occasional misconception is that we have no visibility into how some ML models compute predictions, which renders them uninterpretable. Outside of proprietary algorithms, all common ML models, including neural networks, gradientboosted trees, and random forests, are well studied and have large bodies of literature documenting their inner workings. As an example, a fitted feedforward neural network is simply a composition of linear transformations followed by nonlinear activation functions. As in Equation 2, one can write down the mathematical equation for calculating the prediction given some inputs, but it may be difficult for a human to reason about it. We show later that we can still provide explanations of completely blackbox models, but it is important to note that ML model predictions are still governed by mathematical rules, and are deterministic in most cases.
4.3. Explanations Are Contextual
Hilton (1990) proposed a framework, later interpreted by Miller (2017) in the context of ML, for understanding model explanations as conversations or social interactions. One consequence of this identification is that explanations need to be relevant to the audience. This framework is consistent with ASOP 41, which formulates a similar requirement in terms of an intended user of the actuarial communication. In developing, filing, and operationalizing a pricing model, one needs to accommodate a variety of stakeholders, each of whom has a different set of questions, assumptions, and technical capacity. First, there are internal stakeholders at the company, a group that includes management and underwriters. While some of the individuals in this audience may have technical expertise, they are likely less familiar with predictive modeling techniques than the actuaries and data scientists who build the models. Next, we have the regulators, who may have limited resources to review the models, and will focus on a specific list of questions motivated by statute and public policy. Finally, we have potential policyholders, who have an interest (perhaps more so than the other parties) as they are responsible for paying the premiums.
It is interesting to note that the modelers, who are most familiar with the models, tend to be the same people designing and communicating the explanations. This poses a challenge that Miller, Howe, and Sonenberg (2017) call “inmates running the asylum,” where the modelers design explanations for themselves rather than the intended audience. For example, they may be interested in technical questions, such as extrapolation behavior, and shape the explanations accordingly, which may be irrelevant to a prospective policyholder.
Another point outlined by Miller (2017) is that explanations are contrastive. In other words, people are often interested in not why something happened, but rather why it happened instead of something else. For example, policyholders might not care exactly how their auto premiums are computed but would like to know why they are being charged more than their coworkers who drive similar vehicles. As an extension, policyholders may want to know what they can change in order to obtain lower premiums.
4.4. Asking and Answering the Right Questions
With the preceding considerations in mind, we propose a potential framework for interpreting ML models for insurance pricing: the actuarial profession, in collaboration with regulators and representatives of the public, define a set of questions to be answered by explanations accompanying ML models, along with acceptance criteria and examples of successful explanations. In other words, interpretability for our purposes is defined as the ability of a model’s explanations to answer the posed questions.
It should be noted that no ideal set of questions exists that would encompass all potential models. Rather, the actuary must consider what aspects of the model would raise questions from the perspective of the model’s intended users. We propose that relevant stakeholders, by providing example questions and answers, would inherently provide guidance by which actuaries can reasonably anticipate the kinds of specific questions most important to those stakeholders and address them proactively.
These questions should relate to existing guidelines, such as those described in the NAIC’s “Regulatory Review of Predictive Models” (2019) and outlined earlier in Section 3, standards of practice, and regulation, and in fact should not be specific only to ML models. By conceptualizing a set of questions, we reduce the burden on both companies and regulators; this is especially important for the latter, who are already resource constrained facing an increasing variety of models being filed. This format should also be familiar to actuaries who are accustomed to adhering to specific guidelines in, for example, ASOPs. As is true of the ASOPs, we envision that the questions and guidelines will be continually updated to reflect feedback obtained and advances in research.
Whereas the realization of a set of such guidelines is an ambitious undertaking beyond the scope of this paper, we present in the next section a sample set of questions and techniques one can leverage to answer them. The goal of our case study is twofold: to more concretely illustrate the proposed framework, and to expose the actuarial audience to modern ML interpretation techniques.
5. Applying Model Interpretation Techniques
Now that we have established a framework for model interpretation in the form of asking and answering relevant questions, we demonstrate examples of such exchanges via an illustrative case study. Analytically, our starting point is a fitted deep neural network model for predicting loss costs. As the modeling details are of secondary importance, they are available in Appendix A. The questions we ask of the model are as follows:

What are the most important predictors in the model? Put another way, to what extent do the predictors improve the accuracy of the model?

How does the predicted loss cost change, on average, as we change an input?

For a particular policyholder, how does each characteristic contribute to the loss cost prediction?
In our discussion, we adopt the organization of techniques and some notation presented in Molnar and others (2018) and Biecek and Burzykowski (2019), which are comprehensive references on the most established ML interpretation techniques.
5.1. A Simplified View of Interpretation Techniques
Before we dive into answering the questions, we present a brief taxonomy of ML interpretation techniques. Rather than attempting an exhaustive classification, the goal is to orient ourselves among broad categories of techniques, so we can map them to tasks indicated by the questions being asked. For our purposes, model interpretation techniques can be categorized across two dimensions: intrinsic versus post hoc and global versus local.
5.1.1. Intrinsic versus Post Hoc
Intrinsic model interpretation draws conclusions from the structure of the fitted model and is what we typically associate with “interpretable” classes of models. This is viable only with models with simple structures, such as the sparse linear model and shallow decision tree we saw in Section 4.1, where we arrive at explanations by reading off parameter estimates or a few decision rules. For algorithms that produce models with complex structure that do not lend themselves easily to intrinsic exploration, we can appeal to post hoc techniques. This class of techniques interrogates the model by presenting it with data for scoring and observing the prediction behavior of the model. Such techniques are concerned with only the inputs and outputs, and hence are agnostic of the model itself, which means they can also be applied to simple models. Because most useful ML models have a level of complexity beyond the threshold of intrinsic interpretability, we focus on modelagnostic techniques in our case study. As we will see later on, the data that we present to the models are usually some perturbed variations of test data.
5.1.2. Global versus Local
Along the other dimension, we categorize model interpretations as global, or model level, and local, or instance level. The former class provides insights with respect to the model as a whole. Some examples of these explanations include variable importances and sensitivities, on average, of the predicted response with respect to individual predictors. Note that these methods may be compared to the methods described in Goldburd, Khare, and Tevet (2016, ch. 7), which focus on global interpretation of GLMs.
In our case study, questions 1 and 2 are associated with global interpretations. On the other hand, question 3 pertains to an individual prediction, which would fall in the local, or instancelevel, category. In addition to individual variable attribution, we can also inquire about what would happen to the current predicted response if we were to perturb specific predictor variables.
5.2. Answering the Questions
Having aligned the questions with the categories of interpretation techniques, we now introduce a selection of appropriate techniques to answer them.
5.2.1. Variable Importance
“What are the most important predictors in the model?”
For linear models and their generalizations, and some ML models, measures of variable importance can be obtained from the fitted model structure. In the case of GLMs, one might observe the magnitudes of the estimated coefficients or
statistics, whereas for random forests, one might use outofbag errors (Breiman 2001). For more complex models, such as the neural network in our case study, we need to devise another approach.We follow the methodology of permutation feature importance as described in Fisher, Rudin, and Dominici (2018), and use the notation introduced by Biecek and Burzykowski (2019). The gist of the technique is as follows: to see how important a variable is, we make predictions without it and see how much worse off we are in terms of accuracy. One way to achieve this would be to refit the model many times (as many times as the number of variables). However, this may be intractable with lengthy model training times or large numbers of variables, so a more popular approach is to instead keep the same fitted model but permute the values of each predictor.
More formally, let
denote the vector of responses, denote the matrix of predictor variables, denote the fitted model, and where applies to rowwise, denote the value of the loss function, which is mean squared error in the case of regression. Now, if denotes the predictor matrix where the th variable has been permuted, then we can compute the loss with the permuted data set as With this, we define the variable importance asIn Figure 3, we show a plot of variable importances. In our particular example, we see that the “make” variable contributes most to the accuracy of the model with “sex” contributing the least. This provides a way for the audience to quickly glance at the most relevant variables, and ask further questions as necessary.
Note that these measures do not provide information regarding the directional sensitivity of the predictors on the response. Also, when there are correlated variables, one should be careful about interpretation, as the result may be biased by unrealistic records in the permuted data set. Another ramification of a group of correlated variables is that their inclusion may cause each to appear less important than if only one is included in the model.
5.2.2. Partial Dependence Plots
“How does the predicted loss cost change, on average, as we change an input?”
For this question, we again consider first how it would be answered in the GLM setting. When the input predictor in question is continuous, we can answer the question by looking at the estimated coefficient, which provides the change in the response per unit change in the predictor (on the scale of the linear predictor). For nonparametric models and neural networks, where no coefficients are available, we can appeal to partial dependence plots (PDPs), first proposed by Friedman (2001) for gradientboosting machines.
To describe the PDP, we need to introduce some additional notation. Let
denote the input variable of interest. Then we define the partial dependence function as\[h(z) = E_{X^{j}}[\widehat{f}(xx^j = z)],\tag{3}\]
where the expectation is taken over the distribution of the other predictor variables. In other words, we marginalize them out so we can focus on the relationship between the predicted response and the variable of interest. Empirically, we estimate
by\[\widehat{h}(z) = \frac{1}{N}\sum_{i = 1}^{N}\widehat{f}(x_ix_i^{j} = z),\tag{4}\]
where
is the number of records in the data set.In Figure 4, we exhibit the PDP for the “vehicle age” variable. We see that the average predicted loss cost decreases with vehicle age until the latter is around 18. Note that the accompanying histogram shows that the data are quite thin for vehicle age greater than 18, so the apparent upward trend to the right is driven by just a few data points.
This information allows the modeler and stakeholders to consider whether it is reasonable for the anticipated loss cost to follow this shape.
The question posed here is particularly important for regulators, who would like to know whether each variable affects the prediction in the direction that is expected, based on intuition, experience, and existing models. During the model development stage, the PDP can also be used as a reasonableness test for candidate models by identifying unexpected relationships for the analyst to investigate.
As with permutation feature importance, one should be careful when interpreting the PDP when there are strongly correlated variables. Since we average over the marginal distribution of the rest of the variables, we may take into account unrealistic data (e.g., high vehicle age for a model that is brand new).
5.2.3. Variable Attribution
“For a particular policyholder, how does each characteristic contribute to the loss cost prediction?”
In the previous two examples, we look at modellevel explanations; now we move on to one where we investigate one particular prediction instance. As before, we consider how we would approach the question for linear models. For a GLM with a log link common in ratemaking applications, for example, we start with the base rate, and then the exponentiated coefficients would have multiplicative effects on the final rate. Similar to the previous examples, for ML models in general we do not have directly interpretable weights. Instead, one way to arrive at variable contributions is calculating the change in the expected model prediction for each predictor, conditioned on other predictors.
Formally, for a fitted model
and a specific instance we would like to decompose the model prediction into\[ \widehat{f}(x_* ) = v_0 + \sum_{j=1}^p v(j, x_*),\tag{5}\]
where
denotes the average model response, and denotes the contribution of the th variable in instance defined as\[\small{\begin{aligned} v(j, x_* ) &= E_X[\widehat{f}(X)  X^1 = x_* ^1, \dots, X^j = x_* ^j] \nonumber\\ & \quad  E_X[\widehat{f}(X)  X^1 = x_* ^1, \dots, X^{j1} = x_*^{j  1})].\end{aligned}\tag{6}}\]
Hence, the contribution of the
th variable to the prediction is the incremental change in the expected model prediction when we set assuming the other variables take their values in Note here that this definition implies that the order in which we consider the variables affects the results. Empirically, the expectations in (6) are calculated by sampling the test data set.In Figure 5, we exhibit a waterfall plot of variable contributions. The “intercept” value denotes the average model prediction and represents the term in Equation (5). The predicted loss cost for this particular policyholder is slightly less than average. The characteristics that make this policyholder more risky are that he is a male between the ages of 18 and 25; counteracting the risky driver characteristics are the vehicle properties: it is a GM vehicle built domestically and is seven years old.
Instancelevel explanations are useful for investigating specific problematic predictions generated by the model. Regulators and model reviewers may be interested in variable contributions for the safest and riskiest policyholders to see if they conform to intuition. A policyholder with a particularly high premium may wish to find out which of his or her characteristics contribute to it, and may follow up with a question about how he or she can lower it, which would require another type of explanation.
As noted earlier, the ordering of variables has an impact on the contributions calculated, especially for models that are nonadditive, which could cause inconsistent explanations. Several approaches exist to ameliorate this phenomenon, including selecting variables with the largest contributions first, including interaction terms, and averaging over possible orderings. The last of these ideas is implemented by Lundberg and Lee (2017) using Shapley values from cooperative game theory, an approach referred to as Shapley additive explanations. See Biecek and Burzykowski (2019) and that paper’s references for further discussion of these approaches. Shapley values were also used in Mango (1998), which has appeared on the CAS syllabus since 2004, in the context of determining how to allocate catastrophe risk loads between multiple accounts.
5.3. Other Techniques
In this paper, we demonstrate just a few modelagnostic ML interpretation techniques. They represent a small subset of existing techniques, each of which has additional variations. In the remainder of this section, we point out a few common techniques not covered in our case study.
Individual conditional expectation plots disaggregate PDPs into their instancelevel components for a more granular view into predictor sensitivities (Goldstein et al. 2015). To accommodate correlated variables in PDP, accumulated local effect plots compute expected changes in model response over the conditional, rather than marginal, distribution of the other variables (Apley and Zhu 2016).
Local interpretable modelagnostic explanations (Ribeiro, Singh, and Guestrin 2016) build simple surrogate models using model predictions, with higher training weights given to the point of interest, in effect replacing the complex ML model with an easily interpretable linear regression or decision tree in neighborhoods of specific points for the purpose of explanation. Taking the concept further, one can also train a global surrogate model across the entire domain of interest.
6. Conclusion
Actuarial standards of practice, most notably ASOP 41, place responsibility on the actuary to clearly communicate actuarial work products, including insurancepricing models. Those responsibilities create special challenges for communicating ML models, which may appear as black boxes due in part to their complexity, nonlinearity, flexible construction, and ad hoc nature.
In this paper, we discuss particular questions of model validation that are of key importance in explaining models and that may present particular difficulty when explaining ML models as compared with GLMs or traditional pricing models. Specifically,

How does the model affect individual insurance consumers?

How are the predictor variables related to expected losses?
We contextualize these questions in terms of different frameworks for defining interpretability. We conceptualize interpretability in terms of the ability of a model (or modeler) to answer a set of idealized questions that would be refined. We then offer potential (families of) modelagnostic techniques for providing answers to the questions.
Much work remains to be done to define the role of ML algorithms in actuarial practice. Lack of interpretability has been a key barrier preventing wider adoption and exploration of these techniques. The methods proposed in this paper could therefore represent important strides in unlocking the potential of machine learning within the insurance industry.
Acknowledgments
We thank Navdeep Gill for his presentation on interpretable ML at SDSS and helpful followup discussions, and Daniel Falbel for his assistance with R and TensorFlow. This work is supported by the Casualty Actuarial Society.