Modelling in the case of Incomplete Data (3)

This article is the third in a series in which I discuss techniques that can be used when presented with incomplete data. The focus of this article is on the use of the likelihood for prediction. In particular, we will discuss the Bayesian approach to generating predictive distributions.

Re-cap

In the first article of this series we discussed a problem involving incomplete data. The problem was to estimate the amount of future payments on all claims reported to an insurer given data on historical payments. Below is the plot of historical payments that we discussed.

claim_development

As a first step towards constructing an estimate of future payments, we looked at a model that could be used to describe the payment process. The model consisted of two elements, a state transition element and a payment amount element.

Having built a model, in the second article we considered the problem of assessing the goodness of fit of different parameter values. We found that an appealing measure of goodness of fit was the likelihood. Furthermore, of relevance to our problem, we saw that we could calculate the likelihood even for problems involving incomplete data.

Here we will discuss how Bayesian methods leverage the likelihood in order to produce predictions. We will tackle this in two stages. First, we will look at the issue that is resolved by the Bayesian approach and how the likelihood contributes to the solution. Second, we will discuss why using the likelihood in this way makes sense from a more technical point of view. Throughout the article we will use our claim payment problem as a case study.

Please note that this article is necessarily abbreviated. It would be more complete if we discussed the simulation methods I used. However, simulation is not the main theme of this article, hence the omission.

If you haven’t already, you may find reading the first two articles in this series helpful. In particular, as I will refer to them later, you will want to know that our model has three parameters: ‘p’ and ‘q’ parameterise the state transition model; ‘mu’ parameterises the payment amount model (we assume claim payments follow an Exponential distribution).

The Problem & A Solution

The output of a Bayesian analysis is an exact predictive distribution of what we are interested in given what we know. In our case, this means the approach produces a probability distribution of future claim payments given what we have learned from historical payments.

As a first step to understanding the approach, it is useful to consider how we would estimate the probability of a future outcome if we knew what the “correct” values for the parameters in our model were.

For instance, how would we estimate the probability of future payments exceeding $10m if we knew the correct values for ‘p’, ‘q’ and ‘mu’?

To answer this question we could use simulation.

By using the “correct” parameter values, we could simulate future outcomes from our model and then estimate the required probability by calculating the proportion of simulations in which future payments exceeded $10m.

The uncertainty that this approach captures is the uncertainty in the statistical process. Let’s refer to this as “process uncertainty”.

Unfortunately, we don’t actually know the correct values for the parameters. Therefore, not only do we have to allow for process uncertainty but we also have to allow for the uncertainty regarding the parameters.

From a practical perspective, as with process uncertainty, Bayesian methods allow for “parameter uncertainty” by simulating different values for the parameters. The distribution we simulate from is known as the “posterior” distribution.

In other words, with our Bayesian hat on, we can augment our previous simulation approach (which involved fixed “correct” parameters) by simulating parameter values from the posterior before using these to simulate future outcomes from our model.

As you can probably tell, the trick to this approach is in constructing a posterior distribution that makes sense. It needs to capture our general ignorance about the parameter values as well as what we have learned from the data.

One relatively coherent approach to doing this is to use the likelihood.

Below is a plot of the likelihood for our claim transition model, based on the data shown in the plot above.

levelplot_observeddata

As we discussed in the previous article, the likelihood measures the goodness of fit of different parameter values. Those with a high value (the red region) represent values that are a relatively good fit and those with a low value (the blue region) are those with a poor fit.

With a leap of faith, we might wonder whether we could use the likelihood to represent the posterior. For instance, if the total likelihood in the red region is twenty times higher than that of the blue region, could we reasonably say that the probability of the parameter falling in the red region is twenty times that of it falling in the blue region?

Although some may wince at this derivation, it so happens that we can–more on this in the next section.

In particular, the normalised likelihood (the likelihood scaled so that it sums to one) gives us a posterior distribution of the parameters. With this we are able to generate predictions by using the simulation approach mentioned above.

The plot below represents the result of applying this approach to our problem.

PredictiveFan

The plot displays historical payments as well as predictions. Historical cumulative payments up to the present time (0) are represented by the black line. The median predicted cumulative payment as well as prediction intervals are shown for each future quarter for the next thirty five years (times 1 to 140).

From this we might say that we expect total future payments to be ~$10m. Furthermore, we could say that we think there is a 95% probability that total payments will fall between ~$8m and ~$12m.

Why the Solution makes sense

As you may have noticed, the critical stage in a Bayesian analysis is in constructing the posterior distribution. In the previous section I raced past our choice of posterior, here I’ll quickly discuss why the approach makes sense.

In our case, the posterior distribution produces answers to questions like, “Given what you have seen, what do you believe the probability is that the true values for ‘p’, ‘q’, and ‘mu’ are equal to 0.1, 0.5, and $10k?”. In general, through probabilities, the posterior represents our beliefs regarding the true parameter values.

Using symbols, the posterior probability that the true values for the parameters ( \Lambda ) are equal to some value ( \lambda ) given the data ( D ) is denoted  P(\Lambda = \lambda | D) .

This definition may remind you of the likelihood. Indeed, in the previous article we introduced the likelihood as the probability of the model generating data equal to the observed data–otherwise denoted as P(D | \Lambda = \lambda).

Because of the symmetry between the Likelihood and the Posterior, we can make use of Bayes theorem to write:

     \begin{gather*} P(\Lambda = \lambda | D) \propto P(D|\Lambda = \lambda) \cdot P(\Lambda = \lambda)  \end{gather*}

In words, the posterior is proportional to the product of the likelihood and some a priori probability for the parameters. The a priori probability is referred to as the “prior”.

With this formula we can see what assumption I had made in the previous section. I assumed that the posterior was proportional to the likelihood, hence I implicitly assumed that the prior was constant for all values of the parameters.

Please note that other prior assumptions are valid. However, we can be satisfied that the approach we arrived at has some technical merit.

For those who will apply Bayesian methods to their problems, it is important to recognise that any choice of prior is ultimately subjective. This is not necessarily a weakness but if you need to maintain a degree of objectivity, it is worth taking a look at articles concerning “Objective Bayes” and testing a variety of reasonable priors.

Final Word

The Bayesian approach is very powerful. Given some basic building blocks it enables us to generate predictive distributions of future outcomes. With these distributions we can attempt to answer questions like the one presented in the first article of this series.

Before becoming completely smitten by Bayesian methods, however, it is important to recognise that this power does not come for free. In particular, employing Bayesian methods involves a subjective selection of the prior distribution.

Given that we now have a method for solving the problem posed in the first article, I will let this series sit for a while. However, there are many other approaches we have not discussed and in future I may revisit this series to discuss them.

If you have any questions or requests for future topics please leave them in the comments section.