# Applied Probability Models in Marketing

Recently, I had a chance to take a look at the course material for Pete Fader’s “Applied Probability Models in Marketing”, taught at the Wharton School. I was completely unfamiliar with the academic traditions of marketing research, so it was an illuminating experience to go through a number of papers in the field.

The course revolves around the research of Fader and Bruce Hardie, a marketing researcher at the London Business School. This post is an overview of the papers covered in the course.

## 1. How To Project Customer Retention (Fader–Hardie, 2007)

In the subscription-based business model, each customer signs a contract that lasts for a fixed amount of time. Once the contract expires, it is up to the customer to renew their subscription.

The existence of contracts implies that the firm is able to observe when a customer leaves, i.e., declines to renew the subscription. This leads to the notion of the retention rate, which provides a measure of customer departure. Formally, we assume that each contract lasts one time unit and define $P(t)$ to be the percent of subscribing customers at the end of period $t$, with respect to the initial group of customers. The retention rate for period $t$ is the ratio

Here we have assumed that each customer can have at most one contract with the firm. To ensure that $r_t \leq 1$, we assume that customers may not rejoin the service once they leave.

Assuming that the customer makes a decision whether to renew at the end of each period independently of their previous decisions, the probability that a customer remains subscribed until the end of period $t$ is given by the survivor function

The sum

denotes the expected length of retention of a customer, i.e., the expected tenure.

We, of course, cannot observe infinitely many time periods, so we need a way to estimate the expected tenure. The naïve approach of fitting a simple curve—such as linear, quadratic, or exponential functions—to observed retention does not work for predictive purpose (Berry–Linoff, 2004).

Fader–Hardie proposes a stochastic model of customer retention, with the following assumptions:

1. For each customer, the probability of contract renewal at the end of each period is constant.
2. The contract renewal probability can vary across customers.

Assumption 1 implies that the survivor function is of the form

where $r$ is the fixed retention rate. Moreover, if we denote by $T$ the subscription length of a customer, we see that the probability of $T = t$ is $r^{t-1}(1-r).$

Setting $\theta = 1-r$, we see that the random variable $T$ has a geometric distribution

We note that we do not, in general, have the luxury to observe the value of $\theta$ before studying $S$ or $T$. This means that we must condition them on unknown $\theta$:

Fader–Hardie models Assumption 2 with a beta distribution probability density function

where $B$ denotes the beta function

Since $\theta$ assumes a beta distribution,

and computing the integral gives us the identity

Similar calculation yields

Fader–Hardie dubs this customer retention model the shifted-beta-geometric distribution.

The shifted-beta-geometric distribution can be computed without the beta function. Recalling that

where $\Gamma$ denotes the Gamma function, we see that

Now, for each fixed $t$,

It thus follows that

for all $t \geq 1$.

What is the retention rate under this model? Let us denote by $r(t;\alpha, \beta)$ the retention rate at the end of period $t$ given the parameters $\alpha$ and $\beta$. Observe that

We observe that $r(t; \alpha, \beta)$ is an increasing function, even though the true, unobserved retention rate is constant. Fader–Hardie explains as follows: “There are no underlying time dynamics at the level of the individual customer; the observed phenomenon of retention rates increasing over time is simply due to heterogeneity …”

With simple closed-form formulas for the retention rate, it is not difficult to compute the maximum likelihood estimate of $\alpha$ and $\beta$. Indeed, under the above model, Fader–Hardie a computes a close estimate of the dataset in Berry–Linoff, which deals with yearly percentage of surviving customers at an unspecified firm. Another paper (Lee–Fader–Hardie, 2007) produces a good shifted-beta-geometric-model estimate of patient persistency rate, i.e., the percentage of patients continuing to refill a prescription.

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