There's a 1982 book I keep coming back to. It's called A River of Time, written by a man named Jerome Greene for an audience of marketers who didn't want to read statistics. It is not famous. It was a popularization, the kind of book that exists to make a hard idea usable by people with a quota to hit. I'm fond of it anyway, mostly for one paragraph buried in chapter ten that I think about more than almost anything else in my field.
The book's central image is a river. Picture every one of your customers as a lily pad drifting on it. Each pad moves at its own speed, and that speed is the customer's underlying purchase rate, how often, on average, they buy from you. Fast pads buy often, slow pads rarely. What you actually observe, sitting on the bank, is not the speed. It's the moment a pad bobs up and breaks the surface, which is a purchase. The daily life underneath, the real rhythm, you never see directly. You infer it from the occasional surfacing.
A purchase log isn't the customer. It's the splashes.
Greene's math, borrowed from Ehrenberg in 1959, did one thing beautifully. It described the differences between pads. Some drift fast, some slow, and the spread of those speeds across your whole base follows a tidy distribution. Hand it a customer's purchase history and it will tell you, with real skill, whether you're looking at a fast pad or a slow one. For a certain kind of question, who are my heavy buyers, this is still most of what a company needs.
Then chapter ten. Greene, to his credit, wrote down what his own framework could not do. The gist: the way any single person's rate shifts inside the window you're studying slips through the model entirely, treated as noise and thrown away. In plain terms, his math assumes each pad keeps its speed. It cannot see a pad speed up or slow down. If a customer's life changes, a kid, a move, a new sport, a slow falling-out with your brand, the model files the change under noise and discards it.
He knew. He wrote it down in 1982, and then, because there was no tool to do better, the field mostly moved on.
I find that paragraph moving in a way that's hard to explain to people who don't spend their days inside customer data. It's a man telling you exactly where the floor of his own work is, in a book meant to sell that work. Most of us are not that honest about our blind spots.
Here is the part that should bother anyone running a modern CRM. The blind spot Greene named in 1982 is, for the majority of commercial systems, still the blind spot today.
Walk the timeline. 1959, Ehrenberg gives the statistics. 1982, Greene gives the metaphor and the confession. 1987, Schmittlein and colleagues add the idea that a pad can sink for good, that customers don't just go quiet, sometimes they leave forever, and you can estimate the odds they're already gone. 2005, Fader and Hardie make that math cheap enough to run at scale, and it becomes the quiet engine inside a lot of the "customer lifetime value" you've heard pitched. 2008 is the year the blind spot finally gets an instrument: Netzer and colleagues bring hidden Markov models into the field, which is a precise way of saying a customer lives in one of a few hidden states, each with its own speed, and moves between them, and you can recover those moves from the order of what they did. The same pad, changing speed, becomes something you can compute. 2018, Ascarza adds a sharp and unwelcome finding: a large share of the customers your model marks as "about to churn" will not respond to anything you do, so the rescue budget you aim at them is simply gone.
SHOW THE MATH · the 2008 model
A hidden Markov model assumes the customer is in one of a few hidden states at any time, each with its own purchase rate, and moves between them month to month. Two matrices define it.
The transition matrix is the probability of moving from one state to another in a month (rows are the state you're in, columns the state you move to):
| → Dormant | → Transitioning | → Active | |
|---|---|---|---|
| Dormant | 0.92 | 0.07 | 0.01 |
| Transitioning | 0.18 | 0.52 | 0.30 |
| Active | 0.04 | 0.16 | 0.80 |
The emission matrix is the probability of what you observe in a month, given the state (here: no purchase, a core purchase, or a purchase in a new category):
| no purchase | core | new category | |
|---|---|---|---|
| Dormant | 0.80 | 0.19 | 0.01 |
| Transitioning | 0.50 | 0.32 | 0.18 |
| Active | 0.30 | 0.50 | 0.20 |
Feed in a customer's purchase sequence and run the forward algorithm, and you get the belief over states at each month: the probability the customer is dormant, transitioning, or active right now. That running belief is exactly what the 847 chart below plots. The dormant state is sticky (0.92 stays dormant), so a lone purchase barely moves it; a new-category purchase, rare for a dormant customer, moves it sharply.
Sixty-some years. Each generation drew one more stroke of the picture. And if you opened the CRM at most brands tomorrow and asked it a plain question, which of my customers is changing direction right now, it could not answer. It can give you the tier. It is parked, more or less, in 1987.
I built a talk around this recently, and the spine of it was an old prayer, the one about accepting what you can't change, finding the courage to change what you can, and the wisdom to tell the two apart. It maps onto customer data almost too cleanly. What you can't change is the difference between people, the fast and slow pads, what an older language would call fate. What you can change is the movement inside one person over time, fortune. The wisdom, the hard part, is telling which is which, because the same rising number can mean "this person was always valuable and we missed it" or "this person is having a good quarter and will drift back," and those two readings send your money in opposite directions.
A small story makes it concrete. There was a customer, call him 847, who looked like nothing for two years. Four purchases a year, filed under low value, a label that never moved. In month fourteen he did something a little out of pattern. A snapshot model can't react to that, since one odd purchase barely moves a twelve-month average. By month eighteen he'd tripled his buying and the system finally noticed. A model that read him as a sequence saw the turn back in month fourteen.
Four months is a long time in a relationship with a customer. It's the difference between meeting someone where they're heading and chasing them after they've arrived.
People want to talk about AI in all of this, and I get why, but the honest version is duller than the pitch. No model, however large, changes fate; a naturally low-rate customer stays low, and the math that says so is sixty years old and still right. What the new tools buy you is speed and resolution on fortune. They see the state change sooner. They can estimate how much of a campaign's revenue was going to land anyway, the part you don't get to take credit for. The value isn't a smarter answer. It's finally being able to ask the question Greene couldn't.
So that's why I keep going back to that book. Not for the river, lovely as it is, but for the paragraph in chapter ten where a man tells you plainly what he can't see, forty-odd years before most of the industry catches up to caring. If you work with customer data, here's the question I now put to everyone: when you call someone a high-value customer, do you actually know whether that's fate or fortune? Pull the time series. Flat is one, bumpy is the other. Most people have never looked.