We often come up with solutions to problems that lend themselves to various interpretations. Exploring the different interpretations can be useful, as such exploration has the potential to provide deeper insight into the problem at hand, or lead to an extension of the result to an entirely different class of problems. A particularly gruesome example came up this week in a course that I am taking.
In this course, we are studying a category of processes called Poisson processes. Poisson processes are a way of predicting or modeling how many random events occur in a given time period. A standard example from economics is a model for how many customers have entered a store by a given time. In a somewhat idealized world, customers enter the store at random, and our count of customers will be a Poisson process.
Having created such a model, we might be interested in several other statistics. For instance, how long do we have to wait between customers? how long will it take until we see our one millionth customer? what is the average rate at which customers enter the store?
It turns out that most (if not all) of the questions that we could ask about a Poisson process are related to the rate at which things happen. If you know that you get an average of 30 customers per hour, then you know just about everything that there is to know. This rate tells you that you can expect to see 30 customers in any given hour, but it also tells you that you are going to have to wait an average of 1/30 hours (or two) between customers.
In this problem, there is a clear connection between the rate at which customers arrive, and the amount of time that we have to wait before we see a customer. They are reciprocals of each other. If we expect to see one customer every two minutes, we can equivalently state that we expect there to be two minutes between customers.
Sometimes the wait time and rate at which events happen are not so clearly linked. In the problem that we worked this week, we were asked to determine whether or not a very sick person lives long enough to get a kidney transplant. This person is in a hospital, where she is waiting for a kidney to arrive. Kidneys arrive according to a Poisson process, at a given rate λ (the Greek letter “lambda”).
So far, this problem makes sense. We can either think about λ kidneys arriving every day, or 1/λ days passing between the arrival of two kidneys. The first interpretation is a rate, while the second is a waiting time. For instance, a sensible, “real world” value for λ might be something like 1/25. We could think of this as meaning that 1/25th of a kidney arrives every day, or that we have to wait an average of 25 days for a kidney to arrive.
Here’s the part where it gets a little weird: the patient will live for an average of μ (the Greek letter “mu”) days without a kidney. However, this life-expectancy is random, and the patient could die at any moment, or live to be 100. The amount of time that the patient lives can be modeled by the same equation that models the amount of time that she has to wait before a kidney arrives, except that the time waited is related to μ, rather than 1/λ. But if the waiting time is μ, this implies that there is some related rate, 1/μ.
But what does it mean for a patient to die with rate 1/μ? Normally, we think of rates as telling us how quickly events happen, when there are many events that could happen. Going back to our kidneys again, lots of kidneys come into the hospital, so it makes sense to talk about a rate of one kidney per month (or something similar). But our patient can only die once, so it doesn’t really make sense to talk about the patient dying at a rate of one death every week.
And now we enter Herald’s House of Horrors.1Named for Dr. Christopher Herald, the professor who (as far as I know) came up with the morbid little scenario. Imagine a not-too-distant future when the art of cloning has been perfected. You work in a strange little hospital where you only have one patient. This patient has kidney problems, and has a life-expectancy of μ days. On the bright side, you have an inexhaustible supply of clones of this patient. Unfortunately, they all have the same bad kidneys, and all have a life-expectancy of μ days. In order to get the most out of your patients, only one of them is ever defrosted at a time. Every time a patient dies, you thaw out another clone, and wait for it to die. Your job is to count the number of times that your patient dies.
In this little scenario, it does make sense to talk about a rate of death. If your patient lives for an average of μ days, then clones are going to die at a rate of 1/μ per day. It requires some rather morbid thinking, but we can come up with a way of interpreting life-expectancy as a rate—a person’s life-expectancy is simply the reciprocal of the rate at which their clones would die in this odd little clinic.
The most obvious next question is, “Why would anyone ever want to imagine such a horrible scenario?” The answer is that it leads to a quick-and-easy way to determine the probability that our original kidney patient lives.
Returning to our store for a moment, suppose that we are interested not only in the rate at which customers arrive, but in the probability that our next customer will be a man. It turns out that if a total of 30 customers arrive per hour, and 5 men arrive per hour, then the probability that a man is the next customer is the ratio of these two rates. That is, the probability that a man is the next customer to arrive is 5/30 = 1/6. Thinking in terms of rates means that we can avoid doing a great deal of computation.
So if we want to know whether our kidney patient will live or die (or, more specifically, the probability that she will get a new kidney before she dies), it is helpful to be able to think in terms of rates, rather than waiting times. If we know the patient’s rate of death, and we know the rate at which kidneys arrive, then we can figure out how likely she is to get a new kidney. If we sum the two rates, then we get the rate at which something happens (either a new kidney arrives, or the patient dies). Taking the ratio of the kidney arrival rate to the rate at which something happens gives us the probability that a kidney arrives before the patient dies. No computation necessary.
All it required was that we try to find a way to interpret a life-expectancy as a rate of death in a dystopian world of disposable clones.