The goal is to provide a simple & clear process such that:
Given the history of your system (number of visitors/signups/conversions each day), you can see a best guess of the conversion/cancellation rates and how they’ve changed over time.
Given hypothetical future conversion/cancellation rates, what will your system look like at some point $t^\prime$ in the future?
(this includes number of free/paid/cancelled users, and revenue totals and rates, etc.)
For the first question, we will use statistics. For the second, we will use simulations.
It’s usually easy to get a listing of events (“user subscribed to plan A”, “user cancelled from plan C”). We wish to take this list of events, and produce a estimate for the transition rates between each “bucket” at any given time in the past or present.
This method, DSE, can be used to to simulate, the movement of people across an online business, or to simulate the changing concentration of chemicals in a reaction.
“Discrete” means the objects jump from one type to another, but can’t have values in between. For example, a customer is either a “free trial” customer or a “monthly subscription” customer, nothing in between.
The method works even when the number of objects (people/molecules/etc) is very small, and when their change from one type to another happens randomly over time.
In general, it can simulate any system whose “state” (a list of numbers that completely describe the system) can only change discretely (by whole numbers).
Given event data, get a plot like the following for the rate of events as a function of time:
nonparametric regression
(locally adaptive) kernel density estimation
smoothing estimators
Bayesian statistics of time series
Inhomogenous Poisson process. (or “time-varying Poisson process”)
spike rate estimation (requires multiple “runs” ?)
Notes:
Bayesian probability is a way of updating your best guess distribution of an uncertain result, when you come across new data. There are two main steps:
So far, the procedure that seems most applicable, and most straightforward.
A probability distribution is a function that tells you how likely a given result is.
Here are three example distributions for the value $x$. In the blue distribution, $x$ can be anywhere, but is very likely to be at $0$ (the center peak). In the green one, $x$ could still be anywhere, but is likely to be $-2$.