Most times I go to update a forecast, nothing has happened. There are several ways to update on this, which I explore below.

This post was written after Ozzie Gooen suggested I play around with Squiggle, following up on his earlier Multivariate estimation & the Squiggly language. Squiggle is an experimental domain-specific language for working with probabilities, which is currently in the very early stages of its existence. It can probably do anything Guesstimate can do, while being more easily copied and pasted, at the cost of not having a graphical interface. It currently uses vim as a text editor, so the user has to press i to switch to insert mode and edit the code, and in general it's on the very early stages, e.g., syntax and functionality are still being figured out.

1. Naïve decay

Perhaps the most straightforward way of updating on the passage of time is as follows: If one assigns a probability $p_{y e a r l y} = 0.9$ to an event happening in a year, then the monthly probability $p_{m o n t h l y}$ is such that $(1 - p_{m o n t h l y})^{12} = 1 - 0.9$ , or $p_{m o n t h l y} = 1 - (1 - 0.9)^{1 / 12}$ . Now, if a month has passed, then the remaining probability is $p_{u p d a t e d} = 1 - (1 - p_{m o n t h l y})^{11}$ .

In Squiggle, this would look as follows:

## Press i to edit. 
## Initial setup
yearly_probability=0.9
period_probability_function(epsilon)=1-(1-yearly_probability)^(1/epsilon)
probability_decayed(t, total_time, period_probability) = 1-(1-period_probability)^(total_time-t)

## Monthly decomposition
total_time_monthly=12 # months in a year
monthly_probability=period_probability_function(total_time_monthly)
probability_decayed_monthly(t)=probability_decayed(t, total_time_monthly, monthly_probability) 
probability_decayed_monthly
### probability_decayed_monthly(6)

And this would look like so:

The type signature of this function is $p \to (t \to p)$ . That is, I take a probability (0.9), and output a function that goes from time passed, t, to probability remaining after time t. If I wanted to know what the probability is after 6 months, I'd uncomment the last line, probability_decayed_monthly(6), in this case equal to 0.684.

2. Unprincipled decay.

A strategy which works well as a forecasting technique in practice, but which is totally unprincipled, is to consider probabilities over probabilities. I can ask myself what the lowest and upper bets I'd be willing to take would be, and they might be something like

1:9, where I receive 9 if the event

...

Lowry

Lowry