Problem solved: Found what I was looking for in: An Axiomatic Characterization Causal Counterfactuals, thanks to Evan Lloyd.

Basically, making every endogenous variable a deterministic function of the exogenous variables and of the other endogenous variables, and pushing all the stochasticity into the exogenous variables.

Old post:

A problem that's come up with my definitions of stratification.

Consider a very simple causal graph:

In this setting, A and B are both booleans, and A=B with 75% probability (independently about whether A=0 or A=1).

I now want to compute the counterfactual: suppose I assume that B=0 when A=0. What would happen if A=1 instead?

The problem is that P(B|A) seems insufficient to solve this. Let's imagine the process that outputs B as a probabilistic mix of functions, that takes the value of A and outputs that of B. There are four natural functions here:

• f₀(x) = 0
• f₁(x) = 1
• f₂(x) = x
• f₃(x) = 1-x

Then one way of modelling the causal graph is as a mix 0.75f₂ + 0.25f₃. In that case, knowing that B=0 when A=0 implies that P(f₂)=1, so if A=1, we know that B=1.

But we could instead model the causal graph as 0.5f₂+0.25f₁+0.25f₀. In that case, knowing that B=0 when A=0 implies that P(f₂)=2/3 and P(f₀)=1/3. So if A=1, B=1 with probability 2/3 and B=1 with probability 1/3.

And we can design the node B, physically, to be one or another of the two distributions over functions or anything in between (the general formula is (0.5+x)f₂ + x(f₃)+(0.25-x)f₁+(0.25-x)f₀ for 0 ≤ x ≤ 0.25). But it seems that the causal graph does not capture that.

Owain Evans has said that Pearl has papers covering these kinds of situations, but I haven't been able to find them. Does anyone know any publications on the subject?

Hi, lesswrong.

I just finished my application for android devices, LifeTracking, which has been motivated by the discussions here; primarily discussions about akrasia and measuring/tracking your own actions. I don't want to make this sound like an advertisement (the application is completely free anyway), but I would really really like to get feedback from you and hear your comments, criticism, and suggestions. If there are enough LessWrong-specific feature requests, I will make a separate application just for that.

Here is a brief description of the app:

LifeTracking application allows you to track any value (like your weight or your lesswrong karma), as well as any time-consuming activities (like sleeping, working, reading Harry Potter fanfic, etc). You can see the data visually, edit it, and analyze it.

The goal of the application is to help you know yourself and your schedule better. Hopefully, when you graph various aspects of your life side-by-side you will come to a better understanding of yourself. Also, this way you will not have to rely on your faulty memory to remember all that data.

You can download the app from the Market (link only works from Android devices) or download .apk directly. Screenshots: [1], [2], [3], [4], [5], [6].

Edit: LifeTracking website

And while we are on topic of mobile apps, what other applications would you like to see made? (For example, another useful application would be "your personal prediction tracker", where you enter various short-term predictions, your confidence interval, and then enter the actual result. You can classify each prediction and then see if you are over- or under-confident in certain areas. (I remember seeing a website that does something similar, but can't find it now.))