## Integration & Correlation

Claims by AI chatbots, engineers and Nobel prize winners notwithstanding, absence of correlation does not prove absence of causation, any more than presence of correlation proves presence of causation. Bard outlines several reasons from noise and nonlinearity, but missed a key one: bathtub statistics.

Here’s a really simple example of how this reasoning can go wrong. Consider a system with a stock Y(t) that integrates a flow X(t):

X(t) = -t

Y(t) = ∫X(t)dt

We don’t need to simulate to solve for Y(t) = -1/2*t^2 +C.

Over the interval t=[-1,1] the X and Y time series look like this:

The X-Y relationship is parabolic, with correlation zero:

Zero correlation can’t mean “not causal” because we constructed the system to be causal. Even worse, the sign of the relationship depends on the subset of the interval you examine:

This is not the only puzzling case. Consider instead:

X(t) = 1

Y(t) = ∫X(t)dt = t + C

In this case, X(t) has zero variance. But Corr(X,Y) = Cov(X,Y)/σ(X)σ(Y) which is 0/0. What are we to make of that?

This pathology can also arise from feedback. Consider a thermostat that controls a heater that operates in two states (on or off). If the heater is fast, and the thermostat is sensitive with a narrow temperature band, then σ(temperature) will be near 0, even though the heater is cycling with σ(heater state)>0.

## AI Chatbots on Causality

Having recently encountered some major causality train wrecks, I got curious about LLM “understanding” of causality. If AI chatbots are trained on the web corpus, and the web doesn’t “get” causality, there’s no reason to think that AI will make sense either.

TLDR; ChatGPT and Bing utterly fail this test, for reasons that are evident in Google Bard’s surprisingly smart answer.

## Google Bard: PASS

Google gets strong marks for mentioning a bunch of reasons to expect that we might not find a correlation, even though x is known to cause y. I’d probably only give it a B+, because it neglected integration and feedback, but it’s a good answer that properly raises lots of doubts about simplistic views of causality.

## Causality in nonlinear systems

Sugihara et al. have a really interesting paper in Science, on detection of causality in nonlinear dynamic systems. It’s paywalled, so here’s an excerpt with some comments.

Abstract: Identifying causal networks is important for effective policy and management recommendations on climate, epidemiology, financial regulation, and much else. We introduce a method, based on nonlinear state space reconstruction, that can distinguish causality from correlation. It extends to nonseparable weakly connected dynamic systems (cases not covered by the current Granger causality paradigm). The approach is illustrated both by simple models (where, in contrast to the real world, we know the underlying equations/relations and so can check the validity of our method) and by application to real ecological systems, including the controversial sardine-anchovy-temperature problem.

Identifying causality in complex systems can be difficult. Contradictions arise in many scientific contexts where variables are positively coupled at some times but at other times appear unrelated or even negatively coupled depending on system state.

Although correlation is neither necessary nor sufficient to establish causation, it remains deeply ingrained in our heuristic thinking. … the use of correlation to infer causation is risky, especially as we come to recognize that nonlinear dynamics are ubiquitous. Continue reading “Causality in nonlinear systems”