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If we use the example of the correlation between frequency of cricket chirps and temperature, where there is a causal relationship between temperature and crickets' chirping rates; it seems to me we could trust this kind of predictive model only in the case in which the temperature is the only factor impacting cricket chirping rate.

For example, let's say that when crickets are sick they chirp at much lower rates than usual to preserve energy, and that this instinct supersedes the instinct to match chirping rates with outside temperature; such that even if it is very hot, if the crickets are sick they will chirp at rates corresponding to much lower temperatures based on our predictive model.

Are we just hoping that our sample data covers so many cases that some of them will take into account all of the hidden variables in some way?

In the end, how much can you ever trust a predictive model, even when it describes a causal relationship?

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First of all, let's distinguish prediction from causality. A model may be predictively accurate without correctly specifying causal relationships, or even attempting to specify causal relationships. For example, if $X$ causes $Y$, and $Y$ has little other variation, then you may be able to accurately predict $X$ using $Y$ although the casual relationship goes in the other direction. Conversely, a model that specifies causal relationships accurately may not be predictively accurate. For example, the model may be very complex and hence have its parameters readily overfit.

Now, how is the predictive accuracy of a model affected if an important variable is left out, as in the case of cricket health? As you note, it may perform worse than it would if it had access to the left-out variable. But this isn't a reason not to trust a predictive model. When you assess a predictive model (in the right way), you get an estimate of its predictive accuracy that already includes whatever problems might be reducing predictive accuracy, such as an omitted variable. The only scenario where you could worry that an obtained estimate of predictive accuracy is too optimistic because it fails to account for an omitted variable is where you've estimated the model's predictive accuracy on a dataset where the variable is constant and then applied the model to data where the variable is non-constant. And that's just one manifestation of the trouble with extrapolation.

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