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Very interesting topic. Glad the OP took it on. I’ve unsuccessfully tried to teach non-statisticians to think this way. Brings to mind the quote attributed to

Very interesting topic. Glad the OP took it on. I’ve unsuccessfully tried to teach non-statisticians to think this way. Brings to mind the quote attributed to Samuelson and Keynes: “When events change, I change my mind. What do you do?”

alexpetralia 6y ago
I agree with this, but to play devil's advocate (and something I have not resolved internally yet): Assume a model states there is a 99% likelihood something will occur. Now the data changes, and the likelihood drops to 1%. Was the original model "correct" insofar that there was a 99% likelihood of something occurring (given the information it had at the time)? Or should it have "priced in" the fact that data may change substantially, and 99% was far too overconfident? How are we supposed to interpret variability in model estimates? Do we throw up our hands and say "the data changed"? Or do we hold the models somewhat accountable, saying - no, you weren't "right at the time, given your data". If your estimates are changing so strongly, you are wrong. A 99% estimate that drops to 1% is simply, undeniably "unreliable." In this case, we somewhat care about "model robustness", but how does this extrapolate to situations where the data changing _should in fact_ impact the model substantially? I suspect the answer necessitates a deeper look into the nature of probability, risk and uncertainty.
- fractionalhare 6y ago
  In general there are rigorous ways to quantify those kinds of model deficiencies. From an informal perspective: if the change in data is small and the change in model output is large, your model is poor (and in particular, possibly overfit). Formally what you're describing is the bias-variance tradeoff.[1] You can assess this by looking at the conditioning of your model, which measures how sensitive it is to changes.[2] Roughly speaking, the condition number of an estimator (or generally, function) measures how large the change f(x) -> f(y) in the range for the change x -> y in the domain, where x and y are close. That will give you variance. If you try to minimize bias too much, you may overfit your model and it would exhibit high variance in cross validation. If you try to minimize variance, you may fail to capture relations in your underlying data, which would exhibit higher bias. Practically speaking, for your specific example: if a relatively small change in the sample data resulted in the model adjusting its prediction to 99% from 1%, I would assume your model is severely overfit (I can't quantify exactly how small without more context, but let's agree it's small). From a meta Bayesian perspective, it would take quite a lot of further cross validation for me to drop that belief ;) _____________________ 1. https://en.m.wikipedia.org/wiki/Bias–variance_tradeoff https://en.m.wikipedia.org/wiki/Bias–variance_tradeoff 2. https://mathworld.wolfram.com/ConditionNumber.html https://mathworld.wolfram.com/ConditionNumber.html
  - alexpetralia 6y ago
    What a great answer. Thank you so much!
- Emphere 6y ago
  Yep. What you're getting at in a sense is "Knightian uncertainty", the presence of unknown unknowns (aka "black swans"). And the heart of the matter is certainty, which we cannot hope to achieve. So what exactly do these tools for handling risk help us achieve? I think this is still a question of active investigation and at any rate, if good answers do exist for it, they certainly haven't filtered down to the common masses.

alexpetralia 6y ago

I agree with this, but to play devil's advocate (and something I have not resolved internally yet): Assume a model states there is a 99% likelihood something will occur. Now the data changes, and the likelihood drops to 1%. Was the original model "correct" insofar that there was a 99% likelihood of something occurring (given the information it had at the time)? Or should it have "priced in" the fact that data may change substantially, and 99% was far too overconfident? How are we supposed to interpret variability in model estimates? Do we throw up our hands and say "the data changed"? Or do we hold the models somewhat accountable, saying - no, you weren't "right at the time, given your data". If your estimates are changing so strongly, you are wrong. A 99% estimate that drops to 1% is simply, undeniably "unreliable." In this case, we somewhat care about "model robustness", but how does this extrapolate to situations where the data changing _should in fact_ impact the model substantially? I suspect the answer necessitates a deeper look into the nature of probability, risk and uncertainty.

fractionalhare 6y ago
In general there are rigorous ways to quantify those kinds of model deficiencies. From an informal perspective: if the change in data is small and the change in model output is large, your model is poor (and in particular, possibly overfit). Formally what you're describing is the bias-variance tradeoff.[1] You can assess this by looking at the conditioning of your model, which measures how sensitive it is to changes.[2] Roughly speaking, the condition number of an estimator (or generally, function) measures how large the change f(x) -> f(y) in the range for the change x -> y in the domain, where x and y are close. That will give you variance. If you try to minimize bias too much, you may overfit your model and it would exhibit high variance in cross validation. If you try to minimize variance, you may fail to capture relations in your underlying data, which would exhibit higher bias. Practically speaking, for your specific example: if a relatively small change in the sample data resulted in the model adjusting its prediction to 99% from 1%, I would assume your model is severely overfit (I can't quantify exactly how small without more context, but let's agree it's small). From a meta Bayesian perspective, it would take quite a lot of further cross validation for me to drop that belief ;) _____________________ 1. https://en.m.wikipedia.org/wiki/Bias–variance_tradeoff https://en.m.wikipedia.org/wiki/Bias–variance_tradeoff 2. https://mathworld.wolfram.com/ConditionNumber.html https://mathworld.wolfram.com/ConditionNumber.html
- alexpetralia 6y ago
  What a great answer. Thank you so much!
Emphere 6y ago
Yep. What you're getting at in a sense is "Knightian uncertainty", the presence of unknown unknowns (aka "black swans"). And the heart of the matter is certainty, which we cannot hope to achieve. So what exactly do these tools for handling risk help us achieve? I think this is still a question of active investigation and at any rate, if good answers do exist for it, they certainly haven't filtered down to the common masses.