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Earlier quoted context omitted.
> And it turns out that a strong positive correlation can be a good place to hunt for possible causality I don't disagree, but I have yet to see a mathematically clean definition of "causality".
Do you take issue with Pearl?
His intro to the topic bodes well: http://bayes.cs.ucla.edu/BOOK-2K/why.html
Correlation and convolution are adjoints.
Can you expand a little further? I don't see the adjunction (between what categories?)
That said convolution and correlation are not linear operators but bilinear operators, so I'm struggling a bit to see in what sense they're adjoint.
Maybe something like:
=
which does actually work if you let the inner product be the integral of the product and the correlation the time-correlation (so the inner product is the correlation at time 0).
Correlations are a profound part of our universe. When observing the universe, humans can never prove any facts about the universe. We can only establish correlations. Correlations between events that occur in our universe is the furthest "truth" we can establish about the universe short of a full on proof. What this means is that nothing in the physical universe can be proven. Proof is the domain of maths and logic,…
Note that there are ways to test causal hypotheses without intervening. For example, suppose we wish to test the hypothesis that smoking causes lung cancer via the main mechanism of tar buildup in the lungs, against the alternative hypothesis that smoking is correlated with lung cancer because of a gene that predisposes people to both smoking and lung cancer (this example comes from Judea Pearl’s Book of Why ). If th…
y = x*b + e
b = cov(x,y)/var(x) = corr(x,y) * (std(y)/std(x))
This is like... intro-to-intro statistics, summarized in the last bit of the article: "yes, corr is like a rescaled regression coefficient." y = x*b + e b = cov(x,y)/var(x) = corr(x,y) * (std(y)/std(x)) It's too bad the article did not mention omitted variable bias. The two principal sources of spurious correlation are (1) measurement error, (2) omitted variable bias. It is much easier to grok omitted variable bias i…
Is it that an ommitted variable isn't part of a causal chain of an included variable, and a confounder is?
This is like... intro-to-intro statistics, summarized in the last bit of the article: "yes, corr is like a rescaled regression coefficient." y = x*b + e b = cov(x,y)/var(x) = corr(x,y) * (std(y)/std(x)) It's too bad the article did not mention omitted variable bias. The two principal sources of spurious correlation are (1) measurement error, (2) omitted variable bias. It is much easier to grok omitted variable bias i…
I last studied this stuff 20 years ago. What's the difference between ommitted variable and a confounding variable? Seems like a similar idea. Is it that an ommitted variable isn't part of a causal chain of an included variable, and a confounder is?
Omitted variable bias tells it like it is. A bias in a regression coefficient that results from an incorrectly specified model.
Earlier quoted context omitted.
I last studied this stuff 20 years ago. What's the difference between ommitted variable and a confounding variable? Seems like a similar idea. Is it that an ommitted variable isn't part of a causal chain of an included variable, and a confounder is?
Confounding variable is a more confusing, less precise name for the same phenomenon. Omitted variable bias tells it like it is. A bias in a regression coefficient that results from an incorrectly specified model.
I wonder why Wikipedia has separate entries for them:
The article's question: "Why are correlations meaningful?" Here's a simplified answer. Correlations are meaningful because they indicate how strongly variables can predict other variables, in some circumstances. This is valuable because some variables are easier to use than others. A positive correlation between two variables means they tend to move in the same direction (relative to their mean). A negative correlati…
I don't quite like this answer, because it simplifies a lot of things about correlations which are just not really accurate. > Corrleations are meaningful because they indicate how strongly variables can predict other variables. This is not wrong per se, but may be wrongly interpreted. 1) I think it is important to stress that the reverse is not true. If a correlation is 0, it does not mean the variables are independ…
Earlier quoted context omitted.
Confounding variable is a more confusing, less precise name for the same phenomenon. Omitted variable bias tells it like it is. A bias in a regression coefficient that results from an incorrectly specified model.
Thanks. Assuming they refer to the exact same phenomenon, I agree that ommitted variable bias is better terminology. I wonder why Wikipedia has separate entries for them: https://en.wikipedia.org/wiki/Omitted-variable_bias https://en.wikipedia.org/wiki/Confounding
The picture at the top of the confounding article gives it away. Those kinds of diagrams are common in "hierarchical Bayesian models" like LDA.
In the simple linear setting, they are the same thing. Teaching people the "confounding variable" concept in a general setting before teaching them about "omitted variable" in a linear setting is like teaching people about Riemannian manifolds before teaching them about vector spaces.
Correlation isn't a particularly useful concept outside of simple linear models.
The article's question: "Why are correlations meaningful?" Here's a simplified answer. Correlations are meaningful because they indicate how strongly variables can predict other variables, in some circumstances. This is valuable because some variables are easier to use than others. A positive correlation between two variables means they tend to move in the same direction (relative to their mean). A negative correlati…
I don't quite like this answer, because it simplifies a lot of things about correlations which are just not really accurate. > Corrleations are meaningful because they indicate how strongly variables can predict other variables. This is not wrong per se, but may be wrongly interpreted. 1) I think it is important to stress that the reverse is not true. If a correlation is 0, it does not mean the variables are independ…
The data may be flawed in many ways, so there is no automatic assumption that any conclusion is correct.
And - as this answer says - this is a completely different process to discovering a robust causal mechanism that can reliably explain a correlation and predict its persistence into the future.