Black Swans

Batman

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I don't think he makes the point brilliantly, but the point is:

Imagine that I have a probability distribution $p_x(x)$, and a value $\hat{x}$ such $p_x(\hat{x})$ gives the maximum value for $p_x$. I.e. $\hat{x}$ represents the most likely outcome.

Now imagine that I perform a change of variable operation. E.g. $y = \sin(x)$. It's not necessarily true that $\sin(\hat{x})$ will also represent the most likely the maximum value of $p_y(y) = p_x(g(y))|g'(y)|$, because $|g'(y)|$ is itself a function, not a constant.

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