Moritz Schauer

It’s not only easy to spot. It’s hard to unsee.

p=0.048, but in an enlightened way

If your niche is small enough, every post is a viral post.

I think it’s not supposed to be spelled correctly en.wikipedia.org/wiki/Diff%C3...

wikipedia turns 25 today! the last unenshittified major website! backbone of online info! triumph of humanity! powered by urge of unpaid randos to correct each other! somehow mostly reliable! "good thing wikipedia works in practice, because it sure doesn't work in theory" - old wiki adage

Yeah, the Kalman gain K is the regression coefficient, so the conditional mean is old mean plus observations scaled by K. If you write K (H Σ⁻ Hᵀ + Σ_ε) = Σ⁻ Hᵀ you see how it is aligned with the normal equations A Σ₂₂ = Σ₁₂ from above.

In general, residuals of linear regression are only uncorrelated with the predictors, not independent, so their conditional mean need not vanish. Gaussianity upgrades uncorrelatedness to independence; once this happens, the linear predictor becomes the mean of the conditional distribution.

Deriving the conditional distributions of a multivariate normal distribution We have a multivariate normal vector ${\boldsymbol Y} \sim \mathcal{N}(\boldsymbol\mu, \Sigma)$. Consider partitioning $\boldsymbol\mu$ and ${\boldsymbol Y}$ into $$\boldsymbol\mu = \begin{bmatrix} \

Have a look here: stats.stackexchange.com/a/30600

The trick: choose A by the normal equation A Σ₂₂ = Σ₁₂ and see that X₁ − A X₂ is uncorrelated with X₂, and by Gaussianity also independent. So E[X₁∣X₂] = A X₂. Even works in the singular case.

At a technical university the steps of Pearl’s ladder are called stochastics, stochastic control and optimal transport

Yeah, more oil and less integrals

Mostly echoing your statement bsky.app/profile/p-hu... The do-operator formalizes how a system acts to interventions, so certain statements about interventions become propositions in a calculus, but you still have to argue how this maps to the system you want to describe.

Pearl is maybe also dismissive of this meta level, whereas people do make clean meta-level arguments for RCTs etc, in fact it is unavoidable, cf @p-hunermund.com

In classical approaches, correctness of causal claims is argued at the meta level, by appealing to design or understanding. In the do-calculus, that burden is shifted into a mathematical formalism.

Causal inference is often hidden in plain sight. In a randomized clinical trial, the setup is such that interventional and conditional distributions coincide.

That is E(X | do(T = t)) = E(X | T = t).

Love it. Adding Sid Meier's Beta Centauri

REMEMBERING HARRY VAN ZANTEN

Botond Szabó and Aad van der Vaart in the ISBA Bulletin.

(and point null is the worst case for an error in the directional statements)

By the way, I am quite okay with users drawing directional conclusions after rejecting a two-sided null hypothesis; because the error rate under the point null is the same as that of the original test.

It’s giving late-game vibes of Sid Meier’s Civilization, where the player is bored and just trying to see what happens if they declare some wars before they abandon the game.

Feels like our two-sided tests make things better or worse.

How do you do cookie banners per fax? I fear there is a way

Atatürk: The Biography of the Founder of Modern Turkey Mustafa Kemal Atatürk was virtually unknown until 1919,…

www.goodreads.com/book/show/19...

Ultraviolet radiation as a predictor of sex hormone levels in postmenopausal women: A European multi-center study (ECRHS) Solar ultraviolet radiation (UVR) affects the body through pathways that exhibit positive as well as negative health effects such as immunoregulation …

*) There is a causal pathway candidate between estrogen and UV exposure www.sciencedirect.com/science/arti...

This conclusion did not rely on biological knowledge, experiments, or interventions. It followed entirely from the pattern of independence and conditional dependence. The real question is when such patterns force causal direction — and when they do not... Slides: github.com/mschauer/Cau...

This pattern leaves essentially one causal interpretation: low estrogen and lack of sunlight exposure are independent causes of bone mineral density loss. The conclusion follows from the correlation structure alone, not from prior biological knowledge.

Now restrict attention to individuals with low bone mineral density. Within this group, sunlight exposure and estrogen level are no longer independent. If one is normal, the other is more likely to be deficient. Conditioning has created dependence.

So consider two variables: sunlight exposure and estrogen level. A priori, it is not obvious that either is related to bone mineral density. Empirically, in the population, sunlight exposure and estrogen level are independent*. No causal assumptions are made beyond that.

Students learn early that correlation does not imply causation. Correct, but incomplete... Certain patterns of independence and conditional dependence constrain #causal structure very strongly. A simple example:

Vaguely funny expression: a serious title

God made not only the numbers but also a sequence of independent standard normal random variables.