Moritz Schauer
@mschauer.bsky.social
Statistician, Associate Professor (Lektor) at University of Gothenburg and Chalmers; inference and conditional distributions for anything
https://mschauer.github.io
http://orcid.org/0000-0003-3310-7915
[ˈmoː/r/ɪts ˈʃaʊ̯ɐ]
https://mschauer.github.io
http://orcid.org/0000-0003-3310-7915
[ˈmoː/r/ɪts ˈʃaʊ̯ɐ]
Reposted by Moritz Schauer
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
January 15, 2026 at 1:47 PM
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.
January 15, 2026 at 2:23 PM
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.
January 15, 2026 at 1:37 PM
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.
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.
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.
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}
\
stats.stackexchange.com
January 15, 2026 at 9:57 AM
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.
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
January 14, 2026 at 11:32 AM
At a technical university the steps of Pearl’s ladder are called stochastics, stochastic control and optimal transport
Yeah, more oil and less integrals
January 13, 2026 at 9:36 AM
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.
January 13, 2026 at 8:18 AM
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
January 13, 2026 at 7:27 AM
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.
January 12, 2026 at 9:34 AM
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).
That is E(X | do(T = t)) = E(X | T = t).
January 12, 2026 at 7:48 AM
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).
That is E(X | do(T = t)) = E(X | T = t).
Love it. Adding Sid Meier's Beta Centauri
January 11, 2026 at 11:47 AM
Love it. Adding Sid Meier's Beta Centauri
REMEMBERING HARRY VAN ZANTEN
Botond Szabó and Aad van der Vaart in the ISBA Bulletin.
Botond Szabó and Aad van der Vaart in the ISBA Bulletin.
The ISBA Bulletin
isba-bulletin.github.io
January 9, 2026 at 2:52 PM
REMEMBERING HARRY VAN ZANTEN
Botond Szabó and Aad van der Vaart in the ISBA Bulletin.
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)
January 7, 2026 at 4:11 PM
(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.
January 7, 2026 at 3:36 PM
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.
Reposted by Moritz Schauer
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.
June 17, 2025 at 4:56 PM
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.
January 7, 2026 at 12:45 PM
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
January 7, 2026 at 10:25 AM
How do you do cookie banners per fax? I fear there is a way
*) There is a causal pathway candidate between estrogen and UV exposure www.sciencedirect.com/science/arti...
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 …
www.sciencedirect.com
December 19, 2025 at 10:13 AM
*) 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...
github.com
December 19, 2025 at 10:12 AM
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.
December 19, 2025 at 10:12 AM
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.
December 19, 2025 at 10:12 AM
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.
December 19, 2025 at 10:12 AM
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:
December 19, 2025 at 10:01 AM
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
December 2, 2025 at 9:46 AM
Vaguely funny expression: a serious title