As far as I'm aware nobody claims trans fats aren't bad.

 

See comment by Gilch, allegedly Vaccenic acid isn't harmful. The particular trans-fats produced by isomerization of oleic and linoleic acid, however, probably are harmful. Elaidic acid for example is a major trans-fat component in margarines, which were banned.

As far as I'm aware nobody claims trans fats aren't bad.

Yeah i was unaware of vaccenic acid. I've edited the post to clarify.

I've also realized that it might explain the anomalous (i.e. after adjusting for confounders) effects of living at higher altitude. The lower the atmospheric pressure, the less oxygen available to oxidize the PUFAs. Of course some foods will be imported already full of oxidized FAs and that will be too late, but presumably a McDonalds deep fryer in Colorado Springs is producing less PUFAs/hour than a correspondingly-hot one in San Francisco.

This feels too crazy to put in the original post but it's certainly interesting.

That post is part of what spurred this one

I uhh, didn't see that. Odd coincidence! I've added a link and will consider what added value I can bring from my perspective.

Thanks for the feedback. There's a condition which I assumed when writing this which I have realized is much stronger than I originally thought, and I think I should've devoted more time to thinking about its implications.

When I mentioned "no information being lost", what I meant is that in the interaction $A\to B$, each value $b\in B$ (where $B$ is the domain of $P_B$) corresponds to only one value of $a\in A$. In terms of FFS, this means that each variable must be the maximally fine partition of the base set which is possible with that variable's set of factors.

Under these conditions, I am pretty sure that $A\perp C⟹A\perp C|B$

I was thinking about causality in terms of forced directional arrows in Bayes nets, rather than in terms of d-separation. I don't think your example as written is helpful because Bayes nets rely on the independence of variables to do causal inference: $X\to Y\to Z$ is equivalent to $X\leftarrow Y\leftarrow Z$.

It's more important to think about cases like $X\to Y\leftarrow Z$ where causality can be inferred. If we change this to $\hat{X},\hat{Y},\hat{Z}$ by adding noise then we still get a distribution satisfying $\hat{X}\to \hat{Y}\leftarrow \hat{Z}$ (as $\hat{X}$ and $\hat{Z}$ are still independent).

Even if we did have other nodes forcing $X\to Y\to Z$ (such as a node $U$ which is parent to $Y$, and another node $V$ which is parent to $Z$), then I still don't think adding noise lets us swap the orders round.

On the other hand, there are certainly issues in Bayes nets of more elements, particularly the "diamond-shaped" net with arrows $W\to X,W\to Y,X\to Z,Y\to Z$. Here adding noise does prevent effective temporal inference, since, if $\hat{X}$ and $\hat{Y}$ are no longer d-separated by $\hat{W}$, we cannot prove from correlations alone that no information goes between them through $\hat{Z}$.

I had forgotten about OEIS! Anyway Ithink the actual number might be 1577 rather than 1617 (this also gives no answers). I was only assuming agnosticism over factors in the overlap region $ABCD$ if all pairs $AB,AC,...,CD$ had factors, but I think that is missing some examples. My current guess is that any overlap region like $ABCD$ should be agnostic iff all of the overlap regions "surrounding" it in the Venn diagram ($ABC$, $ABD$, $ACD$, $BCD$) in this situation either have a factor present or agnostic. This gives the series 1, 2, 15, 1577, 3397521 (my computer has not spat out the next element). This also gives nothing on the OEIS.

My reasoning for this condition is that we should be able to "remove" an observable from the system without trouble. If we have an agnosticism, in the intersection $ABCD$, then we can only remove observable $B$ if this doesn't cause trouble for the new intersection $ABD$, which is only true if we already have an factor in $ABD$ (or are agnostic about it). 

I know very, very little about category theory, but some of this work regarding natural latents seem to absolutely smack of it. There seems to be a fairly important three-way relationship between causal models, finite factored sets, and Bayes nets.

To be precise, any causal model consisting of root sets $B$, downstream sets $X$, and functions mapping sets to downstream sets like $f_4:(B_1\otimes B_3\otimes X_2)\to X_4$ must, when equipped with a set of independent probability distributions over B, create a joint probability distribution compatible with the Bayes net that's isomorphic to the causal model in the obvious way. (So in the previous example, there would be arrows from only $B_1$, $B_3$, and $X_2$ to $X_4$) The proof of this seems almost trivial but I don't trust myself not to balls it up somehow when working with probability theory notation.

In the resulting Bayes net, one "minimal" natural latent which conditionally separates $X_i$ and $X_j$ is just the probabilities over just the root elements from $B$ which both $X_i$ and $X_j$ depend on. It might be possible to show that this "minimal" construction of $\Lambda$ satisfies a universal property, and so other ${\Lambda }^{\prime }$ which is also "minimal" in this way must be isomorphic to $\Lambda$.