Thank you, that is very clarifying!

I've been doing a deep dive on this post, and while the main theorems make sense I find myself quite confused about some basic concepts. I would really appreciate some help here!

• So 'latents' are defined by their conditional distribution functions whose shape is implicit in the factorization that the latents need to satisfy, meaning they don't have to always look like $P\left[\Lambda |X\right]$, they can look like $P\left[\Lambda \right],P\left[X|\Lambda \right]$, etc, right?
• I don't get the 'standard form' business. It seems like a procedure to turn one latent variable $\Lambda$ into another relative to $X$? I don't get what the notation ${\Lambda }^{\ast }=(x\mapsto P[X=x\mid \Lambda \left]\right)$ means—does it mean that it takes $\Lambda$ defined by some conditional distribution function like $P\left[\Lambda \right],P\left[X|\Lambda \right],P\left[\Lambda |X\right]$ and converts it into $P\left[X|\Lambda \right]$? That doesn't seem so, the notation looks more like a likelihood function than a conditional distribution. But then what conditional distribution defines this latent ${\Lambda }^{\ast }$?

The Resampling stuff is a bit confusing too:

if we have a natural latent $\Lambda$, then construct a new natural latent by resampling $\Lambda$ conditional on $X$ (i.e. sample from $P\left[\Lambda |X\right]$), independently of whatever other stuff ${\Lambda }^{\prime }$ we’re interested in.

• I don't know what operation is being performed here - what CPDs come in, what CPDs leave.
• "construct a new natural latent by resampling $\Lambda$ conditional on $X$ (i.e. sample from $P\left[\Lambda |X\right]$), independently of whatever other stuff ${\Lambda }^{\prime }$ we’re interested in." isn't this what we are already doing when stating a diagram like $X_1\leftarrow \Lambda \to X_2$, which implies a factorization $P\left[\Lambda \right]P\left[X_1|\Lambda \right]P\left[X_2|\Lambda \right]$, none of which have ${\Lambda }^{\prime }$! What changes when resampling? aaaaahhh I think I'm really confused here.
• Also does all this imply that we're starting out assuming that $\Lambda$ shares a probability space with all the other possible latents, e.g. $P[X,\Lambda ,{\Lambda }^{\prime },{\Lambda }^{\prime \prime },\dots ]$? How does this square with a latent variable being defined by the CPD implicit in the factorization?

And finally:

In standard form, a natural latent is always approximately a deterministic function of $X$. Specifically: $\Lambda \left(X\right)\approx {\prod }_i(x^{\prime }\mapsto P[X_i=x_i^{\prime }|X_{\overline{i}}\left]\right)$.

...

Suppose there exists an approximate natural latent over $X_1,\dots ,X_n$. Construct a new random variable $X^{\prime },$ sampled from the distribution $(x^{\prime }\mapsto {\prod }_{i}P[X_i=x_i^{\prime }|X_{\overline{i}}\left]\right)$. (In other words: simultaneously resample each $X_i$ given all the others.) Conjecture: $X^{\prime }$ is an approximate natural latent (though the approximation may not be the best possible). And if so, a key question is: how good is the approximation?

Where is the top result proved, and how is this statement different from the Universal Natural Latent Conjecture below? Also is this post relevant to either of these statements, and if so, does that mean they only hold under strong redundancy?

Does anyone know if Shannon arrive at entropy from the axiomatic definition first, or the operational definition first?

I've been thinking about these two distinct ways in which we seem to arrive at new mathematical concepts, and looking at the countless partial information decomposition measures in the literature all derived/motivated based on an axiomatic basis, and not knowing which intuition to prioritize over which, I've been assigning less premium on axiomatic conceptual definitions than i used to:

• decision theoretic justification of probability > Cox's theorem
• shannon entropy as min description length > three information axioms
• fernando's operational definition of synergistic information > rest of the literature with its countless non-operational PID measures

The basis of comparison would be its usefulness and ease-of-generalization to better concepts:

• at least in the case of fernando's synergistic information, it seems far more useful because i at least know what i'm exactly getting out of it, unlike having to compare between the axiomatic definitions based on handwavy judgements.
• for ease of generalization, the problem with axiomatic definitions is that there are many logically equivalent ways to state the initial axiom (from which they can then be relaxed), and operational motivations seem to ground these equivalent characterizations better, like logical inductors from the decision theoretic view of probability theory

(obviously these two feed into each other)

Just finished the local causal states paper, it's pretty cool! A couple of thoughts though:

I don't think the causal states factorize over the dynamical bayes net, unlike the original random variables (by assumption). Shalizi doesn't claim this either.

• This would require proving that each causal state is conditionally independent of its nondescendant causal states given its parents, which is a stronger theorem than what is proved in Theorem 5 (only conditionally independent of its ancestor causal states, not necessarily all the nondescendants)

Also I don't follow the Markov Field part - how would proving:

if we condition on present neighbors of the patch, as well as the parents of the patch, then we get independence of the states of all points at time t or earlier. (pg 16)

... show that the causal states is a markov field (aka satisfies markov independencies (local or pairwise or global) induced by an undirected graph)? I'm not even sure what undirected graph the causal states would be markov with respect to. Is it the ...

• ... skeleton of the dynamical Bayes Net? that would require proving a different theorem: "if we condition on parents and children of the patch, then we get independence of all the other states" which would prove local markov independency
• ... skeleton of the dynamical Bayes Net + edges for the original graph for each t? that would also require proving a different theorem: "if we condition on present neighbors, parents, and children of the patch, then we get independence of all the other states" which would prove local markov independency

Also for concreteness I think I need to understand its application in detecting coherent structures in cellular automata to better appreciate this construction, though the automata theory part does go a bit over my head :p

I thought if one could solve one NP-complete problem then one can solve all of them. But you say that the treewidth doesn't help at all with the Clique problem. Is the parametrized complexity filtration by treewidth not preserved by equivalence between different NP-complete problems somehow?

All NP-complete problems should have parameters that makes the problem polynomial when bounded, trivially so by the => 3-SAT => Bayes Net translation, and using the treewidth bound.

This isn't the case for the clique problem (finding max clique) because it's not NP-complete (it's not a decision problem), so we don't necessarily expect its parameterized version to be polynomial tractable — in fact, it's the k-clique problem (yes/no is there a clique larger than size k) that is NP-complete. (so by the above translation argument, there certainly exists some graphical quantity that when bounded makes the k-clique problem tractable, though I'm not aware of it, or whether it's interesting)

To me, the interesting question is whether:

• (1) translating a complexity-bounding parameter from one domain to another leads to quantities that are semantically intuitive and natural in the respective domain.
• (2) and whether easy instances of the problems in the latter domain in-practice actually have low values of the translated parameter.
  • rambling: If not, then that implies we need to search for a new $k$ for this new domain. Perhaps this can lead to a whole new way of doing complexity class characterization by finding natural $k$-s for all sorts of NP-complete problems (whose natural $k$-s don't directly translate to one another), and applying these various "difficulty measures" to characterize your NP-complete problem at hand! (I wouldn't be surprised if this is already widely studied.)

Looking at the 3-SAT example ($Q_i$ are the propositional variables, $C_i$ the ORs, and $X$ the AND with $A_i$ serving as intermediate ANDs):

• re: (1), at first glance the treewidth of 3-SAT (clearly dependent on the structure of the $Q_i$ - $C_j$ interaction) doesn't seem super insightful or intuitive, though we may count the statement "a 3-SAT problem gets exponentially harder as you increase the formula-treewidth" as progress.
• ... but even that wouldn't be an if and only if characterization of 3-SAT difficulty, because re: (2) there exist easy-in-practice 3-SAT problems that don't necessarily have bounded treewidth (i haven't read the link).

I would be interested in a similar analysis for more NP-complete problems known to have natural parameterized complexity characterization.

You mention treewidth - are there other quantities of similar importance?

I'm not familiar with any, though ChatGPT does give me some examples! copy-pasted below:

• Solution Size (k): The size of the solution or subset that we are trying to find. For example, in the k-Vertex Cover problem, k is the maximum size of the vertex cover. If k is small, the problem can be solved more efficiently.
• Treewidth (tw): A measure of how "tree-like" a graph is. Many hard graph problems become tractable when restricted to graphs of bounded treewidth. Algorithms that leverage treewidth often use dynamic programming on tree decompositions of the graph.
• Pathwidth (pw): Similar to treewidth but more restrictive, pathwidth measures how close a graph is to a path. Problems can be easier to solve on graphs with small pathwidth.
• Vertex Cover Number (vc): The size of the smallest vertex cover of the graph. This parameter is often used in graph problems where knowing a small vertex cover can simplify the problem.
• Clique Width (cw): A measure of the structural complexity of a graph. Bounded clique width can be used to design efficient algorithms for certain problems.
• Max Degree (Δ): The maximum degree of any vertex in the graph. Problems can sometimes be solved more efficiently when the maximum degree is small.
• Solution Depth (d): For tree-like or hierarchical structures, the depth of the solution tree or structure can be a useful parameter. This is often used in problems involving recursive or hierarchical decompositions.
• Branchwidth (bw): Similar to treewidth, branchwidth is another measure of how a graph can be decomposed. Many algorithms that work with treewidth also apply to branchwidth.
• Feedback Vertex Set (fvs): The size of the smallest set of vertices whose removal makes the graph acyclic. Problems can become easier on graphs with a small feedback vertex set.
• Feedback Edge Set (fes): Similar to feedback vertex set, but involves removing edges instead of vertices to make the graph acyclic.
• Modular Width: A parameter that measures the complexity of the modular decomposition of the graph. This can be used to simplify certain problems.
• Distance to Triviality: This measures how many modifications (like deletions or additions) are needed to convert the input into a simpler or more tractable instance. For example, distance to a clique, distance to a forest, or distance to an interval graph.
• Parameter for Specific Constraints: Sometimes, specific problems have unique natural parameters, like the number of constraints in a CSP (Constraint Satisfaction Problem), or the number of clauses in a SAT problem.

I like to think of treewidth in terms of its characterization from tree decomposition, a task where you find a clique tree (or junction tree) of an undirected graph.

Clique trees for an undirected graph is a tree such that:

• node of a tree corresponds to a clique of a graph
• maximal clique of a graph corresponds to a node of a tree
• given two adjacent tree nodes, the clique they correspond to inside the graph is separated given their intersection set (sepset)

You can check that these properties hold in the example below. I will also refer to nodes of a clique tree as 'cliques'. (image from here)

My intuition for the point of tree decompositions is that you want to coarsen the variables of a complicated graph so that they can be represented in a simpler form (tree), while ensuring the preservation of useful properties such as:

• how the sepset mediates (i.e. removing the sepset from the graph separates the variables associated with all of the nodes on one side of the tree with another) the two cliques in the original graph
• clique trees satisfy the running intersection property: if (the cliques corresponding to) the two nodes of a tree contain the same variable X, then X is also contained in (the cliques corresponding to) each and all of the intermediate nodes of the tree.
  • Intuitively, information flow about a variable must connected, i.e. they can't suddenly disappear and reappear in the tree.

Of course tree decompositions aren't unique (image):

So we define $\text{treewidth}\left(G\right)={min}_{\text{clique tree}T\text{of}G}{max}_{\text{clique of}T}\text{size}\left(T\right)-1$.

• Intuitively, treewidths represent the degree to which nodes of a graph 'irreducibly interact with one another', if you really want to see the graph as a tree.
  • e.g., the first clique tree of the image above is a suboptimal coarse graining in that it misleadingly characterizes $G$ as having a greater degree of 'irreducible interaction' among.

Bayes Net inference algorithms maintain its efficiency by using dynamic programming over multiple layers.

Level 0: Naive Marginalization

• No dynamic programming whatsoever. Just multiply all the conditional probability distribution (CPD) tables, and sum over the variables of non-interest.

Level 1: Variable Elimination

• Cache the repeated computations within a query.
• For example, given a chain-structured Bayes Net $A⟶B⟶C⟶D$, instead of doing $P\left(D\right)={\sum }_A{\sum }_B{\sum }_{C}P(A,B,C,D)$, we can do $P\left(D\right)={\sum }_{C}P\left(D|C\right){\sum }_{B}P\left(C|B\right){\sum }_{A}P\left(A\right)P\left(B|A\right)$. Check my post for more.

Level 2: Clique-tree based algorithms — e.g., Sum-product (SP) / Belief-update (BU) calibration algorithms

• Cache the repeated computations across queries.
• Suppose you have a fixed Bayes Net, and you want to compute the marginalization not only $P\left(D\right)$, but also $P\left(A\right)$. Clearly running two instances of Variable Elimination as above is going to contain some overlapping computation.
• Clique-tree is a data structure where, given the initial factors (in this case the CPD tables), you "calibrate" a tree whose nodes correspond to a subset of the variables. Cost can be amortized by running many queries over the same Bayes Net.
  • Calibration can be done by just two passes across the tree, after which you have the joint marginals for all the nodes of the clique tree.
  • Incorporating evidence is equally simple. Just zero-out the entries of variables that you are conditioning on for some node, then "propagate" that information downwards via a single pass across the tree.

Level 3: Specialized query-set answering algorithms over a calibrated clique tree.

• Cache the repeated computations across a certain query-class
• e.g., computing $P(X,Y)$ for every pair of variables can be done by using yet another layer of dynamic programming by maintaining a table of $P(C_i,C_j)$ for each pair of clique-tree nodes ordered according to their distance in-between.

Perhaps I should one day in the far far future write a sequence on bayes nets.

Some low-effort TOC (this is basically mostly koller & friedman):

• why bayesnets and markovnets? factorized cognition, how to do efficient bayesian updates in practice, it's how our brain is probably organized, etc. why would anyone want to study this subject if they're doing alignment research? explain philosophy behind them.
• simple examples of bayes nets. basic factorization theorems (the I-map stuff and separation criterion)
• tangent on why bayes nets aren't causal nets, though Zack M Davis had a good post on this exact topic, comment threads there are high insight
• how inference is basically marginalization (basic theorems of: a reduced markov net represents conditioning, thus inference upon conditioning is the same as marginalization on a reduced net)
• why is marginalization hard? i.e. NP-completeness of exact and approximate inference worst-case
  what is a workaround? solve by hand simple cases in which inference can be greatly simplified by just shuffling in the order of sums and products, and realize that the exponential blowup of complexity is dependent on a graphical property of your bayesnet called the treewidth
• exact inference algorithms (bounded by treewidth) that can exploit the graph structure and do inference efficiently: sum-product / belief-propagation
• approximate inference algorithms (works in even high treewidth! no guarantee of convergence) - loopy belief propagation, variational methods, etc
• connections to neuroscience: "the human brain is just doing belief propagation over a bayes net whose variables are the cortical column" or smth, i just know that there is some connection