Difference between graphical model and markov chain

Question

Representing causality using fuzzy cognitive maps presents a cognitive model which is a graphical model consisting of weighted directed graph. To me it looks like a state transition machine. Can somebody explain the difference between causal map and markov chain? Thank you

Answer 1

The fact that a graphical model and a state transition machine both use directed graphs doesn't of itself imply they're similar.

What is critical is how we understand the meaning of the directed graph.

State transition machines represent the way a state can change through time. The graphical model needn't have any time component*, it's about relationships between variables.

*(though if it's causal there's an implication of causes preceding effects, we're not observing their progression through a sequence of times; the entire model may represent a cross section at a single instant)

That is, saying that they "look like" isn't much of an argument, you can't rely on the similarity of appearance to tell you anything about the time component. You have to look at what the directed graph represents.

For example, Bayesian graphical models use directed graphs. The statistical model doesn't have a time component -- at least not normally; it's cross-sectional. All variables may well be measured at the same instant; indeed the start node of an arrow could even be measured before the end node. Speaking logically, the underlying conception of the situation - if conceived causally - may well have a time component (albeit effectively instantaneous), simply because we conceive of causes as preceding effects.

In a state machine every arrow implies a time step from observed state to observed state -- explicitly as part of the model.

By contrast in general in a graphical model there's no equivalent implication about movement in time from one time unit to the next, since graphical models don't usually have that.

The fact that both share nodes and directed arrows in their pictorial representation doesn't of itself imply anything special about how they see time. So I'd say "generally, no" for graphical models per se, but 'in at least a weak sense' when discussing how we conceive of a causal graphical model.

Your argument about what the graphical model represents - specifically $C_i(t) = f(C_i(t-1)*w_ij)$ in this particular case - is critical to deciding whether they're similar in the sense we're discussing.

The graphical model described in your links does appear to have a specific time component (since your formula shows a progression of $C_i$ through time), and that does indeed make them similar. However, that similarity is not inherent in the superficial similarity of the use of directed graphs to represent the model, it's explicitly a result of what those graphs represent.

Hopefully that's now clear.

Answer 2

Another, more concise, way to put it is to say that a state transition machine is a type of graphical model. Namely one, where the conditional probabilities (links) are probabilities of a state-change in time.

Answer 3

This an old question, but I want to add what I think other answers lack.

Graphical Models: They describe how a probability distribution over some $N$ variables factorizes using a parent-child relationship. Given parents, the distribution of the random variable is independent of its non-descendants. Example:

This graph describes a distribution $f$ over $X_1,X_2,X_3,X_4,X_5,X_6$ which factorizes as, \begin{equation} f(X_1,X_2,X_3,X_4,X_5,X_6) = p(X_1)\cdot p(X_3)\cdot p(X_2 | X_1,X_3)\cdot p(X_5| X_1)\cdot p(X_4|X_5,X_3)\cdot p(X_6 | X_1,X_3) \end{equation}

Markov Chains: A discrete-time Markov chain defines a sequence of random variables that can take several discrete values called "states." A graph for a Markov chain usually represents how these states evolve into each other. Example:

Here a random variable, say $X_0$ can take 4 values $\{A,B,C,D\}$. Given we know the value $X_0$ takes, we describe a probability distribution over states for the random variable $X_1$, where the corresponding weighted edges give the probability of each state.

However, we can draw a different graph for Markov Chains, one which describes a probability distribution over $\{X_t\}, t\geq 0$

So, the distribution factorizes as

\begin{equation} f(X_0,X_1,\dots,X_{t-1},X_t,\dots) = p(X_0)\cdot \prod_{t>0}p(X_t | X_{t-1}) \end{equation}