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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

SKM
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  • The fact that they both use directed graphs doesn't imply they're similar. State transition machines represent the way a state can change through time. The graphical model needn't have a time component\*, it's about relationships between variables. $\quad$ \*(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) – Glen_b Jun 11 '14 at 02:13
  • The way the above cognitive map works is through the same inference mechanism as NN wherein each concept/node = neuron. A node(neuron) influences another concept through C_i(t) = f(C_i(t-1)*w_ij) (http://sipi.usc.edu/~kosko/FCM.pdf) – SKM Jun 12 '14 at 18:34
  • So, does the formula not imply that the graphical model has a time component. The map is a deterministic system and evolves to generate a time series for each concept/node. Is this the same mechanism for markov chain ( I am unaware of the details about Markov chain). To me both look like a state diagram.I shall be really grateful for a detailed insight. – SKM Jun 12 '14 at 18:36

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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.

Glen_b
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  • Thank you for the reply, although it is quite high level. Please correct me if my understanding from your explanation is correct or not - Bayesian graphical model does is not dynamical from the perspective that it does not evolve w.r.t time. Markov model is a state machine which is dynamical where the dynamical perspective is evaluated from the viewpoint that how each state transitions w.r.t time. So, it does not tell us what the value of a particular state influences another state.Plz let me know if my understanding is correct? – SKM Jun 16 '14 at 22:42
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    That's usually the situation, but exceptions can exist as your example shows. The point wasn't that they can't be similar, it's that the graph *of itself* doesn't make them similar. – Glen_b Jun 16 '14 at 23:52
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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.

Blaubaer
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  • Right now this reads more like a comment than an answer and doesn't seem to add very much to what's already here. Perhaps you could expand a bit on your thoughts and make this a proper answer? – einar May 12 '17 at 15:29
  • Honestly, there isn't much more to say. I just wanted to give a more "concise" version of the above answer. I know it's more like a comment, but unfortunately my reputation count did not allow to "comment" ... – Blaubaer May 12 '17 at 15:46
  • That means you need to gain the required reputation before you can post this. That's just the way the SE system works. – gung - Reinstate Monica May 12 '17 at 16:11
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    I know how the system works. I nevertheless thought it's valuable input as I find Glen_b's answer rather overcomplicating. But if you feel it's not worth a comment, or a post, please feel free to delete. – Blaubaer May 12 '17 at 16:17
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    I agree with Blaubaer that this is a valuable contribution, in that it gets to the heart of the matter more quickly than the other answer. – Chill2Macht Dec 10 '17 at 16:36