Dear Peter and Thomas, Thanks for your useful comments:
Peter, I take your point that economics may not be a good domain of application for this type of causal model, though I think this needs to be decided empirically and the issue is not yet settled. Note though that Dynamic Bayesian nets can be used to deal with situations where relationships between variables change. Recursive Bayesian nets can model actions of a decision maker. See Z. Ghahramani, 1998. Learning Dynamic Bayesian Networks In C.L. Giles and M. Gori (eds.), Adaptive Processing of Sequences and Data Structures . Lecture Notes in Artificial Intelligence, 168-197. Berlin: Springer-Verlag. Jon Williamson & Dov Gabbay[2003]: `Recursive Causality in Bayesian Networks and Self-Fibring Networks', in D.A. Gillies (ed.): `Laws and models in the sciences', forthcoming; Peter and Thomas, you both point out that causal Bayesian nets also make strong assumptions. This is certainly true: both require the causal Markov condition to hold and that causal relations are known. However the causal Bayesian net and the structural equation model formalisms are not equivalent in the following practical sense. It is much easier to determine the probability distribution of a variable conditional on its parents (required in a causal Bayesian net) than to determine a functional equation for a variable in terms of its parents (required in a structural equation model). Proponents of structural equation models recongnise this difficulty and tend to use simplifying assumptions such as linearity of functional form. I suggest that it would be better just to use causal Bayesian nets. Thanks again for your comments, All the best, Jon ------------------- Jon Williamson Department of Philosophy, King's College, Strand, London, WC2R 2LS, UK http://www.kcl.ac.uk/jw - ----- Original Message ----- From: "Thomas Richardson" <[EMAIL PROTECTED]> To: "Jon Williamson" <[EMAIL PROTECTED]> Cc: <[EMAIL PROTECTED]> Sent: Friday, September 12, 2003 2:29 AM Subject: "causal" vs. "functional" models > > 6. Slogan: "causal net before functional model". > > As described in Spirtes, Glymour and Scheines (1993,2001) and Pearl > (2000), these formalisms are equivalent for the purposes of determining > the population distributions resulting from the implementation of > policies,. > > A little more discussion follows: > > It is true that traditionally functional relationships (e.g. linearity) > have been assumed in structural equation models, but more modern > approaches are non-parametric, in which case, at a conceptual level, there > is no difference between the two. > > [As a side note: Structural equation models can be interpreted as relating > to individual causal effects - i.e. inferring causes from effects, "Would > John have a headache had he not drunk coffee this morning given that he > did drink coffee and does have a headache", but this has not been done by > econometricians - whereas this is not true of "causal nets". However, such > inferences are based on assumptions that cannot in general be tested even > in experimental contexts. Further Econometricians have typically not drawn > such inferences from their models, hence this is not directly relevant > here. See Pearl (2000) Ch.5 and 7 for further discussion.) > > > > 4. Consequently it would be better to use causal Bayesian networks instead, > > since in that case we only need to estimate the probability distribution of > > each variable conditional on its direct causes. Such models should be > > checked (one should check that the causal Markov condition holds for the > > model and that the model is robust for forcasting and modeling > > interventions) and refined as necessary. > > If "only" we ever knew the direct causes and could directly check the > Markov property! > > However, such conditional independence assumptions are equivalent to the > assumptions made by the coresponding structural equation model. > > Note that in general there will be more than one "causal net" obeying a > given set of conditional independence relations, hence the fact that our > data are compatible with the independence relations given by a given > causal model does not verify the model, to the extent that the data > confirms or supports the proposed model, it also supports every other > model in the given Markov equivalence class. > > Even this inference requires the (untestable) assumption that conditional > independence relations present in the population distribution only arise > through causal structure and not through cancellation of parameters > (so-called violations of faithfulness or stability). > > In general it is often unclear whether all of the relevant variables have > been included, which means that models with hidden variables (or > "correlated errors" or "semi-Markovian") must also be considered, thereby > expanding further the class of causal models that are not ruled out by the > data. This problem is equally present for both formalisms. (Again see the > above references for more discussion.) > > In general, on finite samples, it is often unrealistic to hope to reliably > determine the conditional independence structure holding in the population > non-parametrically. For this reason functional forms are often assumed > (e.g. "noisy-or" in the causal net literature; linearity in > Econometrics). Such problems have been particularly acute in Econometrics > where datasets have traditionally been rather small (this is starting to > change). > > To summarize, there is no significant distinction between the two > approaches. The primary practical problems present in applications of > structural equation models - lack of data, large numbers of possible > variables and possible models, are also present for causal nets. It is > true that users of structural equation models have often been unclear or > even unaware of the causal interpretation of their models and this has led > to much confusion and obscurity. In this regard, the community of causal > net modellers may have some advantage (though this issue is also often not > addressed in many Bayes Net applications). > > The issues of stationarity raised by Peter McBurney are of course also of > concern in many Econometric applications. Feedback is another possibility > in many econometric settings For more discussion of this see Lauritzen and > Richardson "Chain graph models and their causal interpretations (with > discussion)", (2001) Journal of the Royal Statistical Society Series B. > > Thomas Richardson > > Department of Statistics > University of Washington > >
