## Fabio Cozman's Errata (as of January 2017)

Across the years I have noticed some errors in papers I have published (well, there are remaining typos here and there, but unfortunately here I refer to real mistakes).

Here is a list:

1) There is an incorrect result on divergence of posterior distributions that appears in two papers (Theorem 4 fails as its proof relies on an incorrect inequality): F. G. Cozman, Computing posterior upper expectations, International Journal of Approximate Reasoning, 24:191-205, 2000; and F. G. Cozman, Computing Posterior Upper Expectations, First International Symposium on Imprecise Probabilities and Their Applications (ISIPTA), pp. 131-140, 1999. If you are interested in asymptotic dilation, the best reference I know of is: Timothy Herron; Teddy Seidenfeld; Larry Wasserman, The Extent of Dilation of Sets of Probabilities and the Asymptotics of Robust Bayesian Inference, PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association, Volume One: Contributed Papers. (1994), pp. 250-259.
2) Theorem 3 in the paper:

F. G. Cozman, Computing lower expectations with Kuznetsov's independence condition, Third International Symposium on Imprecise Probabilities and Their Applications, pp. 177-187, Carleton Scientific, 2003.

gives a number of constraints that must be satisfied by a Kuznetsov extension; the theorem claims that the list is exhaustive, but the proof only contains necessary conditions, not sufficient ones. Indeed the Kuznetsov extension requires more constraints. Besides, the proof of Theorem 3 is used to prove weak union of Kuznetsov independence, which may lead to problems there too; it is an open question whether weak union holds or not.

3) The paper:

J. C. F. da Rocha, F. G. Cozman, Inference with separately specified sets of probabilities in credal networks, XVIII Conference on Uncertainty in Artificial Intelligence, editors, pp. 430-437, Morgan Kaufmann, San Francisco, California, 2002.

has several problems. First, the result on complexity of inferences for credal networks is correct, but its proof is not (the result was latter proved correctly by Cassio Polpo de Campos and appeared in IJCAI 2005). Second, the inference algorithm is presented so tersely that several key steps are not given, and these steps are actually the ones that have the most complexity. Third, the theorem on why separately specified sets decompose the inference is incomplete; the decomposition happens as stated but not in all cases. And finally, the connection with the 2U algorithm is correct, but it is so complicated to write that you should use the 2U algorithm, not the variable elimination version sketched in this paper. Well, I hope I never write anything else with so many mistakes. In any case, everything in this paper has been superseded by papers at UAI2003, UAI2004 and IJCAI2005.

4) The paper:

D. Kikuti, F. G. Cozman, C. P. de Campos. Partially ordered preferences in decision trees: computing strategies with imprecision in probabilities, IJCAI Workshop on Advances in Preference Handling, Edinburgh, United Kingdom, 2005.

contains a few problems with examples and calculations; an errata has been produced with corrected versions.

5) The paper:

Fabio G. Cozman. Sets of probability distributions, independence, and convexity, Synthese, 186(2):577-600, 2012.

defines epistemic irrelevance as equality of conditional and unconditional lower expectations, where the conditional expectations are conditioned in every event B that contains values of Y. However, epistemic irrelevance only requires equality given events Y=y for every possible value of Y. Likewise, confirmational irrelevance must be defined only given events Y=y (in this case the examples showing failure of Decomposition and Weak Union hold; they are not valid for the definition that uses conditioning on every B containing values of Y). Additionally, the paper mentions that regular independence fails Decomposition and Weak Union, but the counterexamples it mentions are not appropriate for this. Finally, after publication it has been found that strong independence fails the Contraction property, a surprising fact that contradicts a statement in the paper.

6) The paper:

Fabio G. Cozman, Denis D. Maua. Bayesian networks specified using propositional and relational constructs: Combined, data, and domain complexity, AAAI Conference on Artificial Intelligence, pp. 3519-3525, 2015.

states that complexity of inference with Bayesian networks specified using function-free first-order logic is PP-complete (Lemma 2); in fact it is PSPACE-complete. The data complexity is PP-complete as stated in the paper, but the proof of this fact requires more work than described in the paper.

7) The paper:

Fabio G. Cozman, Denis D. Maua. On the complexity of propositional and relational credal networks, International Symposium on Imprecise Probability: Theories and Applications, pp. 97-106, 2015.

states in Theorem 3 that complexity of inference with credal networks (strong extensions) specified using function-free first-order is complete for NP with oracle PP; in fact it is PSPACE-complete. The data complexity (Theorem 4) is in fact complete for NP with oracle PP, but the proof of this fact requires more work than described in the paper. Another problem with this paper is that it presents Theorem 5 on the complexity of credal networks with coupled parameters, but to prove this it assumes that lower and upper expectations are attained only at vertices of local credal sets --- this is not true. To obtain the stated PP-completeness of inference for credal networks with coupled parameters one must add more assumptions for instance assuming that only a finite number of probabilities are possible.

Finally, some of the algorithms for credal networks in my PhD thesis were not written with all necessary steps, and a couple of lemmas were correct but had problems in their proofs; these problems were corrected in a subsequent journal paper (F. G. Cozman, Credal networks, Artificial Intelligence Journal, vol. 120, pp. 199-233, 2000). Actually, I don't see any reason for you to read my PhD thesis, as the relevant results appeared in papers published in 2000.
Fabio Cozman, 2017/01/31.