Abstract
Inference for Probabilistic Argumentation has been focusing on computing the probability that a given argument or proposition is acceptable. In this paper, we formalize such tasks as computing marginal acceptability probabilities given some evidence and learning probabilistic parameters from a dataset. We then show that algorithms for them can be composed by finely joining a basic PA inference algorithm and existing algorithms for the corresponding tasks in Probabilistic Logic Programming or even Bayesian networks.
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Notes
- 1.
credulous/grounded/stable semantics. There are many other semantics. For a review, readers are referred to, e.g. [1].
- 2.
For convenience, define \(head(r) = l_0\) and \(body(r) = \{l_1,\dots l_n\}\).
- 3.
Elements of \(\lnot \mathcal A_p = \{ \lnot x \mid x \in \mathcal A_p\}\) are called negative probabilistic assumptions.
- 4.
G is a directed acyclic graph over \(\mathcal X = \{X_1, \dots , X_m\}\) and \(\varTheta \) is a set of conditional probability tables (CPTs), one CPT \(\varTheta _{X\mid par(X)}\) for each \(X \in \mathcal X\).
- 5.
Probabilistic parameters are made up for the sake of illustrations and so is the dependency of burglaries on earthquakes.
- 6.
We shall make use of usual notations in FOL such as atoms, literals, Herbrand base, interpretations, etc. without precise definitions.
- 7.
When discussing a PABA inference task, we always refer to an arbitrary but fixed PABA framework \(\mathcal P = (\mathcal A_p, \mathcal N, \mathcal F)\) if not explicitly stated otherwise.
- 8.
That is, a partial world is interpreted as a conjunction of probabilistic assumptions, while a frame is interpreted as a disjunction of partial worlds (In other words, a DNF over probabilistic assumptions).
- 9.
Note that \(\mathcal F_{s'}\) is the ABA framework obtained from \(\mathcal F\) by adding a set of facts \(\{p \leftarrow \mid p \in s'\}\).
- 10.
Note that if \(s = s_1 \cup \dots \cup s_n\) is inconsistent, then s is not a partial world and hence \(s \not \in \mathcal S\).
- 11.
Readers are referred to http://problog.readthedocs.io/en/latest/cli.html for details about ProbLog concrete syntax.
- 12.
obj(.) maps evidences to sentences of the underlying language.
- 13.
Download link of this implementation: http://ict.siit.tu.ac.th/~hung/Prengine/2.0.
- 14.
Prolog-based PLP languages using SLDNF resolution such as ProbLog fail to learn this dataset because SLDNF resolution does not terminate if queried ?-bark, howl.
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Hung, N.D. (2017). Inference and Learning in Probabilistic Argumentation. In: Phon-Amnuaisuk, S., Ang, SP., Lee, SY. (eds) Multi-disciplinary Trends in Artificial Intelligence. MIWAI 2017. Lecture Notes in Computer Science(), vol 10607. Springer, Cham. https://doi.org/10.1007/978-3-319-69456-6_1
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