We show that while standard first-order inductive logic is not decidable, a large class of inferences in objective Bayesian inductive logic is decidable. Decidability is achieved by reducing the general inference problem to a quantifier-free problem. We show that for any inference, if the quantifier-free reduction of the premisses is satisfiable, then the original inference is decidable. We go on to show that Bayesian networks offer the potential to provide a computationally tractable inference procedure for objective Bayesian inductive logic. We also consider inferences with infinitely many premisses and explore some properties of the logic.

Explanatory relationships between data and hypotheses have been suggested to play a role in the formation of posterior probabilities. This suggestion was tested in a toy environment and supported by simulations by David H. Glass. We here put forward a variety of inference to the best explanation approaches for determining posterior probabilities by intertwining Bayesian and inference to the best explanation approaches. We then simulate their performances for the estimation of parameters in the Brock and Hommes agent-based model for asset pricing in finance. We find that performances depend on circumstances and also on the evaluation metric. However, most of the time our suggested approaches outperform the Bayesian approach. • Bayesian posteriors and Inference to the Best Explanation are intertwined. • Inference to the Best Explanation posteriors outperform the Bayesian posterior. • Parameters are estimated for the Brock and Hommes model for asset pricing. • Comparison of explanation-based and probabilistic inference.

This paper addresses a data integration problem: given several mutually consistent datasets each of which measures a subset of the variables of interest, how can one construct a probabilistic model that fits the data and gives reasonable answers to questions which are under-determined by the data? Here we show how to obtain a Bayesian network model which represents the unique probability function that agrees with the probability distributions measured by the datasets and otherwise has maximum entropy. We provide a general algorithm, OBN-cDS, which offers substantial efficiency savings over the standard brute-force approach to determining the maximum entropy probability function. Furthermore, we develop modifications to the general algorithm which enable further efficiency savings but which are only applicable in particular situations. We show that there are circumstances in which one can obtain the model (i) directly from the data; (ii) by solving algebraic problems; and (iii) by solving relatively simple independent optimisation problems.

The maximum entropy principle is widely used to determine non-committal probabilities on a finite domain, subject to a set of constraints, but its application to continuous domains is notoriously problematic. This paper concerns an intermediate case, where the domain is a first-order predicate language. Two strategies have been put forward for applying the maximum entropy principle on such a domain: (i) applying it to finite sublanguages and taking the pointwise limit of the resulting probabilities as the size n of the sublanguage increases; (ii) selecting a probability function on the language as a whole whose entropy on finite sublanguages of size n is not dominated by that of any other probability function for sufficiently large n. The entropy-limit conjecture says that, where these two approaches yield determinate probabilities, the two methods yield the same probabilities. If this conjecture is found to be true, it would provide a boost to the project of seeking a single canonical inductive logic—a project which faltered when Carnap's attempts in this direction succeeded only in determining a continuum of inductive methods. The truth of the conjecture would also boost the project of providing a canonical characterisation of normal or default models of first-order theories. Hitherto, the entropy-limit conjecture has been verified for languages which contain only unary predicate symbols and also for the case in which the constraints can be captured by a categorical statement of Σ1 quantifier complexity. This paper shows that the entropy-limit conjecture also holds for categorical statements of Π1 complexity, for various non-categorical constraints, and in certain other general situations.