The investigation reported in this paper aims at clarifying an important yet subtle distinction between (i) the logical objects on which measure theoretic probability can be defined, and (ii) the inter- pretation of the resulting values as rational degrees of belief. Our central result can be stated informally as follows. Whilst all subjective degrees of belief can be expressed in terms of a probability measure, the converse doesn’t hold: probability measures can be defined over linguistic objects which do not admit of a meaningful betting interpretation. The logical framework capable of expressing this will allow us to put forward a pre cise formalisation of de Finetti’s notion of event which lies at the heart of the Bayesian approach to uncertain reasoning.
Second-order uncertainty, also known as model uncertainty and Knightian uncertainty, arises when decision-makers can (partly) model the parameters of their decision problems. It is widely believed that subjective probability, and more generally Bayesian theory, are ill-suited to represent a number of interesting second-order uncertainty features, especially “ignorance” and “ambiguity”. This failure is sometimes taken as an argument for the rejection of the whole Bayesian approach, triggering a Bayes vs anti-Bayes debate which is in many ways analogous to what the classical vs non-classical debate used to be in logic. This pa- per attempts to unfold this analogy and suggests that the development of non-standard logics offers very useful lessons on the contextualisa- tion of justified norms of rationality. By putting those lessons to work I will flesh out an epistemological framework suitable for extending the expressive power of standard Bayesian norms of rationality to second- order uncertainty in a way which is both formally and foundationally conservative. Contents
This paper presents the results of an experiment on mutual versus common knowledge of advice in a two-player weak-link game with random matching. Our experimental subjects play in pairs for thirteen rounds. After a brief learning phase common to all treatments, we vary the knowledge levels associated with external advice given in the form of a suggestion to pick the strategy supporting the payoff-dominant equilibrium. In the mutual knowledge of level 1 treatment, the suggestion appears on every subject's monitor at the beginning of every round, with no common knowledge that everybody sees the same suggestion. In the mutual knowledge of level 2 treatment, the same suggestion appears on each subject's monitor, accompanied by the request to "send" the suggestion to the partner in the round, followed by a notification that the message has been read. Finally, in the common knowledge treatment, the suggestion is read aloud by the experimenter at the end of the learning phase. Our results are somewhat surprising and can be summarized as follows: in all our treatments both the choice of the efficiency-inducing action and the percentage of efficient equilibrium play are higher with respect to the control treatment, revealing that even a condition as weak as mutual knowledge of level 1 is sufficient to significantly increase the salience of the efficient equilibrium with respect to the absence of advice. Furthermore, and contrary to our hypothesis, mutual knowledge of level 2 (as the one occurring in our "message" treatment) induces successful coordination more frequently than common knowledge.
Keywords:
Coordination games, experimental philosophy, epistemic attitudes, weak-link game, conventions
Whilst supported by compelling arguments, the representation of uncertainty by means of (subjective) probability does not enjoy a unanimous consensus. A substantial part of the relevant criticisms point to its alleged inadequacy for representing ignorance as opposed to uncertainty. The purpose of this paper is to show how a strong justification for taking belief as probability, namely the Dutch Book argument, can be extended naturally so as to provide a logical characterization of coherence for imprecise probability, a framework which is widely believed to accommodate some fundamental features of reasoning under ignorance. The appropriate logic for our purposes is an algebraizable logic whose equivalent algebraic semantics is a variety of MV-algebras with an additional internal unary operation representing upper probability (these algebras will be called UMV-algebras).
Keywords:
Subjective probability; Probabilistic logic; Coherence; Betting framework; Imprecise probability; Many-valued logic
The aim of this note is to investigate a very general problem of (radical) interpretation in terms of a simple coordination game: the conformity game. We show how, within our mathematical framework, the solution concept for the conformity game does indeed provide an algorithmic procedure facilitating triangulation, in the sense of Davidson.
KEYWORDS
Rationality, Strategic interaction, Focal points, Radical Interpretation, Selection of multiple Nash-equilibria.
The paper considers the problem of characterizing an inference process for reasoning under uncertainty in Geographic Information Systems (GIS). By focusing on a representative case study we outline the crucial aspects of the management of uncertainty in GIS. This enables us to argue, on methodological rather than practical grounds, in favour of the Maximum Entropy (ME) inference process. Specifically, we show how this constitutes a theoretically well-founded solution to the problems that arise naturally in GIS facing imperfect information. We also put forward how, as a consequence of the encouraging developments on computational techniques for reasoning under maximum entropy, the latter must be considered as a most crucial approach to uncertainty management in various fields of GIS science.
We argue in favour of identifying one aspect of rational choice
with the tendency to conform to the choice you expect another
like-minded, but non-communicating, agent to make and study this
idea in the very basic case where the choice is from a non-empty
subset $K$ of $2^A$ and no further structure or knowledge of $A$
is assumed.
KEYWORDS: Rationality; Common Sense; Coordination Games; Uncertainty; Principle of Charity; Social Choice Theory; Reasons.
Review and critical note on the newly published book by David Makinson, Bridges from Classical to Nonmonotonic Logic, King's College Text in Computing, volume 5, 2005 ISBN 1-904987-00-1. Published February 2005. Paperback.
Comments are very much welcome!
It has been suggested that AI investigations of mechanical learning undermine sweeping anti-inductivist views in the theory of knowledge and the philosophy of science. In particular, it is claimed that some mechanical learning systems perform epistemically justified inductive generalization and prediction. Contrary to this view, it is argued that no trace of such epistemic justification is to be found within a rather representative class of mechanical learning agents. An alternative deductive analysis of mechanical learning from examples is outlined.
First drawing on behaviour-based robotics, we examine relatively simple autonomous robots that learn from experience, insofar as they acquire new sensorimotor capabilities and generalize from observation. Analysis of a representative behaviour-based architecture reveals that even these rudimentary learning mechanisms embody crucial assumptions about their environment. In this context, the epistemic problem of induction is not ?solved?. Rather it is reformulated as the problem of assessing whether projections based on such background assumptions are reasonable to believe.
This epistemic problem is more informatively addressed by reference to the symbolically richer, ID3-style learning algorithms. Pervasive overfitting of training data jeopardizes the idea that epistemically justified induction is at work there: overfitting reminds one that good approximation to the target concept or rule on training data is not, in itself, diagnostic of good approximation over the whole instance space of that concept or rule. Once again, both ID3 learning and successful post-pruning of overfitting trees rely on the conjectural representativeness of concept instance collections.
Having found no trace of epistemically justified induction, an alternative deductivist account is outlined, drawing on some families of non-monotonic consequence relations, and emphasizing deductive trial and error-elimination processes in autonomous learning mechanisms. These processes interleave default-based introduction of projective hypotheses from observed samples, retraction of falsified hypotheses, and heuristic selection of new background assumptions for more effective learning.
KEYWORDS:Induction, machine learning, behavior-based robotics,
non-monotonic consequence relations, artificial intelligence..
Rationality as conformity, Doctoral dissertation: School of Mathematics, The University of Manchester, (Successfully examined on 22 June 2005.)
Outline -- Full text (pdf)
Rationality-as-conformity begins with the idea that a ``rational", ``commonsensical",
``natural", or simply ``logical" choice is one which corresponds to the choice other
similar agents would come up with in similar situations. Our aim is to model the
choice processes leading to this sort of conformity. Consider the following example.
[Supermarket shelf arrangement.] There are numerous ways in which a
supermarket manager might choose to arrange the shelves in her store, for example by
alphabetical order of product name, by product size or weight, by price, by the
package's colours, and so on indefinitely (not to mention the astronomic number of
random orderings!). However when stepping into a new supermarket (i.e. one we have
never visited before, and about which nothing is known to us, apart from the fact it
is a supermarket) we expect to find teas close to coffees, pastas close
to rices, nappies near to toilet rolls. At least we argue that it would surely seem
natural to hold expectations of this sort. In fact, if after ten minutes
searching we finally located the sugar among the washing powders, we might well be
inclined to question the store manager's rationality! After all, we see this as a
situation where, for mutual convenience, the store manager and ourselves are trying
to conform on the selection of a common world, i.e. shelf arrangement.
Although this is the sort of situation we intend to model within our
framework, it is not hard to see how rapidly the complications would arise, if we
were to work with this informal problem. For instance it could be put forward that in
fact there is no choice process to be modelled, but rather the appropriate use of
common knowledge. The objection here would be that that there are in fact rules or
conventions (arising in all probability from marketing research) that regulate what a
``rational" shelf arrangement is. [Surely there seems to be some cold-blooded form of
logicality when it comes to shelving sweets right at the eyes-height of an
invariantly bored child queueing at the till!] Hence, the objection would conclude,
those rules or conventions are all that a ``rational" customer would have to learn
to shop conveniently. This would surely work if there was something as a ``universal
shelving rule" around. Yet, needless to say, this is utterly unreasonable. Therefore
it is not hard to see that this possible objection just begs the question for ``the
new customer" would still have to figure out which shelving convention the
supermarket manager is in fact adopting.
The next objection then, might be to notice that supermarket managers might indeed
fill up the store with signs and maps indicating to the unlearned customer where is
what. An account of ``rational shelving" pursuing this line, however, would seem to
be easily exposed to two sorts of shortcomings. Firstly, this rule would require an
enormous amount of computational effort to be learnt. And surely the vast majority of
this effort could in fact be spared if the manager would adopt the same rule she
thinks the customer would utilize in the same situation. Secondly, what if the
customer doesn't happen speak the language(s) chosen by the manager for the sings?
What if the manager and the customer didn't in fact share any language at all?
In this thesis we shall consider an idealised and mathematically abstract situation
where communication or knowledge of such rules and conventions are not available to
the agents. Since it is assumed that agents' ultimate goal is that of conforming to
the expectations of their peers, we shall refer to the overall approach as
Rationality-as-conformity.
Rationality (or common sense - we shall not make a distinction among
these terms) is, in its full generality an extremely complex and widely debated
subject. Yet within the scope of our simple mathematical formulation we shall be able
to provide what amounts to a characterization or definition of what it means
to choose rationally. The hope is that such investigations will ultimately provide a
way of viewing and understanding these notions in a much more general real world
context.
Dear Sir,
We (Mr. Rosen and I) had sent you our manuscript for publication and had not authorized you to show it to specialists before it is printed. I see no reason to address the — in any case erroneous — comments of your anonymous expert. On the basis of this incident I prefer to publish the paper elsewhere.
Respectfully,
Albert Einstein