A Case Where Chance and Credence Coincide
David Atkinson (Groningen, The Netherlands)
Attempts have been made to provide a bridge between chance and
credence, such as David Lewis's Principal Principle and Howson and
Urbach's transmogrification of von Mises' frequency theory into a
subjectivist account. I propose a contribution to the general project of
connecting chance and credence by showing that Reichenbach's objectivistic
approach is intimately linked to subjectivistic Jeffrey
conditionalization.
Reichenbach defended his method of the "appraised posit" in his
frequentistic theory of chance, according to which a chain of objective
probabilities was provisionally grounded on a posited chance that could be
partly determined by empirical success. Jeffrey's technique of updating
credences by means of incoming evidence whose reliability remains
uncertain resembles Reichenbach's method in purpose if not in form.
I will first derive a compact formula for the probability of an event
that is conditioned by an arbitrary, finite chain of events, given the
relevant conditional probabilities and a posit. Next I will show that, in
the event that the posit is "ideal", in the sense that the probability it
generates is exactly right, the Jeffrey update of the posit would be
"invariant" (i.e. unchanged). The converse is also proved, namely that an
invariant Jeffrey update implies an ideal Reichenbach posit.
Certainly Probable: Belief and Aleatory Probability in Hume's Theory
of Induction
Conor Barry (Paris, France)
The emergence of the mathematical concept of probability in the decades
before the publication of both the 'Treatise of Human Nature' and the
'Enquiry Concerning Human Understanding' had, as Ian Hacking has most
perceptively noted, a critical influence upon the development of HumeÕs
conception of induction. Although pure mathematics had traditionally been
regarded by philosophers as the touchstone of epistemic certitude,
probability constituted a sub-branch of mathematics that was, by nature,
indeterminate.
Inspired by Hacking's researches and the critical responses of Larry
Lauden and Robert Brown in philosophy of science, as well as comparative
researches of ancient and modern skepticism undertaken by Malcolm
Schofield, Hume scholars such as Dorothy Coleman and Kevin Meeker have
drawn attention, in Hume scholarship, to the ambiguities and
inconsistencies in Hume's use of the term "probable". Derived from the
Latin probare, meaning to approve or believe, the word "probable", for
Hume's contemporaries, was simply a non-mathematical synonym for
"credible". Certain passages of Hume make definite use of the term
"probable" in the strictly mathematical sense. However, other passages
waver between the two meanings of the term. This ambiguity may be
resolved, I maintain, if we recognize the equivalence of both.
This essay, thus, unites two uncontroversial points concerning Hume's
epistemology into a potentially controversial one. The first is that,
according to Hume, the aleatory probability or chance that an event will
occur in the future is the ratio of positive outcomes that the event has
occurred in the past. The second is that, according to Hume, induction
is the process by which a belief forms, involuntarily, from the repeated
impressions of sense. In other words, the belief that arises in a human
subject is an expression of qualitative, sense vivacity rather than
quantitative, reflective ratiocination. The potentially controversial
point is as follows. According to Hume, the positive frequency with which
an effect follows a cause determines the qualitative vivacity with which
we hold belief in that specific cause-effect relationship. Therefore,
HumeÕs notion of induction is an epistemic formulation of mathematically
probabilistic justification. Hume regards an inductively arrived at belief
as the credence in the future occurrence of a particular event based on
the ratio of positive outcomes with which that particular event has
previously occurred.
Employing the theory of chance, Hume manages to codify, rationally,
indefinite qualitative belief. In ordinary life, the human subject may
not, of course, take stock, enumerating, precisely, the proportion of
confirming and disconfirming instances with which a specific effect
follows from a given cause. However, the force and vivacity of sense
impressions repeated with a high positive frequency involuntarily
generates, within him or her, a corresponding degree of qualitative
belief. Probability is, therefore, the operation of subjective credence
based on the ratio of past positive outcomes of an event. Hume thinks of
probability as a rigorous codification of sense belief, that probability
defines, with quantitative precision, a cognitive process that is, in
ordinary experience, purely qualitative.
Distribute white and black hats in a dark room to a group of three
rational players with each player having a fifty-fifty chance of receiving
a hat of one colour or the other. Clearly, the chance that, as a result
of this distribution, (A) "Not all hats are of the same colour" is 3/4.
The light is switched on and all players can see the hats of the other
persons, but not the colour of their own hats. Then no matter what
combination of hats was assigned, at least one player will see two hats of
the same colour. For her the chance that not all hats are of the same
colour strictly depends on the colour of her own hat and hence equals 1/2.
On Lewis's principal principle, a rational player will let her degrees
of belief be determined by these chances. So before the light is switched
on, all players will assign degree of belief of 3/4 to (A) and after the
light is turned on, at least one player will assign degree of belief of
1/2 to (A). Suppose a bookie offers to sell a single bet on (A) with
stakes $4 at a price of $3 before the light is turned on and subsequently
offers to buy a single bet on (A) with stakes $4 at a price of $2 after
the light is turned on. If, following Ramsey, the degree of belief equals
the betting rate at which the player is willing to buy and to sell a bet
on a given proposition, then any of the players would be willing to buy
the first bet and at least one player would be willing to sell the second
bet. Whether all hats are of the same colour or not, the bookie can make
a Dutch book - she has a guaranteed profit of $1.
However, it can be shown that a rational player whose degree of belief
in (A) equals 1/2 would not volunteer to sell the second bet on (A),
neither when her aim is to maximise her own payoffs, nor when she wants to
maximise the payoffs of the group. The argument to this effect shares a
common structure with models (i) for the tragedy of the commons and (ii)
for strategic voting in juries.
Non-pragmatic arguments for Bayesianism
Timothy Childers (Prague, Czech Republic)
Dutch Book arguments purport to establish that degrees of beliefs
should correspond to probabilities. However, they are, at least in their
traditional form, linked to notions of actual behaviour. Since betting
behaviour is influenced by myriad factors including willingness to bet,
the capacity to understand odds and risk-aversion, the arguments become
untenable.
Recent attempts to remove these links have been provided by Colin
Howson (2003 and in Howson and Urbach 2006). We explore HowsonÕs account
of Dutch Book arguments as soundness and completeness proofs, contrasting
his "Fregean" account with the traditional "pragmatist" account. We also
note that similar contrasting accounts are offered in other areas, namely
current philosophy of language and logic.
We conclude that there are good reasons to accept Howson's account of
the purpose of Dutch Book arguments, but that there are better arguments
which will serve that purpose. In particular we employ a representation
theorem developed by de Groot 1970 and French 1982 on the basis of some
ideas found in de Finetti. The theorem does not rely on a preference
ordering, but does rely on the notion of a flat distribution, assuming
directly a likelihood ordering. We argue that this assumption is not "too
strong", and we further argue that such theorems provide a better
framework for non-pragmatic accounts of subjective probability.
DeGroot, Morris H., 1970, Optimal Statistical Decisions (New York:
McGraw-Hill).
French, Simon, 1982, "On the axiomatisation of subjective probabilities",
Theory and Decision 14, 19-33.
Howson, Colin, 2003, "Probability and Logic", Journal of Applied Logic 1,
151-165.
Howson, Colin and Peter Urbach, 2006, Scientific Reasoning: The Bayesian
Approach, 3rd ed., Open Court, La Salle, IL.
The role of intuitive probabilistic reasoning in scientific theory
formation
Helen De Cruz (Brussels, Belgium)
Experimental evidence from cognitive science has revealed a
discrepancy between intuitive and formal modes of probabilistic reasoning.
When asked to make a probabilistic judgement, laypeople fail to think in
statistical terms Instead, they seem to rely on mental modelling of
concrete situations (e.g., Johnson-Laird et al., 1999), or on fast and
shallow heuristics (e.g., Kahneman & Tversky, 1996). These strategies
often lead to errors in probability estimates, e.g., smokers
tend to believe that smoking does not significantly increase their chance
of getting cancer, because they extrapolate from a very limited data set
(i.e., the covariance of cancer and smoking in family members and
friends). The cognitive science literature has hitherto mainly focused on
the 'negative' aspects of intuitive probabilistic reasoning, in particular
its inability to predict the likelihood of single events. However, despite
these epistemological limitations, I
will defend the view that intuitive probabilistic reasoning plays an
important role in scientific practice. I illustrate this with historical
examples, including Darwin's Origin of Species (1859), early evolutionary
psychology (1970s to early 1990s), and Zahavi's handicap theory (1975).
Intuitive probabilistic reasoning plays a legitimate role in guiding
scientific creativity, as it can help develop novel ideas in the absence
of adequate mathematical models in which such ideas can be phrased more
precisely. This is especially important in the early
creative stages of theory formation. In the examples mentioned above, the
theories were developed well in advance of their mathematical description,
and all made extensive use of intuitive probabilistic arguments to bolster
claims. For example, statistical methods in population genetics became
available more than 50 years after the publication of the Origin of
Species, only then enabling evolutionary biologists to discriminate the
influence of natural selection on organic evolution.
References
Darwin, C. (1859). On the origin of species by means of natural selection
or the preservation of favoured races in the struggle for life. London:
John Murray.
Johnson-Laird, P.N., Legrenzi, P., Girotto, V., & Legrenzi, M.S. (1999).
Naive probability: A mental model theory of extensional reasoning.
Psychological Review, 106, 62-88.
Kahneman, D. & Tversky, A. (1996). On the reality of cognitive illusions.
Psychological Review, 103, 582-591.
Zahavi, A. (1975). Mate selection: A selection for a handicap. Journal of
Theoretical Biology, 53, 205-214.
Model Checking Knowledge in Probabilistic Systems
Carla A.D.M. Delgado (Rio de Janeiro, Brazil) and Mario Benevides (Rio de Janeiro, Brazil)
Our aim is to develop models, languages and algorithms to formally
verify knowledge properties of concurrent Multi-Agents Systems. Two
models are proposed and their adequacy is discussed with respect
representation
and verification of knowledge properties.
First, we address the issue of model checking knowledge in concurrent
systems. The work benefits from many recent results on model checking
and combined logics for time and knowledge, and focus on the way
knowledge relations can be captured from automata-based system
specifications.
We present a formal language with compositional semantics
and the corresponding Model Checking algorithms to model and verify
Multi-Agent Systems (MAS) at the knowledge level, and a process for
obtaining the global automaton for the concurrent system and the knowledge
relations for each agent from a set of local automata that represents
the behavior of each agent. Our aim is to describe a model suitable for
model checking knowledge in a pre-defined way, but with the advantage
that the knowledge relations for this would be extracted directly from the
automata-based model.
Finally, we extend the previous approach to reasoning about
probabilistic
and non-deterministic model of discrete time. Instead of using
automata we use Markov decision process. The probabilities measures
are defined and the non-determinism are dealt with adversaries. Thus,
the global MDP becomes probabilistic. In each state of the system is
possible to verify probabilistic temporal formulas PCTL. We extend the
language of PCTL with knowledge operators, one for each agent, and
provide algorithms for model checking formulas in this language. The
semantics associates distribution of probabilities to each set of
indistinguishable
states.
Can there be a propensity interpretation of conditional
probabilities?
Isabelle Drouet (Paris, France)
The propensity interpretation of probability provides us with our sole
concept of objective singular probability. To this extent it seems
theoretically indispensable. At the same time, it is the only
interpretation that is available for the probabilities attached to quite a
wide range of natural phenomena Š especially phenomena resorting to
fundamental physics.
Yet, the propensity interpretation has experienced many criticisms
since it was introduced by Popper in the 1950s. Amongst those criticisms,
the most serious one is probably the one known as "Humphreys's Paradox".
The paradox states that it is impossible to give a propensity
interpretation of conditional probabilities. Humphreys's argument in
favour of this statement is roughly as follows: propensity candidates for
the interpretation of conditional probabilities --what Humphreys calls
"conditional propensities"-- provably do not behave as conditional
probabilities do.
The first aim of the present talk is to call Humphreys's analysis into
question. More precisely, my criticism deals with the idea according to
which what would be the propensity interpretation of conditional
probabilities, is analytically contained in the propensity proposition for
the interpretation of absolute probabilities. Against this idea, I claim
that conditionalisation needs an interpretation of its own, and that this
interpretation can in no way be reduced to the interpretation of absolute
probabilities. This claim is based on an analysis of the concept of
conditional probability, leading to a general account of what it is to
interpret conditional probabilities Š and especially of how the
interpretation of conditional probabilities is to be articulated with an
interpretation of absolute probabilities. The account is shown to be
corroborated by the subjectivist and frequentist interpretations of the
probability calculus -- considered as a calculus of both conditional and
absolute probabilities. It leads to reject Humphreys's argument as relying
on a misconception as to what an interpretation of conditional
probabilities is.
Now, giving a general account of what it is to interpret conditional
probabilities does not serve only as a tool to criticize Humphreys's
argument. Positively, it allows to reformulate the question of the
propensity interpretation of conditional probabilities. As an answer to
the question thus reformulated, I actually propose a propensity
interpretation of conditional probabilities. According to this
proposition, relative to a system S, the conditional probability
p(A | B)
is to be interpreted as the propensity for A in the system which is
the
most similar to S amongst those for which the propensity for
B has
value
1. Obvious possible difficulties with this proposition are envisaged and
discussed. It is concluded that none of them is fatal to the proposition.
Therefore my conclusion is that the proposition may be accepted -- at
least
for the time being.
Bibliographical indications :
Humphreys 1985 : "Why propensities cannot be probabilities", The
Philosophical Review, 94
Humphreys 2004 : "Some considerations on conditional chances", BJPS,
55
McCurdy 1996 : "Humphreys's paradox and the interpretation of inverse
conditional propensities", Synthese, 108
Popper 1959 : "The propensity interpretation of probabilities", BJPS,
10.
Probability: One or many?
Maria Carla
Galavotti
(Bologna, Italy)
After more than three centuries and a half since the ?official? birth of
probability with the work of Blaise Pascal and Pierre Fermat, and about
two centuries after Pierre Simon de Laplace endowed probability with a
univocal meaning and started a tradition in this sense, the dispute on the
interpretation of probability is far from being settled. While the
classical interpretation forged by Laplace is outdated, there are at least
four interpretations still on the market: frequentism, propensionism,
logicism and subjectivism, each of which admits of a number of variants.
Upholders of one or the other of these interpretations have traditionally
been quarrelling over the "true: meaning of probability and the "right"
method for calculating initial probabilities.
According to a widespread opinion, the natural sciences call for
an objective notion of probability and the subjective interpretation is
better suited to the social sciences. This viewpoint is shared by authors
of different inspiration - not just frequenstists but also logicists like
Harold Jeffreys. Even an upholder of a pluralistic tendency like Donald
Gillies, maintains that the natural sciences - physics in the first place
- and the social sciences call for different notions of probability.
Gillies makes the further claim that the propensity theory is apt to
represent physical probabilities, while probabilities encountered in the
social sciences are better interpreted along subjectivist lines.
As a matter of fact, subjectivism seems to be surrounded by a halo
of arbitrariness that makes it unpalatable even to those operating in
various areas of the social sciences; for instance, law scientists tend to
discard it in favour of the logical interpretation, reassured by the
promise of objectivity involved in the term "logical".
By contrast, the paper will argue that the subjective
interpretation has the resources to account for all applications of
probability, in the natural as well as the social sciences.
Statistics without Stochastics
Peter
Grünwald (Amsterdam, The Netherlands)
Consider a set of experts that sequentially predict the future given
the
past and given some side information. For example, each expert may be a
weather(wo)man who, at each day, predicts the probability that it will
rain the next day.
We describe a method for combining the experts' predictions that
performs well *on every possible sequence of data*. In marked contrast,
classical statistical methods only work well under stochastic
assumptions ("the data are drawn from some distribution P") that are
often violated in practice.
Nonstochastic prediction schemes can be used as a basis for
robust, nonstochastic versions of standard statistical problems such as
parameter estimation and *model selection*, a central issue in
essentially all applied sciences: given two structurally different
models that fit a set of experimental data about equally well, how
should we choose between them?
The resulting theory is closely related to Bayesian statistics, but
avoids some of its conceptual problems, essentially by replacing "prior
distributions" by "luckiness functions".
This work is based on Dawid's Prequential Analysis,Vovk's work on
universal prediction and Rissanen and Barron's work on the Minimum
Description Length Principle.
Redoing the Foundations of Decision Theory
Joe Halpern
(Ithaca NY, United States of America)
In almost all current approaches to decision making, it is assumed that
a decision problem is described by a set of states and set of outcomes,
and the decision maker (DM) has preferences over a rather rich set of
Acts, which are functions from states to outcomes. However, most
interesting decision problems do not come with a state space and an
outcome space. Indeed, in complex problems it is often far from clear
what the state and outcome spaces would be. We present an alternate
foundation for decision making, in which the primitive objects of choice
are not defined on a state space. A representation theorem is proved
that generalizes standard representation theorems in the literature,
showing that if the DM's preference relation on her choice set satisfies
appropriate axioms, then there exist a set S of states, a set
O of
outcomes, a way of viewing choices as functions from S to O,
a
probability on S, and a utility function on O, such that the
DM
prefers choice a to choice b if and only if the expected
utility of a
is higher than that of b. Thus, the state space and outcome space
are subjective, just like the probability and utility; they are not part
of the description of the problem. In principle a modeller can test for
SEU behavior without having access to states or outco mes. A number of
benefits of this generalization are discussed.
Indeterminacy in the Combining of Attributes
Jeffrey
Helzner (New York NY, United States of America)
One point of entry into certain theoretical studies of decision making
begins with a study of set-valued choice functions. Let X be a set of
alternatives. Let S be a collection of subsets of X, e.g. the finite
subsets. If C is a function on S such that, for all Y in S, Y is a subset
of C(Y) and C(Y) is nonempty whenever Y is nonempty, then C is a choice
function on S. The term "choice function" is, though standard, a bit
misleading since C may be set-valued while in the intended interpretation
no more than one alternative can actually be selected from a given menu of
alternatives. By way of avoiding this tension it is sometimes convenient
to say that an element y in C(Y) is "admissible" in Y; one possible
reading of this term takes the admissible elements of Y to be the set of
alternatives that the agent would be willing to choose if offered the
opportunity to make a selection from Y.
Let X be the product X1×...×Xn
One familiar
interpretation takes (x1, ...
, xn) in X to be the "act" that would have outcome xi if the ith state
were to obtain. Normative studies concerning this interpretation often
begin by equating admissibility with maximizing expectation against
numerically precise probabilities and utilities. Though less familiar than
these traditional expected utility models there is now an active study of
choice functions that allow for indeterminacy in the underlying
probabilities or utilities that inform expected utility maximization.
While some of these models remain committed to the usual reduction of
choice to a complete ranking of the alternatives -- or, more suggestively,
a reduction of choice to preference -- some who have allowed for such
indeterminacy, Isaac Levi most notably, have abandoned this reduction.
Abandoning this traditional reduction forces one to rethink the role of
preference in choice and the manner in which preferences are to be
measured within the context of such an account.
A second important interpretation of choice in product sets takes (x1,
..., xn) to be an alternative that has value xi
with respect to the ith
attribute. Applications of this sort of interpretation range from
psychological models of judgment and decision to prescriptive techniques
in decision analysis. The most well-studied models in this setting assume
that admissibility amounts to maximizing a real-valued function that is
additive over the attributes; that is, the relevant index is computed by
summing the alternative's performance over each of the attributes. As in
the act-state interpretation one may allow for indeterminacy, e.g. there
might be a family of appropriate additive models. Again, such
indeterminacy can require us to rethink the manner in which preferences
are to be measured. Techniques that have been developed for the analogous
problem within the act-state framework are not appropriate here since they
impose structural requirements (e.g. convex combinations) that are either
unfounded or undesirable in the multiattribute interpretation. We will
consider the possibility of relaxing assumptions in the theory of additive
conjoint measurement in a way that allows for indeterminacy in the
combining of attributes.
Beliefs: identification and change
Brian Hill (Paris, France)
Beliefs play an important role in human decision. Beliefs change. These
banalities conceal the two most important questions regarding beliefs. The
first is the question of the identification of the role of beliefs in
decision; the second is the question of how they change, of their
dynamics. Both these questions have been posed, and answers have been
proposed, in terms of probabilities, the assumption being that
probabilities can be considered a faithful enough representation of
beliefs. This assumption is retained for the purposes of this
presentation.
The question of the role of beliefs in decision proves difficult
because, as well as the agent's beliefs about the world (or
probabilities), his preferences over the consequences of his actions (or
utilities) are involved in his choice. The problem is to separate the
agent's probabilities from his utilities. Answers have normally come in
the form of representation theorems stating that, for any agent whose
preference relation (over acts) satisfies particular conditions, there is
a unique probability function and an essentially unique utility function
--intuitively the agent's beliefs and preferences over consequences-- such
that the agent prefers acts which have a larger expected utility.
On the other hand, answers to the question of belief change have
generally come in the form of mechanisms (Bayesian update, imaging,
Jeffrey conditionalisation etc.) which, given the prior belief state and a
piece of new information, yields a posterior belief state.
These two questions have generally been dealt with separately. The
assumption underlying this practice is that they can be treated
separately. But what happens in situation where there is change in the
agent's behaviour, but it is an open question whether this change is to be
understood as belief change or as utility change? Such questions will have
to be faced soon, given the current growing interest in preference change.
An interesting example, important in Social Theory, is the phenomenon of
alleged "adaptive preferences" or "sour grapes": is it really the agentÕs
utilities (preferences over consequences) which change, or rather his
beliefs?
In the presentation, it will be argued that the divide-and-conquer
strategy to phenomena where both the distinction of beliefs and
preferences and the change of one or the other are at issue is inadequate.
On two counts: firstly, the conditions demanded by representation theorems
are particularly implausible when they are applied to the agent's
preferences over the acts available in particular situations. Secondly,
this strategy risks making the changes of attitudes between situations
more obscure, if not totally incomprehensible. An alternative strategy for
tackling these questions will be presented, which treats the two questions
simultaneously, using information about the relationship between different
situations to inform the elicitation of beliefs and preferences in
individual situations. Such a strategy places different constraints on
models of decision and change, in so far as these models should be able to
deal fruitfully with both the behaviour in particular decision situations
and the change between situations. Aspects of the application of this
strategy will be considered, and illustrated on the case of sour grapes
phenomena.
Empirical Progress and Truth Approximation by the "Hypothetical
Probabilistic (HP-)method"
Theo
Kuipers
(Groningen, The Netherlands)
Probabilistic approaches to confirmation and testing are usually not
seen as concretizations of deductive approaches. However, as argued
elsewhere (1), if one uses 'positive relevance', i.e. p(E/H)>p(E),
as the
basic criterion of probabilistic confirmation of a hypothesis H by
evidence E, 'deductive confirmation' appears as the common
idealization,
in the minimal sense of an extreme special case (2) , in the rich
landscape of probabilistic confirmation notions.
Now, similarly, if 'p(E/H)>p(E)' is taken as the basic
definition of 'E
is a probabilistic consequence of H', deductive consequences are
idealizations of probabilistic consequences. From this perspective, the
HD-method of testing, based on 'H entails E', resulting
either in
deductive confirmation or falsification, depending on whether the
predicted E turns out to be true or false, is an idealization of
the
Hypothetico-Probabilistic (HP-)method.
According to the HP-method, testing a hypothesis H amounts to
'deriving' probabilistic consequences E, in the sense defined
above. When
such probabilistic predictions come true, H is probabilistically
confirmed, otherwise, that is, when the produced evidence is not-E,
and
hence is such that p(not-E/H) < p(not-E), H is
(said to be)
probabilistically disconfirmed. All this was already anticipated by Ilkka
Niiniluoto in 1973 (3) .
In my "From Instrumentalism to Constructive Realism. On some relations
between confirmation, empirical progress and truth approximation" (4) I
have argued that the comparative evaluation of two (deterministic)
theories, based on repeated application of the HD-method, to be continued
if one or both have already been falsified, results in a plausible
explication of the notion of (deductivistic) empirical progress, with
articulated perspectives on (deductivistic) truth approximation.
Hence, it is plausible to try to argue along similar lines that the
comparative evaluation of theories based on the repeated application of
the HP-method, results in concretized versions of empirical progress and
perspectives on truth approximation. The repeated application amounts to
the systematic comparison of the likelihoods, p(E/X) and
p(E/Y), of two
theories X and Y relative to the successive experimental
results E,
generally called the L(ikelihood)C(omparison)-method. It was already shown
(5) that the HP-/LC-method suggests plausible probabilistic
concretizations up to and including 'being piecemeal more successful'.
This paper deals with the perspectives of the HP-/LC-method on
"empirical progress" and "truth approximation" of successive deterministic
theories, assuming that the truth is also a deterministic theory. The
proposed definition of "Y is probabilistically closer to the truth
than X"
is intuitively rather plausible and turns out to be a concretization of
'deductively closer to' and 'quantitatively closer to'. Moreover, it
entails that, in the long run, Y will show empirical progress
relative to
X, that is, Y will become irreversibly more successful than
X in the
cumulative sense, i.e., the likelihood of Y for the total evidence
exceeds
that of X to a given degree (not < 1). This 'threshold success
theorem'
directly supports the claim that the HP-/LC-method is functional for
probabilistic truth approximation.
(1) Kuipers, T., Structures in Science, Synthese Library 301, Kluwer
AP, Dordrecht, 2001, Ch. 7.1.2.
(2) For, if H entails E, p(E/H)=1 or, if undefined
because p(H)=0, it is
plausible to define it as 1.
(3) See (mainly) his "Towards a non-inductivist logic of induction" in I.
Niiniluoto and R. Tuomela: 1973, Theoretical Concepts and
Hypothetico-Inductive Inference, Synthese Library, Vol. 53, Reidel,
Dordrecht.
(4) Synthese Library 287, Kluwer AP, Dordrecht, 2000.
(5) Kuipers, T. (to appear), "The
hypothetico-probabilistic (HP-)method as
a concretization of the HD-method", to appear in Festschrift in honour
of
Ilkka Niiniluoto
Combining probability and logic: Embedding causal discovery in a
logical framework
Bert Leuridan (Ghent, Belgium)
The meaning and `logic' of the concepts of causation and of causal
discovery have long escaped precise treatment. Since the 1980s, however,
this has changed. By combining probability and graph theory, Pearl
(2000), Spirtes et al. (2000) and others were able to (partly) overcome
these problems. They have developed algorithms for causal discovery,
consisting of syntactic inference rules that allow one to derive causal
structure from purely observational data (e.g. Pearl's IC or Spirtes' and
Glymour's PC algorithm).
However valuable and fruitful these algorithms are, from a logical
point
of view they face several problems. Firstly, while explicitly
incorporating probabilistic and graph theoretic reasoning, they also make
use of classical logic. But these classical inferences remain implicit.
Secondly, although the algorithms are backed with a semantics (which is
formulated graph theoretically), the formulation of this semantics
strongly deviates from those encountered in logic (for instance, the
notion of truth in a model is absent). Thirdly and most importantly,
causal discovery is a form of non-monotonic reasoning. When confronted
with new observational data, a non-omniscient agent may have to drop
causal beliefs previously derived. PC nor IC is designed to handle such
cases. They presuppose the agent has full knowledge (they take a full,
stable distribution over a set of variables as their input).
The aim of this paper is to show how probabilistic theories and
algorithms for causal discovery can be embedded in a logical framework
and to discuss the advantages of doing so.
More specifically, I will embed (a revised version of) the IC algorithm
within a logical framework and show how the resulting logic ALIC solves
the problems in question. Well-formed formulas (wffs) in ALIC are either
atomic probabilistic sentences, or atomic causal statements, or complex
sentences built from such atomic sentences by means of logical
connectives (negation, conjunction, etc.). ALIC is formulated within the
framework of adaptive logics. Adaptive logics are a class of
non-monotonic logics that all have a dynamic proof theory and a (sound
and complete) semantics (Batens, 2004, 2007). ALIC-models assign truth
values (0, 1) to all wffs and represent faithful distributions and
graphs. The proof theory contains non-conditional axioms and inference
rules. These include the rules of classical propositional logic, plus
probabilistic and causal axioms and inference rules. Most importantly, it
also contains a conditional rule to infer causal relations from
correlations. If X and Y are correlated, then one may infer they are
adjacent on the condition that no non-empty set of variables screens them
off (cf. Van Dyck, 2004, where an adaptive logic for causal discovery is
formulated; this logic has no semantics, however, and is not formulated
within the standard format for adaptive logics). A conditional line in a
proof later may need to be marked if its condition is violated, e.g. in
view of newly added premises. Together, the conditional rule and the
marking definition (which governs the marking process) provide the logic
with a dynamics which makes it very suited for formally handling the
non-monotonic characteristics of causal discovery.
I shall proceed as follows. Firstly, I will introduce the reader to the
PC and IC algorithms. I will discuss their value and their main
shortcomings, chiefly focussing on the non-monotonic nature of causal
discovery. Then I will present ALIC's language and proof theory,
focussing on the resemblances and differences with IC. Finally, I will
present its semantics.
References
Batens, D. (2004). The need for adaptive logics in epistemology. In
Gabbay, D., Rahman, S., Symons, J., and Van Bendegem, J.-P., editors,
Logic, Epistemology and the Unity of Science, pages 459-485. Kluwer,
Dordrecht.
Batens, D. (2007). A universal logic approach to adaptive logics. Logica
Universalis, 1:221-242.
Pearl, J. (2000). Causality. Models, Reasoning, and Inference. Cambridge
University Press, Cambridge.
Spirtes, P., Glymour, C., and Scheines, R. (2000). Causation, Prediction,
and Search. MIT Press, Cambridge, Massachusetts.
Van Dyck, M. (2004). Causal discovery using adaptive logics. towards a
more realistic heuristics for human causal learning. Logique et Analyse,
185-188:5-32.
Probability in Evolutionary Theory
Aidan Lyon
(Canberra, Australia)
Evolutionary theory is up to its neck in probability. For example,
probability can be found in our understanding of mutation events, drift,
fitness, coalescence, and macroevolution.
Some authors have attempted to provide a unified realist interpretation
of these probabilities. Or, when that has not worked, some have decided
that this means there is no interpretation available at all, defending a
"no theory" theory of probability in evolution. I will argue that when we
look closely at the various probabilistic concepts in evolutionary theory,
then attempts to provide a unified interpretation of all these
applications of probability appear to be poorly motivated. As a
consequence of this, we need to be more careful in drawing conclusions
from the fact that each interpretation fails for just one particular area
of evolutionary theory. I will also argue that a plurality of
interpretations is much better than no interpretation at all.
On a more positive note, I will outline a particular way that a
plurality of objective and subjective interpretations of probability can
be strung together which provides a suitable understanding of the role
probability plays in evolutionary theory.
Like-Minded Agents as a Foundation for Common Priors
Klaus
Nehring (Davis CA, United States of America)
The Common Prior Assumption (CPA) plays a central role in
information economics and game theory . However, in situations of
incomplete information, that is: in situations in which agents are
mutually uncertain about each others' beliefs, and without a
preceding stage in which beliefs were commonly known, there is a
significant gap between the formal statement of the CPA and its
underlying intuitive content. Indeed, Gul (1998) has even questioned
whether the CPA can be transparently interpreted at all in this
context. This meaningfulness question has by now been successfully
addressed in a number of papers in the literature.
Nonetheless, the extant results leave a major gap in the
normative and positive foundations of the CPA in that all extant
characterizations make assumptions directly on what is commonly
known; that is to say, they make assumptions which, if true, must be
commonly known. By contrast, any norm of "interactive rationality"
must be ``fully local``, i.e. can only constrain the actual beliefs
of actual agents; a normative account of the CPA must therefore be
able to clearly accommodate patterns of interactive beliefs within
which all agents are de facto (at the true state) interactively
rational, but that put positive probability on other agents'
irrationality, or on their belief in others' irrationality. In other
words, interactive rationality must be defined in terms of fully
local properties of beliefs.
The goal of this paper is therefore to derive the CPA from
common knowledge of fully local properties of agents' beliefs,
specifically from common knowledge of their "Like-Mindedness". We
define like-mindedness of two agents formally as equality of their
beliefs, conditional on knowledge of both agents' entire belief
hierarchies; the conditioning ensures that the agents' beliefs are
compared on the basis of the same (hypothetical) information.
Like-mindedness can be viewed as an interpersonal generalization of
the "reflection principle" studied in the philosophical literature.
We show that in the general case of many-sided incomplete
information there is no fully local property whose being commonly
known characterizes the existence of a common prior. Thus any
foundation of the CPA must rely on the existence of auxiliary
assumptions that are not entailed by the CPA itself. In the present
paper, we assume the existence of an "uninformative" outsider to
close this gap. Importantly and in line with the incomplete
information setting, uninformativeness does not require that insiders
know anything about the outsider's beliefs. The main formal result of
the paper shows that common knowledge of like-mindedness with the
outsider together with common knowledge of his uninformativeness
yields a common prior among insiders.
In a concluding section, we also develop a richer framework in
which agents' "epistemic attitudes" are introduced as independent
primitives, and like-mindedness of beliefs is derived from the
recognition of other agents as "equally rational". This framework
allows one to formulate competing normative positions on the content
of intersubjective rationality in terms of alternative restrictions
on these equivalence relations. We distinguish in particular a
rationalist, a pluralist, and a relativist position.
Measuring the Uncertain
Martin Neumann (Osnabrück, Germany)
Throughout the history, the probability calculus was faced with the
problem of its interpretation. Roughly speaking, two major approaches can
be distinguished: a subjective and an objective interpretation. The latter
can be further distinguished into theories of relative frequencies and
propensity theories, most prominently advocated by Karl Popper. Propensity
theories can be characterised as theories of objective single case
probabilities.
In this talk a historical theory of such objective single case
probabilities will be outlined: the so-called "Spielraum" (i.e. range or
scope) theory of probability, which was developed in 1886 by the German
physiologist Johannes von Kries. It will be shown that the style of
probabilistic reasoning is fundamentally different from current propensity
theories: on the one hand, propensities are measured by relative
frequencies and on the other hand, they refer to an ontic indeterminism.
Both claims do not hold for the "Spielraum" theory. This will be shown
behind the backdrop of 19th century science.
The question of measuring objective single case probabilities will be
investigated behind the backdrop of the historical framework of laplacian
probability: the sorest point within laplacian probability theory was the
determination of cases of equal possibility. In the 19th century theories
of so-called objective possibility where of growing interest. Yet, the
question had to be answered, of how to identify cases of equal possibility
in nature. By developing the "Spielraum" theory, von Kries developed an
advice to measure cases of equal possibility. These conditions are called
indifference, comparability and originality. They will be outlined in
detail in the talk. Thereby von Kries developed a sophisticated theory of
measurement. In modern terms it can be characterised as a theory of
extensive measurement. In particular, these conditions have to be
fulfilled in any single case. Thus, a "Spielraum" is not measured by
relative frequencies.
The question of (in)determinism will be investigated behind the
backdrop of developments in the kinetic theory of gases, in particular,
the 2nd law of thermodynamics. Originally, the kinetic theory of gases was
introduced to reduce the phenomenological law of theromdynamics to the
laws of newtonian mechanics. However, these are reversible while the 2nd
law of thermodynamics is irreversible. Ludwig Boltzmann faced this
objection by probabilistic considerations. Yet, this was in contradiction
to the determinism of newtonian mechanics. However, von Kries demonstrated
that Boltzmann's considerations fulfil the measure theoretical conditions
of the "Spielraum" theory. This enabled a formalisation of randomness
which allowed to reconcile probabilistic reasoning with scientific
determinism. Thus, contrary to modern propensities, the "Spielraum" theory
does not call for an indeterministic world view.
Probabilistic Justification and the Regress Problem
Jeanne
Peijnenburg (Groningen, The Netherlands)
Today epistemologists are usually probabilists: they hold that
epistemic justification is mostly probabilistic in nature. If a person
S
is epistemically (rather than prudentially) justified in believing a
proposition E0, and if this justification is inferential
(rather than
noninferential or immediate), then typically S believes a
proposition E1
which makes E0 probable.
How to justify E1 epistemically? Again, if the
justification is
inferential, then there is a proposition E2 that makes
E1
probable.
Imagine that E2 is in turn made probable by
E3, and
that E3 is made
probable by E4, and so on, ad infinitum. Is such a
process possible? Does
the "ad infinitum" makes sense? The question is known as the Regress
Problem and the reactions to it are fourfold. Skeptics have hailed it as
another indication of the fact that we can never be justified in believing
anything. Foundationalists famously argue that the process must come to an
end in a proposition that is itself noninferentially justified.
Coherentists, too, maintain that the infinite regress can be blocked, but
unlike foundationalists they hold that the inferential justification need
not be linear and may not terminate at a unique stopping point. Finally,
infinitists have claimed that there is nothing troublesome about infinite
regresses, the reason being that an infinite chain of reasoning need not
be actually completed for a proposition or belief to be justified.
I will defend a view that is different from all four. Against skeptics,
foundationalists and coherentists I will show that an infinite regress can
make sense; against infinitists I will show that some beliefs are
justified by an infinite chain of reasons that can be formally completed.
Probability Spaces for First-Order Logic
Kenneth Presting
(Chapel Hill NC, United States of America)
Following the pioneering work of Haim Gaifman in 1964, the development
of probability logic in the first-order setting has been advanced by
Kyburg, Bacchus, Halpern, and many others. Most of this work has been in
the context of probability logics or probability semantics. The present
article develops a different approach which is closer to the traditional
measure-theoretic foundation of probability due to Kolmogorov. Our
approach is similar to that of Fenstad, in that we preserve the intuition
that the probability of an expression is the measure of its extension,
that is, the set where the expression is true. We construct probability
spaces from models for the language, and from classes of models. This
allows explicit calculation of objective probabilities and complements the
more subjective approach of probability logic.
We consider a first order language (FOL) L with countably many
variables,
interpreted into a given model M with a countable domain. Every
FOL
contains two varieties of expressions - open formulas and closed
sentences. In the standard Tarski semantics, the extension of an open
sentence is a set of sequences of domain elements from M, {s(i)
| s(i) in
M for every natural i}, such that the sequence satisfies the formula.
The set of all these extensions, denoted [L], generates a
sigma-algebra
s[L]. The Boolean operations in the sigma-algebra are isomorphic
to the
logical conjunction and disjunction in the language. Furthermore, in the
countable models, existential and universal quantification is homomorphic
to countable union and intersection, which is shown by embedding the
cylindrical algebra over the sequences into the infinite-dimensional
product algebra MN. Thus, the set of all sequences is
the domain of a
measurable space (MN,s[L]). These sequences may be
interpreted as the
outcome space for an experiment of random sampling (with replacement) from
the model domain. Assuming exchangeability of the domain members, the
resulting probabilities for open formulas are identical to the
probabilities calculated from traditional sampling models.
Closed sentences are satisfied either by every sequence or by no sequence.
Thus they have an essential role in the space of sequences, although that
space reveals little information to distinguish one generalization from
any other. We construct a probability space in which generalizations have
intermediate probability by appealing to the concept of sampling without
replacement. Every sample from a population is a subset of that
population, and corresponding to the model M there is a class of
sub-models 2M. Letting Lq stand for the set of closed
sentences (i.e.
quantified formulas) in L, we have another measurable space
(2M,
s[Lq])
generated by the quantified formulas Lq, such that the extension of each
sentence is the set of models where it is true. It is an open question
whether the probability of a closed generalization obtained from the space
of sub-models is computable from the probabilities of corresponding open
formulas in the space of sequences. The sample space of sub-models can be
given a Lebesgue measure using a construction due to Halmos, which
corresponds to giving each member of the domain a 50% chance of inclusion
in any random sample. This case has interesting properties, some of which
are explored.
Special progicnet Presentation:
Probabilistic Logics
and
Probabilistic Networks
Rolf Haenni (Bern, Switzerland),
Jan-Willem
Romeijn
(Amsterdam, The Netherlands), Greg
Wheeler
(Lisbon, Portugal), Jon
Williamson (Canterbury, United Kingdom)
In classical logic the question of interest is whether a proposition
ψ is
logically implied by premiss
propositions φ1, φ2, ... ,
φn. A probabilistic logic, or progic for
short, differs in two respects. First, the
propositions have probabilities attached to them. The premisses have the
form φX, where φ is a classical
proposition and X subset [0,1] is a set of probabilities, an where
the premiss is interprested as a restriction of the probability of φ
to X.
Second, the question of
interest is not the direct analogue of the classical
question, namely whether some specific ψY follows
from a set of premisses
φ1X1,
φ2X2,
...,
φnXn.
Rather,
the question of interest is the determination of the smallest possible, or
most informative Y:
(1) φ1X1,
φ2X2,
...,
φnXn models ψ?
That is, what minimal set Y of probabilities should attach to the
conclusion sentence ψ, given the
premisses
φ1X1,
φ2X2,
...,
φnXn.
The first part of the paper shows how a large range of systems for
uncertain inference may be captured
by this general question: the standard probabilistic inference of Ramsey
and Jeffrey, probabilistic
argumentation using degrees of support and possibility, or Dempster-Shafer
belief functions, the partial
entailment of Kyburgian evidential probability, classical and Bayesian
statistical inference, and objective
Bayesian inference. All of these views effectively provide a different
semantics for the terms in Equation
1. Now it must be noted that several of these semantics distance
themselves explicitly from standard
Kolmogorov probability as the expression of uncertainty. However, despite
this disparity they retain a
formal connection to standard probability, and it is this connection that
can be exploited to provide a
syntactic procedure that provides, at least in part, an answer to the
question of Equation 1.
In the second part of the paper, we show that the different systems of
uncertain inference can all be
assisted by employing a procedure based on so-called credal networks. A
credal network is a graphical
representation of a convex set of probability functions. If the sets
Xi
appearing in the premisses are
intervals, then credal networks are applicable, and under some further
assumptions, which may be
motivated by the respective semantics, credal networks appear as
computationally attractive tools for
determining an interval for Y. Thus the progic proposed above can
perform
a double function. It unifies
a number of different views on uncertain inference by showing that they
can be represented in the same
format, and it offers each of these views a useful piece of inference
machinery.
Eclectic Interpretation of the Probability: A Question of
Convenience?
Paolo
Rocchi (Roma, Italy)
Whereas the advocates of the Bayesian view on probability and of the
frequentist view reject the opponent theories as ill-grounded, a large
group of experts is inclined to accept both the interpretations of
probability [1]. Those authors claim that each approach should be used in
accordance to the specific context of the problem to confront. It is often
said that the Bayesian statistics and the classical statistics may be
employed independently from the theoretical convinction, that the truth is
messy and we have to make approximations.
Concluding, the concept of probability ignites fierce debates at one
hand in the scientific community and inspires conciliatory compromise at
the other hand. This astonishing evolution of the mathematical searches
raises a doubt: is the eclectic position a question of convenience or
otherwise may a superior logic be found to conciliate the irreconcilable
opponents?
The pathway we follow does not provide a reply in a direct manner. We
scale to the problem through the argument of the probability which is used
as a key to screen the different theoretical interpretations. This
approach is close to the Popperian school that repeatedly considers the
difference existing between the single event and the long-run event [2].
In detail we examine the argument of the probability from the historical
perspective [3] and then argue about the internal logic of the
subjectivist and frequentist views [4].
[1] J.Berger, Could Fisher, Jeffreys and Neyman have agreed upon
testing?, Statistical Science, 18 (2003),1-32.
[2] D.Gillies, Varieties of propensity, British Journal for the
Philosophy of Science 51 (2000) 807-835.
[3] P.Rocchi, De Pascal à nos jours: Quelques notes sur
l'argument A de
la probabilité P(A), Actes du Congr¸s Annuel de la
Société
Canadienne
d'Histoire et de Philosophie des Mathématiques (CSHPM/SCHPM)
(2006) [in
printing].
[4] P.Rocchi, The Structural Theory of Probability, Kluwer/Plenum, N.Y.
(2003).
Surprise and Evidence in Model Checking
Jan Sprenger (Bonn, Germany; London, United Kingdom)
In various situations we evaluate data merely with respect to a null
model or a family of null models, without specifying any alternative.
Then, p-values are commonly taken as measures of "evidence against the
null". This interpretation is seriously flawed: evidence is a comparative
concept and always relative to an alternative. To the extent that p-values
do not implicitly assume an alternative model, they cannot be taken as a
measure of evidence against the null model simpliciter. Furthermore,
p-values typically depend on the probability of hypothetical,
counterfactual outcomes. That might be useful in a genuine alternative
testing problem, but it precludes their interpretation as quantification
of evidence.
Given the widespread use of p-values in statistical inference, it is
notwithstanding mandatory to clarify their epistemic role. I argue that
they are closely related to measures of surprise and that confounding
surprise and evidence has led to serious misunderstandings about the
significance of p-values. Quantifying surprise in the results is valuable
at preliminary stages of model analysis when a null model is only
tentatively endorsed. At this early stage of the analysis, observation
results are supposed to either license the adoption of the null model or
to indicate the need for modification. In order to describe the relative
expectedness of the actual result under the null model, a measure of
surprise has to depend on the probability of counterfactual outcomes. This
property clearly distinguishes measures of surprise from measures of
evidence. Thus, the epistemic rationale of surprise in the results
consists in guiding and driving the development of alternative models.
Accounting for degrees of surprise and the introduction of new models is
especially challenging for a full Bayesian approach where all conceivable
alternative models are part of a Bayesian supermodel.
This epistemological characterization of surprise enables us to
evaluate various measures of surprise with regard to the relevance of
p-values. Contrary to likelihood-based suggestions, p-values do not ignore
the value of counterfactual considerations in measuring surprise. However,
in the more general case of a whole family of null models, p-values based
on maximum likelihood estimates tend to be overly conservative, concealing
the need for a modification of the model. More refined techniques, e.g.
Bayesian or resampling methods, are required to deal with such cases.
Finally, I propose a measure of surprise which is tightly connected to
classical p-values in the case of a single null model. This recognizes the
significance of p-values but endows them with a more natural scaling and
interpretation.
The preceding results are applied to a comparison of surprise and
evidence. Measuring surprise is required in an exploratory statistical
analysis whereas the intrinsically comparative concept of evidence is
crucial at later stages of the analysis, e.g. for model choice and
validation. Hence, both surprise and evidence play important and distinct
epistemic roles. I further argue that objections to the Likelihood
Principle usually rest on confounding both concepts. These objections can
be convincingly rebutted if the distinction between surprise and evidence
is clearly drawn.
Quality, quantity, and beyond. On nonmonotonic probabilistic
reasoning
Emil Weydert (Luxembourg, Luxembourg)
When artificial intelligence took off in the seventies and eighties,
probabilistic reasoning used to be not so popular. In particular,
quantitative approaches were thought to be too cumbersome for modeling
commonsense reasoning, the holy grail of knowledge representation. This
motivated a lot of research on alternative qualitative approaches, known
as default reasoning, and more generally, nonmonotonic logic.
However, in the last 15 years, probabilistic methods have had their
revenge, mainly through the success story of probabilistic graphical
models. Furthermore, it has turned out that the intuitively most
interesting default formalisms can be actually grounded in probabilistic
considerations, for instance mediated by quotient structures over
nonstandard models of probability theory. In particular, well-motivated
probabilistic choice functions, like entropy maximization (ME), have
inspired defeasible entailment notions for conditional defaults (e.g.
System JLZ).
Although the additional degrees of freedom and slightly different
purposes prevent straightforward one-to-one relationships, it seems
natural to require that any reasonable qualitative approach to reasoning
under uncertainty should be anchored in the probabilistic framework,
understood in a broad sense. This does not just help to provide
justifications and tools for some coarser-grained formalisms, but also to
exploit techniques or insights gained at the qualitative level for the
finer-grained probabilistic level. More generally, we may note that
probabilistic and statistical inference are and always have been instances
of nonmonotonic reasoning, a still hardly explored area.
Taking a look at the complexity of real-world knowledge and of the
corresponding uncertainty management tasks, as confronted e.g. by evolving
information agents, it could therefore be interesting to try to merge the
probabilistic and the default tradition in a more systematic way. This may
result in more transparent and powerful representation and inference
models, less blocked by a uni-dimensional perspective. In the talk, we are
going to discuss some problems and ideas about how to achieve this.