Conditional Probability

A group is often construed as one agent with its own probabilistic beliefs (credences), which are obtained by aggregating those of the individuals, for instance through averaging. In their celebrated “Groupthink”, Russell et al. (2015) require group credences to undergo Bayesian revision whenever new information is learnt, i.e., whenever individual credences undergo Bayesian revision based on this information. To obtain a fully Bayesian group, one should often extend this requirement to non-public or even private information (learnt by not all or (...) just one individual), or to non-representable information (not representable by any event in the domain where credences are held). I pro- pose a taxonomy of six types of ‘group Bayesianism’. They differ in the information for which Bayesian revision of group credences is required: public representable information, private representable information, public non-representable information, etc. Six corre- sponding theorems establish how individual credences must (not) be aggregated to ensure group Bayesianism of any type, respectively. Aggregating through standard averaging is never permitted; instead, different forms of geometric averaging must be used. One theorem—that for public representable information—is essentially Russell et al.’s central result (with minor corrections). Another theorem—that for public non-representable information—fills a gap in the theory of externally Bayesian opinion pooling. (shrink)

We present a general framework for representing belief-revision rules and use it to characterize Bayes's rule as a classical example and Jeffrey's rule as a non-classical one. In Jeffrey's rule, the input to a belief revision is not simply the information that some event has occurred, as in Bayes's rule, but a new assignment of probabilities to some events. Despite their differences, Bayes's and Jeffrey's rules can be characterized in terms of the same axioms: "responsiveness", which requires that revised beliefs (...) incorporate what has been learnt, and "conservativeness", which requires that beliefs on which the learnt input is "silent" do not change. To illustrate the use of non-Bayesian belief revision in economic theory, we sketch a simple decision-theoretic application. (shrink)

The Equation (TE) states that the probability of A → B is the probability of B given A (Jeffrey, 1964: 702–703). Lewis has shown that the acceptance of TE implies that the probability of A → B is the probability of B, which is implausible: the probability of a conditional cannot plausibly be the same as the probability of its consequent, e.g., the probability that the match will light given that is struck is not intuitively the same as the probability (...) that it will light (Lewis, 1976: 299–300). Here I want to counter Lewis’ claim. My aim is to argue that: (1) (TE) doesn’t track the probability of A → B, but instead our willingness to employ it on a modus ponens; (2) the triviality result doesn’t strike us as implausible if our willingness to employ A → B on a modus ponens implies a similar result; (3) (TE) is still inadequate in this limited role given that some conditionals are only employable on a modus tollens or can’t be employed on a modus ponens; (4) (TE) does not have the logical significance that is usually attributed to it, since inferential disposition is a pragmatic phenomenon. (shrink)

The epistemic probability of A given B is the degree to which B evidentially supports A, or makes A plausible. This paper is a first step in answering the question of what determines the values of epistemic probabilities. I break this question into two parts: the structural question and the substantive question. Just as an object’s weight is determined by its mass and gravitational acceleration, some probabilities are determined by other, more basic ones. The structural question asks what probabilities are (...) not determined in this way—these are the basic probabilities which determine values for all other probabilities. The substantive question asks how the values of these basic probabilities are determined. I defend an answer to the structural question on which basic probabilities are the probabilities of atomic propositions conditional on potential direct explanations. I defend this against the view, implicit in orthodox mathematical treatments of probability, that basic probabilities are the unconditional probabilities of complete worlds. I then apply my answer to the structural question to clear up common confusions in expositions of Bayesianism and shed light on the “problem of the priors.”. (shrink)

Pedersen and Wheeler (2014) and Pedersen and Wheeler (2015) offer a wide-ranging and in-depth exploration of the phenomenon of dilation. We find that these studies raise many interesting and important points. However, purportedly general characterizations of dilation are reported in them that, unfortunately, admit counterexamples. The purpose of this note is to show in some detail that these characterization results are false.

Self-locating beliefs cause a problem for conditionalization. Miriam Schoenfield offers a solution: that on learning E, agents should update on the fact that they learned E. However, Schoenfield is not explicit about whether the fact that they learned E is self-locating. I will argue that if the fact that they learned E is self-locating then the original problem has not been addressed, and if the fact that they learned E is not self-locating then the theory generates implausible verdicts which Schoenfield (...) explicitly rejects. (shrink)

Merging of opinions results underwrite Bayesian rejoinders to complaints about the subjective nature of personal probability. Such results establish that sufficiently similar priors achieve consensus in the long run when fed the same increasing stream of evidence. Initial subjectivity, the line goes, is of mere transient significance, giving way to intersubjective agreement eventually. Here, we establish a merging result for sets of probability measures that are updated by Jeffrey conditioning. This generalizes a number of different merging results in the literature. (...) We also show that such sets converge to a shared, maximally informed opinion. Convergence to a maximally informed opinion is a (weak) Jeffrey conditioning analogue of Bayesian “convergence to the truth” for conditional probabilities. Finally, we demonstrate the philosophical significance of our study by detailing applications to the topics of dynamic coherence, imprecise probabilities, and probabilistic opinion pooling. (shrink)

The standard treatment of conditional probability leaves conditional probability undefined when the conditioning proposition has zero probability. Nonetheless, some find the option of extending the scope of conditional probability to include zero-probability conditions attractive or even compelling. This article reviews some of the pitfalls associated with this move, and concludes that, for the most part, probabilities conditional on zero-probability propositions are more trouble than they are worth.

In previous work, we studied four well known systems of qualitative probabilistic inference, and presented data from computer simulations in an attempt to illustrate the performance of the systems. These simulations evaluated the four systems in terms of their tendency to license inference to accurate and informative conclusions, given incomplete information about a randomly selected probability distribution. In our earlier work, the procedure used in generating the unknown probability distribution (representing the true stochastic state of the world) tended to yield (...) probability distributions with moderately high entropy levels. In the present article, we present data charting the performance of the four systems when reasoning in environments of various entropy levels. The results illustrate variations in the performance of the respective reasoning systems that derive from the entropy of the environment, and allow for a more inclusive assessment of the reliability and robustness of the four systems. (shrink)

The use of evidence to resolve uncertainties is key to many endeavours, most conspicuously science and law. Despite this, the logic of uncertainty is seldom taught explicitly, and often seems misunderstood. Traditional educational practice even fails to encourage students to identify uncertainty when they express knowledge, though mark schemes that reward the identification of reliable and uncertain responses have long been shown to encourage more insightful understanding. In our information-rich society the ability to identify uncertainty is often more important than (...) the possession of knowledge itself. In both science and law there are fundamentally different kinds of uncertainty at issue. There is uncertainty whether a particular hypothesis is correct, and there is uncertainty about observable data that may be generated if a particular hypothesis is correct. Both are expressed in terms of probabilities. Each has its own domain of application and its own logic, but the inter-relationship is complex and sometimes misunderstood. Hypothesis probabilities are always open to error through possible failure to take account of realistic alternatives, while the proper inferences that can be drawn from data probabilities (often in the context of significance testing) are quite limited and easily over-interpreted. When considering these two kinds of probability in a court of law it is possible to interpret the phrase 'reasonable doubt' in different ways. It can be seen as addressing data uncertainty: whether such incriminating evidence might with reasonable probability arise to confront an innocent person. Or (the more conventional view) it can be seen as some sort of threshold level on the probability that the defendant is guilty (a hypothesis probability). Each typically involves elements of subjective judgement, but fewer issues and uncertainties arise when considering the data probability and it is argued that this is often the more critical and proper issue for a jury to address. This has particular repercussions for cases involving identification of a suspect through trawl of a DNA or other database. (shrink)

A large number of essays address the Sleeping Beauty problem, which undermines the validity of Bayesian inference and Bas Van Fraassen's 'Reflection Principle'. In this study a straightforward analysis of the problem based on probability theory is presented. The key difference from previous works is that apart from the random experiment imposed by the problem's description, a different one is also considered, in order to negate the confusion on the involved conditional probabilities. The results of the analysis indicate that no (...) inconsistency takes place, whereas both Bayesian inference and 'Reflection Principle' are valid. (shrink)

IBE ('Inference to the best explanation' or abduction) is a popular and highly plausible theory of how we should judge the evidence for claims of past events based on present evidence. It has been notably developed and supported recently by Meyer following Lipton. I believe this theory is essentially correct. This paper supports IBE from a probability perspective, and argues that the retrodictive probabilities involved in such inferences should be analysed in terms of predictive probabilities and a priori probability ratios (...) of initial events. The key point is to separate these two features. Disagreements over evidence can be traced to disagreements over either the a priori probability ratios or predictive conditional ratios. In many cases, in real science, judgements of the former are necessarily subjective. The principles of iterated evidence are also discussed. The Sceptic's position is criticised as ignoring iteration of evidence, and characteristically failing to adjust a priori probability ratios in response to empirical evidence. (shrink)

In this paper I present a precise version of Stalnaker's thesis and show that it is both consistent and predicts our intuitive judgments about the probabilities of conditionals. The thesis states that someone whose total evidence is E should have the same credence in the proposition expressed by 'if A then B' in a context where E is salient as they have conditional credence in the proposition B expresses given the proposition A expresses in that context. The thesis is formalised (...) rigorously and two models are provided that demonstrate that the new thesis is indeed tenable within a standard possible world semantics based on selection functions. Unlike the Stalnaker-Lewis semantics the selection functions cannot be understood in terms of similarity. A probabilistic account of selection is defended in its place. -/- I end the paper by suggesting that this approach overcomes some of the objections often leveled at accounts of indicatives based on the notion of similarity. (shrink)

Systems of logico-probabilistic (LP) reasoning characterize inference from conditional assertions interpreted as expressing high conditional probabilities. In the present article, we investigate four prominent LP systems (namely, systems O, P, Z, and QC) by means of computer simulations. The results reported here extend our previous work in this area, and evaluate the four systems in terms of the expected utility of the dispositions to act that derive from the conclusions that the systems license. In addition to conforming to the dominant (...) paradigm for assessing the rationality of actions and decisions, our present evaluation complements our previous work, since our previous evaluation may have been too severe in its assessment of inferences to false and uninformative conclusions. In the end, our new results provide additional support for the conclusion that (of the four systems considered) inference by system Z offers the best balance of error avoidance and inferential power. Our new results also suggest that improved performance could be achieved by a modest strengthening of system Z. (shrink)

This paper argues that the technical notion of conditional probability, as given by the ratio analysis, is unsuitable for dealing with our pretheoretical and intuitive understanding of both conditionality and probability. This is an ontological account of conditionals that include an irreducible dispositional connection between the antecedent and consequent conditions and where the conditional has to be treated as an indivisible whole rather than compositional. The relevant type of conditionality is found in some well-defined group of conditional statements. As an (...) alternative, therefore, we briefly offer grounds for what we would call an ontological reading: for both conditionality and conditional probability in general. It is not offered as a fully developed theory of conditionality but can be used, we claim, to explain why calculations according to the RATIO scheme does not coincide with our intuitive notion of conditional probability. What it shows us is that for an understanding of the whole range of conditionals we will need what John Heil (2003), in response to Quine (1953), calls an ontological point of view. (shrink)

Starting from a recent paper by S. Kaufmann, we introduce a notion of conjunction of two conditional events and then we analyze it in the setting of coherence. We give a representation of the conjoined conditional and we show that this new object is a conditional random quantity, whose set of possible values normally contains the probabilities assessed for the two conditional events. We examine some cases of logical dependencies, where the conjunction is a conditional event; moreover, we give the (...) lower and upper bounds on the conjunction. We also examine an apparent paradox concerning stochastic independence which can actually be explained in terms of uncorrelation. We briefly introduce the notions of disjunction and iterated conditioning and we show that the usual probabilistic properties still hold. (shrink)

Jonathan Weisberg (2010 ) argues that, given that life exists, the fact that the universe is fine-tuned for life does not confirm the design hypothesis. And if the fact that life exists confirms the design hypothesis, fine-tuning is irrelevant. So either way, fine-tuning has nothing to do with it. I will defend a design argument that survives Weisberg’s critique — the fact that life exists supports the design hypothesis, but it only does so given fine-tuning.