Indifference Principles

In standard probability theory, probability zero is not the same as impossibility. But many have suggested that only impossible events should have probability zero. This can be arranged if we allow infinitesimal probabilities, but infinitesimals do not solve all of the problems. We will see that regular probabilities are not invariant over rigid transformations, even for simple, bounded, countable, constructive, and disjoint sets. Hence, regular chances cannot be determined by space-time invariant physical laws, and regular credences cannot satisfy seemingly reasonable (...) symmetry principles. Moreover, the examples here are immune to the objections against Williamson’s infinite coin flips. (shrink)

Our universe seems to be miraculously fine-tuned for life. Multiverse theories have been proposed as an explanation for this on the basis of probabilistic arguments, but various authors have objected that we should consider our total evidence that this universe in particular has life in our inference, which would block the argument. The debate thus crucially hinges on how Bayesian background and evidence are distinguished and on how indexical or demonstrative terms are analysed. The aim of this article is to (...) take a step back and examine these various aspects of Bayesian reasoning and how they affect the arguments. The upshot is that there are reasons to resist the fine-tuning argument for the multiverse, but the “this-universe-objection” is not one of them. (shrink)

Many philosophers have been attracted to a restricted version of the principle of indifference in the case of self‐locating belief. Roughly speaking, this principle states that, within any given possible world, one should be indifferent between different hypotheses concerning who one is within that possible world, so long as those hypotheses are compatible with one's evidence. My first goal is to defend a more precise version of this principle. After responding to several existing criticisms of such a principle, I argue (...) that existing formulations of the principle are crucially ambiguous, and I go on to defend a particular disambiguation of the principle. According to the disambiguation I defend, how we should apply this restricted principle of indifference sensitively depends on our background metaphysical beliefs. My second goal is to apply this disambiguated principle to classical skeptical problems in epistemology. In particular, I argue that Eternalism threatens to lead us to external world skepticism, and Modal Realism threatens to lead us to inductive skepticism. (shrink)

Brentano never published on Bertrand’s paradox but claimed to have a solution. Adrian Maître has recovered from the Franz Brentano Archive Brentano’s remarks on his solution. They do not give us a worked demonstration of his solution but only an incomplete and in places obscure justification of it. Here I attempt to identify his solution, to explain what seem to me the clearly discernible parts of his justification and to discuss the extent to which the justification succeeds in the light (...) of current work on Bertrand’s paradox. (shrink)

One popular approach to statistical mechanics understands statistical mechanical probabilities as measures of rational indifference. Naive formulations of this ``indifference approach'' face reversibility worries - while they yield the right prescriptions regarding future events, they yield the wrong prescriptions regarding past events. This paper begins by showing how the indifference approach can overcome the standard reversibility worries by appealing to the Past Hypothesis. But, the paper argues, positing a Past Hypothesis doesn't free the indifference approach from all reversibility worries. For (...) while appealing to the Past Hypothesis allows it to escape one kind of reversibility worry, it makes it susceptible to another - the Meta-Reversibility Objection. And there is no easy way for the indifference approach to escape the Meta-Reversibility Objection. As a result, reversibility worries pose a steep challenge to the viability of the indifference approach. (shrink)

Events between which we have no epistemic reason to discriminate have equal epistemic probabilities. Bertrand’s chord paradox, however, appears to show this to be false, and thereby poses a general threat to probabilities for continuum sized state spaces. Articulating the nature of such spaces involves some deep mathematics and that is perhaps why the recent literature on Bertrand’s Paradox has been almost entirely from mathematicians and physicists, who have often deployed elegant mathematics of considerable sophistication. At the same time, the (...) philosophy of probability has been left out. In particular, left out entirely are the philosophical ground of the principle of indifference, the nature of the principle itself, the stringent constraint this places on the mathematical representation of the principle needed for its application to continuum sized event spaces, and what these entail for rigour in developing the paradox itself. This book puts the philosophy and its entailments back in and in so doing casts a new light on the paradox, giving original analyses of the paradox, its possible solutions, the source of the paradox, the philosophical errors we make in attempting to solve it and what the paradox proves for the philosophy of probability. The book finishes with the author’s proposed solution—a solution in the spirit of Bertrand’s, indeed—in which an epistemic principle more general than the principle of indifference offers a principled restriction of the domain of the principle of indifference. (shrink)

It is a consequence of the theory of imprecise credences that there exist situations in which rational agents inevitably become less opinionated toward some propositions as they gather more evidence. The fact that an agent's imprecise credal state can dilate in this way is often treated as a strike against the imprecise approach to inductive inference. Here, we show that dilation is not a mere artifact of this approach by demonstrating that opinion loss is countenanced as rational by a substantially (...) broader class of normative theories than has been previously recognised. Specifically, we show that dilation-like phenomena arise even when one abandons the basic assumption that agents have (precise or imprecise) credences of any kind, and follows directly from bedrock norms for rational comparative confidence judgements of the form `I am at least as confident in p as I am in q'. We then use the comparative confidence framework to develop a novel understanding of what exactly gives rise to dilation-like phenomena. By considering opinion loss in this more general setting, we are able to provide a novel assessment of the prospects for an account of inductive inference that is not saddled with the inevitability of rational opinion loss. (shrink)

An indifference principle says that your credences should be distributed uniformly over each of the possibilities you recognise. A chance deference principle says that your credences should be aligned with the chances. My thesis is that, if we are anti-Humeans about chance, then these two principles are incompatible. Anti-Humeans think that it is possible for the actual frequencies to depart from the chances. So long as you recognise possibilities like this, you cannot both spread your credences evenly and defer to (...) the chances. I discuss some weaker forms of indifference which will allow anti-Humeans to defer to the chances. (shrink)

How do you make decisions under ignorance? That is, how do you decide when you lack subjective probabilities for some of your options’ possible outcomes? One answer is that you follow the Laplace Rule: you assign an equal probability to each state of nature for which you lack a subjective probability (that is, you use the Principle of Indifference) and then you maximize expected utility. The most influential objection to the Laplace Rule is that it is sensitive to the individuation (...) of states of nature. This sensitivity is problematic because the individuation of states seems arbitrary. In this paper, however, I argue that this objection proves too much. I argue that all plausible rules for decisions under ignorance are sensitive to the individuation of states of nature. (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)

The principle of indifference states that in the absence of any relevant evidence, a rational agent will distribute their credence equally among all the possible outcomes under consideration. Despite its intuitive plausibility, PI famously falls prey to paradox, and so is widely rejected as a principle of ideal rationality. In this article, I present a novel rehabilitation of PI in terms of the epistemology of comparative confidence judgments. In particular, I consider two natural comparative reformulations of PI and argue that (...) while one of them prescribes the adoption of patently irrational epistemic states, the other provides a consistent formulation of PI that overcomes the most salient limitations of existing formulations. (shrink)

If the laws of nature are as the Humean believes, it is an unexplained cosmic coincidence that the actual Humean mosaic is as extremely regular as it is. This is a strong and well-known objection to the Humean account of laws. Yet, as reasonable as this objection may seem, it is nowadays sometimes dismissed. The reason: its unjustified implicit assignment of equiprobability to each possible Humean mosaic; that is, its assumption of the principle of indifference, which has been attacked on (...) many grounds ever since it was first proposed. In place of equiprobability, recent formal models represent the doxastic state of total ignorance as suspension of judgment. In this paper I revisit the cosmic coincidence objection to Humean laws by assessing which doxastic state we should endorse. By focusing on specific features of our scenario I conclude that suspending judgment results in an unnecessarily weak doxastic state. First, I point out that recent literature in epistemology has provided independent justifications of the principle of indifference. Second, given that the argument is framed within a Humean metaphysics, it turns out that we are warranted to appeal to these justifications and assign a uniform and additive credence distribution among Humean mosaics. This leads us to conclude that, contrary to widespread opinion, we should not dismiss the cosmic coincidence objection to the Humean account of laws. (shrink)

We develop a Bayesian framework for thinking about the way evidence about the here and now can bear on hypotheses about the qualitative character of the world as a whole, including hypotheses according to which the total population of the world is infinite. We show how this framework makes sense of the practice cosmologists have recently adopted in their reasoning about such hypotheses.

Many theorists have proposed that we can use the principle of indifference to defeat the inductive sceptic. But any such theorist must confront the objection that different ways of applying the principle of indifference lead to incompatible probability assignments. Huemer offers the explanatory priority proviso as a strategy for overcoming this objection. With this proposal, Huemer claims that we can defend induction in a way that is not question-begging against the sceptic. But in this article, I argue that the opposite (...) is true: if anything, Huemer’s use of the principle of indifference supports the rationality of inductive scepticism. (shrink)

It is well known that there are, at least, two sorts of cases where one should not prefer a direct inference based on a narrower reference class, in particular: cases where the narrower reference class is gerrymandered, and cases where one lacks an evidential basis for forming a precise-valued frequency judgment for the narrower reference class. I here propose (1) that the preceding exceptions exhaust the circumstances where one should not prefer direct inference based on a narrower reference class, and (...) (2) that minimal frequency information for a narrower (non-gerrymandered) reference class is sufficient to yield the defeat of a direct inference for a broader reference class. By the application of a method for inferring relatively informative expected frequencies, I argue that the latter claim does not result in an overly incredulous approach to direct inference. The method introduced here permits one to infer a relatively informative expected frequency for a reference class R', given frequency information for a superset of R' and/or frequency information for a sample drawn from R'. (shrink)

Einstein wrote his famous sentence "God does not play dice with the universe" in a letter to Max Born in 1920. All experiments have confirmed that quantum mechanics is neither wrong nor “incomplete”. One can says that God does play dice with the universe. Let quantum mechanics be granted as the rules generalizing all results of playing some imaginary God’s dice. If that is the case, one can ask how God’s dice should look like. God’s dice turns out to be (...) a qubit and thus having the shape of a unit ball. Any item in the universe as well the universe itself is both infinitely many rolls and a single roll of that dice for it has infinitely many “sides”. Thus both the smooth motion of classical physics and the discrete motion introduced in addition by quantum mechanics can be described uniformly correspondingly as an infinite series converges to some limit and as a quantum jump directly into that limit. The second, imaginary dimension of God’s dice corresponds to energy, i.e. to the velocity of information change between two probabilities in both series and jump. (shrink)

I present an argument against the thesis of Uniqueness and in favour of Permissivism. Counterexamples to Uniqueness are provided, based on ‘Safespot’ propositions – i.e. a proposition that is guaranteed to be true provided the subject adopts a certain attitude towards it. The argument relies on a plausible principle: (roughly stated) If S knows that her believing p would be a true belief, then it is rationally permitted for S to believe p. One motivation for denying this principle – viz. (...) opposition to ‘epistemic consequentialism’ – is briefly discussed. The principle is extended to cover degrees of belief and compared with a couple of other well-known constraints on rational degrees of belief. (shrink)

The _Principle of Indifference_ was once regarded as a linchpin of probabilistic reasoning, but has now fallen into disrepute as a result of the so-called _problem of multiple of partitions_. In ‘Evidential symmetry and mushy credence’ Roger White suggests that we have been too quick to jettison this principle and argues that the problem of multiple partitions rests on a mistake. In this paper I will criticise White’s attempt to revive POI. In so doing, I will argue that what underlies (...) the problem of multiple partitions is a fundamental tension between POI and the very idea of _evidential incomparability_. (shrink)

Sometimes different partitions of the same space each seem to divide that space into propositions that call for equal epistemic treatment. Famously, equal treatment in the form of equal point-valued credence leads to incoherence. Some have argued that equal treatment in the form of equal interval-valued credence solves the puzzle. This paper shows that, once we rule out intervals with extreme endpoints, this proposal also leads to incoherence.

Bayesian epistemology tells us with great precision how we should move from prior to posterior beliefs in light of new evidence or information, but says little about where our prior beliefs come from. It offers few resources to describe some prior beliefs as rational or well-justified, and others as irrational or unreasonable. A different strand of epistemology takes the central epistemological question to be not how to change one’s beliefs in light of new evidence, but what reasons justify a given (...) set of beliefs in the first place. We offer an account of rational belief formation that closes some of the gap between Bayesianism and its reason-based alternative, formalizing the idea that an agent can have reasons for his or her (prior) beliefs, in addition to evidence or information in the ordinary Bayesian sense. Our analysis of reasons for belief is part of a larger programme of research on the role of reasons in rational agency (Dietrich and List, Nous, 2012a, in press; Int J Game Theory, 2012b, in press). (shrink)

Sometimes we learn what the world is like, and sometimes we learn where in the world we are. Are there any interesting differences between the two kinds of cases? The main aim of this article is to argue that learning where we are in the world brings into view the same kind of observation selection effects that operate when sampling from a population. I will first explain what observation selection effects are ( Section 1 ) and how they are relevant (...) to learning where we are in the world ( Section 2 ). I will show how measurements in the Many Worlds Interpretation of quantum mechanics can be understood as learning where you are in the world via some observation selection effect ( Section 3 ). I will apply a similar argument to the Sleeping Beauty Problem ( Section 4 ) and explain what I take the significance of the analogy to be ( Section 5 ). Finally, I will defend the Restricted Principle of Indifference on which some of my arguments depend ( Section 6 ). (shrink)

This paper examines three accounts of the sleeping beauty case: an account proposed by Adam Elga, an account proposed by David Lewis, and a third account defended in this paper. It provides two reasons for preferring the third account. First, this account does a good job of capturing the temporal continuity of our beliefs, while the accounts favored by Elga and Lewis do not. Second, Elga’s and Lewis’ treatments of the sleeping beauty case lead to highly counterintuitive consequences. The proposed (...) account also leads to counterintuitive consequences, but they’re not as bad as those of Elga’s account, and no worse than those of Lewis’ account. (shrink)

The principle of indifference is supposed to suffice for the rational assignation of probabilities to possibilities. Bertrand advances a probability problem, now known as his paradox, to which the principle is supposed to apply; yet, just because the problem is ill‐posed in a technical sense, applying it leads to a contradiction. Examining an ambiguity in the notion of an ill‐posed problem shows that there are precisely two strategies for resolving the paradox: the distinction strategy and the well‐posing strategy. The main (...) contenders for resolving the paradox, Marinoff and Jaynes, offer solutions which exemplify these two strategies. I show that Marinoff’s attempt at the distinction strategy fails, and I offer a general refutation of this strategy. The situation for the well‐posing strategy is more complex. Careful formulation of the paradox within measure theory shows that one of Bertrand’s original three options can be ruled out but also shows that piecemeal attempts at the well‐posing strategy will not succeed. What is required is an appeal to general principle. I show that Jaynes’s use of such a principle, the symmetry requirement, fails to resolve the paradox; that a notion of metaindifference also fails; and that, while the well‐posing strategy may not be conclusively refutable, there is no reason to think that it can succeed. So the current situation is this. The failure of Marinoff’s and Jaynes’s solutions means that the paradox remains unresolved, and of the only two strategies for resolution, one is refuted and we have no reason to think the other will succeed. Consequently, Bertrand’s paradox continues to stand in refutation of the principle of indifference. (shrink)

In a recent article, Adam Elga outlines a strategy for “Defeating Dr Evil with Self-Locating Belief”. The strategy relies on an indifference principle that is not up to the task. In general, there are two things to dislike about indifference principles: adopting one normally means confusing risk for uncertainty, and they tend to lead to incoherent views in some ‘paradoxical’ situations. I argue that both kinds of objection can be levelled against Elga’s indifference principle. There are also some difficulties with (...) the concept of evidence that Elga uses, and these create further difficulties for the principle. (shrink)

De Finetti would claim that we can make sense of a draw in which each positive integer has equal probability of winning. This requires a uniform probability distribution over the natural numbers, violating countable additivity. Countable additivity thus appears not to be a fundamental constraint on subjective probability. It does, however, seem mandated by Dutch Book arguments similar to those that support the other axioms of the probability calculus as compulsory for subjective interpretations. These two lines of reasoning can be (...) reconciled through a slight generalization of the Dutch Book framework. Countable additivity may indeed be abandoned for de Finetti's lottery, but this poses no serious threat to its adoption in most applications of subjective probability. Introduction The de Finetti lottery Two objections to equiprobability 3.1 The ‘No random mechanism’ argument 3.2 The Dutch Book argument Equiprobability and relative betting quotients The re-labelling paradox 5.1 The paradox 5.2 Resolution: from symmetry to relative probability Beyond the de Finetti lottery. (shrink)

Nick Bostrom argues that if we accept some plausible assumptions about how the future will unfold, we should believe we are probably not humans. The argument appeals crucially to an indifference principle whose precise content is a little unclear. I set out four possible interpretations of the principle, none of which can be used to support Bostrom’s argument. On the first two interpretations the principle is false, on the third it does not entail the conclusion, and on the fourth it (...) only entails the conclusion given an auxiliary hypothesis that we have no reason to believe. (shrink)

The logical interpretation of probability, or "objective Bayesianism'' – the theory that (some) probabilities are strictly logical degrees of partial implication – is defended. The main argument against it is that it requires the assignment of prior probabilities, and that any attempt to determine them by symmetry via a "principle of insufficient reason" inevitably leads to paradox. Three replies are advanced: that priors are imprecise or of little weight, so that disagreement about them does not matter, within limits; that it (...) is possible to distinguish reasonable from unreasonable priors on logical grounds; and that in real cases disagreement about priors can usually be explained by differences in the background information. It is argued also that proponents of alternative conceptions of probability, such as frequentists, Bayesians and Popperians, are unable to avoid committing themselves to the basic principles of logical probability. (shrink)