A Subjectivist’s Guide to Objective Chance

““ratbag idealism,”(23) Lewis (1994) discusses this worry and tries to offer a solution:”

“15 For more discussion about which propositions are admissible, the reader is referred to Lewis (1980). 16 The example is from van Fraassen (1984, p.”

“With Q being the ‘objective chance’ function, (*) becomes a version of a principle that, suitably finessed, becomes Lewis’ (1980) Principal Principle. Or one might argue, as van Fraassen (1995) does, that epistemic integrity requires one ideally to regard one’s future opinions as being trustworthy—perhaps because of t…”

“So if we want 3 Skyrms (1980) and Lewis (1980) defend Regularity in general, whereas Williamson (2007); Hájek (2010), and Easwaran (2010) argue that it cannot hold in all cases (even when infinitesimal probability values are allowed).”

“For example, David Lewis’s ‘Principal Principle’ says that a rational agent should set his subjective probability equal to the objective chance of an event, unless there is (inadmissible) information about what is actually going to happen (Lewis, 1986).”

“However, it is typically assumed that a rational agent should adopt as her subjective degree of belief objective probabilities (limit frequencies) where these are defined, see e.g., Lewis (1980).”

“11 For discussions of the Principal Principle, see Lewis (1986) and Hall (1994).”

“The most common Humean approach to lawhood is the best-systems view, which holds that laws of nature are simply the axioms of the best systematisation of the Humean mosaic, where the ‘best systematisation’ is supposed to be the one with the best combination of simplicity, robustness, and strength [53].”

“The de Finetti representation theorem shows that an agent whose degrees of belief make a sequence of coin tosses an exchangeable sequence will bet in exactly the same way as someone who believes that there is an objective chance, perhaps imperfectly known, for each toss to come up heads, and who has degrees of belief…”

“Perhaps the best known is David Lewis’s “Principal Principle” (Lewis 1980).”

“The classical source is (Lewis 1980).”

“Some philosophers insist in addition that a rational Pr-function must satisfy further constraints, such as, for instance, the Principal Principle (Lewis 1981), according to which one ought to set one’s credences to conform with known objective chances (with some caveats about admissible evidence).”

“…rational agent attempts to minimise his “losses” insofar as possible, by adopting the credence with the lowest expected inaccuracy (hence, being accurate is for the purposes of this paper simply a matter of not being inaccurate), as calculated by an accuracy measure (see Joyce 1998; Lewis 1980).”

“In fact, there seems to be only two such arguments in the literature: the so-called Laplacean demon argument due to Laplace (1814), and an argument due to Schaffer (2007). The Laplacean demon argument has been sufficiently criticised elsewhere (see Sober 1984, 2010), so that leaves Schaffer’s argument as the only tena…”

“This is also a reason that I think it’s worth exploring alternatives to Best System Analysis chances (Lewis 1980, 1994; Loewer 2001, 2004; Hoefer 2007): Best System probabilities sometimes depend on whatever the frequencies happen to be, without requiring that these frequencies have any causal…”

“aviour of repetitive sequences, rather than to that of individual events. Another individualist term that is very close in spirit to “propensity” is “(objective) chance.” The “Principal Principle” of Lewis (1980), while not defining chance, relates it to personal probability by requiring that, if You learn (don’t ask h…”

“The man needs to know that he is at that position now before he knows where he should go (Perry [1979]).”

“These additional constraints might be principles connecting credences to physical chances such as the Principal Principle (Lewis, 1980), principles connecting credences to symmetry considerations such as the Principle of Indifference (Laplace, 1951; Keynes, 1921) and the Maximum Entropy Principle (Jaynes, 2003; Rosenk…”

“For instance, we have the famous PRINCIPAL PRINCIPLE from (Lewis 1980) and the NEW PRINCIPLE from (Hall 1994) and (Thau 1994) that tell us when to defer to chance.”

“Mutatis mutandis, this property can be seen as a form of the Principal Principle of Lewis [23], an important notion in the Philosophy of Probability that requires that degrees of beliefs be consistent with objective chances, and only with them (any other possible information on events becomes irrelevant once objective…”

“But these constraints lead to a problem: the counting anomaly, introduced by Lewis (1997). Suppose, with Lewis, that a (ridiculously(19)) large number N of distinguishable non-interacting particles are confined within some box.”

“7For instance, the (PP) was originally intended [7] to be applied to agents with “reasonable” — indeed, probabilistic — credence functions.”

“…situation of the universe (or the light-cone) at the time' (Miller, 1994, p. 186), the complete history of the world up till the time in question (Lewis, 1980, p. 99), 9 or 'a complete set of (nomically and/or causally) relevant conditions . . . which happens to be instantiated in that world at…”

“Lewis (1980, 1994) raises the prospect that on certain philosophical views about probability (including frequency views), T itself may be inadmissible in some circumstances, that is, that PT(A ∣TK) ≠ PT(A ∣K).”

“But although thereby the contradiction is evaded, many authors have argued that independently of the paradox it is unintuitive to say that the conjuncts separately can be rationally acceptable (see, among others, Kaplan (1981a, b), Stalnaker (1984), Lehrer (1990), M?her (1993), Ryan (1996), Nelkin (2000), and Douven (…”

“(Lewis 1980, p.92) This characterisation has given rise to controversy, and Lewis himself regarded it as too imprecise, and replaced it with a time-indexed version, in part in order to be able to say that all past events have chance 0 or 1.”

“Recall the idea of Lewis (1980) of indexing chances to times, as carried by his notation P tw ðAÞ, ''the chance, at time t and world w, of A's holding''.”

“This is (in a sense) the complement to Lewis' [21] principal principle that (in the terminology we are using) subjective probability should be constrained by objective probability. be used for establishing the viability of Everettian QM mechanics thus must rely on the assumption that no probabilistic implications of E…”

“12 The original version of PP is more general in that the condition can include additional information as long as it’s admissible (Lewis 1986).”

“Moreover, I will assume that something like Lewis’s Principal Principle holds (Lewis 1980).”

“Perhaps what this mixed state amounts to is simply a credential state that aggregates these several credential states, by taking a weighted average of their assignments (Lewis, 1980): this would reduce the sort of unclarity in this case to simple uncertainty.”

“For example, Humeans of the best-systems school hold that the laws of nature are simply the axioms of the systematisation of the Humean mosaic which best combines simplicity and strength (the term ‘Humean mosaic’ refers to the set of all local matters of particular fact within a given possible world, including things…”

“Graded beliefs are usually conceived as subjective probabilities (Jeffrey, 1965; Howson & Urbach, 1989; Earman, 1992; Oaksford & Chater, 2007), though they can also reflect objective chances (Lewis, 1980).”

“n: a more general statement is that an agent’s subjective probability in X conditional on the objective probability of X being pshould in turn be p(this is known as the Principal Principle, following Lewis (1980)). My main point stands, however: the Principal Principle is postulated, not derived. 5 seems, at least, th…”

“…for striking phenomena does not 4 I do not have a worked-out account of the relevant interpretation of probability, but it seems the notion of probability relevant here is objective chance of the sort Lewis (1981) discusses and connects with subjective credences via his Principal Principle.”

“Non-Archimedean probability theory has been used to generate probability functions on infinite (and even uncountable) sample spaces that reflect the distinction between possibility and impossibility (Lewis, 1980).”

“…the following rule, sometimes called Miller’s Principle after Miller (1966): C(e|h) = Ph(e), where Ph(·) is the physical probability distribution posited by the hypothesis h. Miller’s Principle is a particularly simple version of the rule; a more sophisticated version is worked out in Lewis (1980).”

“…having the best combination of simplicity and informativeness regarding the entire history of instantiations of all properties of an elite sort: the natural, categorical, nonhaecceitistic properties possessed intrinsically by spatiotemporal points or occupants thereof (Lewis 1973, 73; 1986; 1999).”

“That is bad gambling behavior, and violates Lewis’ Principle Principle (Lewis, 1980) (you decide which is worse).”

“For another example, the sort of account of objective chance offered by Lewis (1980) might best be understood as an account of the role that beliefs about chances play in our cognitive economy.”

“For example, the Humean will have to account for chancy laws, which requires adding a third virtue like fit to measure how well a probabilistic system connects to a world (Lewis 1980, 1994).”

“From this principle we can derive the inferential link also, at least in simple cases (see (Lewis 1980, pp. 106–108)).”