Algorithmic probability has shown some promise in dealing with the probability problem in the Everett interpretation, since it provides an objective, single-case probability measure. Many find the Everettian cosmology to be overly extravagant, however, and algorithmic probability has also provided improved models of subjective probability and Bayesian reasoning. I attempt here to generalize algorithmic Everettianism to more Bayesian and subjectivist interpretations. I present a general framework for applying generative probability, of which algorithmic probability can be considered a special case. I apply this framework to two commonly vexing thought experiments that have immediate application to quantum probability: the Sleeping Beauty and Replicator experiments.
Algorithmic probability has shown some promise in dealing with the probability problem in the Everett interpretation, since it provides an objective, single-case probability measure. Many find the Everettian cosmology to be overly extravagant, however, and algorithmic probability has also provided improved models of subjective probability and Bayesian reasoning. I attempt here to generalize algorithmic Everettianism to more Bayesian and subjectivist interpretations. I present a general framework for applying generative probability, of which algorithmic probability can be considered a special case. I apply this framework to two commonly vexing thought experiments that have immediate application to quantum probability: the Sleeping Beauty and Replicator experiments.
The Everett (or relative-state, or many-worlds) interpretation of quantum mechanics has come under fire for inadequately dealing with the Born rule (the formula for calculating quantum probabilities). Numerous attempts have been made to derive this rule from the perspective of observers within the quantum wavefunction. These are not really analytic proofs, but are rather attempts to derive the Born rule as a synthetic a priori necessity, given the nature of human observers (a fact not fully appreciated even by all of those who have attempted such proofs). I show why existing attempts are unsuccessful or only partly successful, and postulate that Solomonoff's algorithmic approach to the interpretation of probability theory could clarify the problems with these approaches. The Sleeping Beauty probability puzzle is used as a springboard from which to deduce an objectivist, yet synthetic a priori framework for quantum probabilities, that properly frames the role of self-location and self-selection (anthropic) principles in probability theory. I call this framework "algorithmic synthetic unity" (or ASU). I offer no new formal proof of the Born rule, largely because I feel that existing proofs (particularly that of Gleason) are already adequate, and as close to being a formal proof as one should expect or want. Gleason's one unjustified assumption--known as noncontextuality--is, I will argue, completely benign when considered within the algorithmic framework that I propose. I will also argue that, to the extent the Born rule can be derived within ASU, there is no reason to suppose that we could not also derive all the other fundamental postulates of quantum theory, as well. There is nothing special here about the Born rule, and I suggest that a completely successful Born rule proof might only be possible once all the other postulates become part of the derivation. As a start towards this end, I show how we can already derive the essential content of the fundamental postulates of quantum mechanics, at least in outline, and especially if we allow some educated and well-motivated guesswork along the way. The result is some steps towards a coherent and consistent algorithmic interpretation of quantum mechanics.
Realist, no-collapse interpretations of quantum mechanics, such as Everett's, face the probability problem: how to justify the norm-squared (Born) rule from the wavefunction alone. While any basis-independent measure can only be norm-squared (due to the Gleason-Busch Theorem) this fact conflicts with various popular, non-wavefunction-based phenomenological measures - such as observer, outcome or world counting - that are frequently demanded of Everettians. These alternatives conflict, however, with the wavefunction realism upon which Everett's approach rests, which seems to call for an objective, basis-independent measure based only on wavefunction amplitudes. The ability of quantum probabilities to destructively interfere with each other, however, makes it difficult to see how probabilities can be derived solely from amplitudes in an intuitively appealing way. I argue that the use of algorithmic probability can solve this problem, since the objective, single-case probability measure that wavefunction realism demands is exactly what algorithmic information theory was designed to provide. The result is an intuitive account of complex-valued amplitudes, as coefficients in an optimal lossy data compression, such that changes in algorithmic information content (entropy deltas) are associated with phenomenal transitions.
The Everett interpretation of quantum mechanics is an increasingly popular alternative to the traditional Copenhagen interpretation, but there are a few major issues that prevent the widespread adoption. One of these issues is the origin of probabilities in the Everett interpretation, which this thesis will attempt to survey. The most successful resolution of the probability problem thus far is the decision-theoretic program, which attempts to frame probabilities as outcomes of rational decision making. This marks a departure from orthodox interpretations of probabilities in the physical sciences, where probabilities are thought to be objective, stemming from symmetry considerations. This thesis will attempt to offer evaluations on the decision-theoretic program.
At the intersection of the foundations of statistics and the foundations of quantum theory, one comes across a hotly-debated question: what are the probabilities in quantum theories? Do they all reflect the subjective opinions of various agents, or do some of them reflect objective features of the world? One answer—Lewisian dualism, a view that takes some quantum probabilities to be subjective and others to be objective—has gained near-consensus among philosophers of science. However, our most sophisticated defense of this view (due to Deutsch, Wallace, Saunders, Myrvold, Greaves, and others) has two significant problems. First, it requires a very particular approach to the measurement problem (namely, the Everett interpretation). Second, it relies on the pragmatic goals of scientists (e.g., their aim to avoid losses in bets) rather than their epistemic goals (e.g., their aim to accurately describe the world). I give a new defense of dualism that rectifies both of these problems. I generalize the Deutsch-Wallace argument to defend the objective probabilities in a host of realist approaches to measurement, and I modify the accuracy-dominance argument for probabilism to defend the subjective probabilities in these approaches. I also critically examine several previous attempts to interpret all quantum probabilities subjectively.
This paper analyses an attempt to reintroduce the probabilistic predictions of quantum mechanics into the Everett interpretation. An overview of the measurement problem and the Everett interpretation are first given, with particular attention paid to the issue of probability within a deterministic theory. An argument is put forth that observers in an Everettian multiverse generically experience a period of post-measurement uncertainty in regards to their location. It is further argued that this uncertainty allows pre-measurement to observers to act as if they are uncertain about future outcomes. This is then the justification for attaching probabilities to the squared moduli of the wave function. It is also noted that there is an assumption of typicality present in discussions of probability within Everett and some arguments to justify this assumption are considered.