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)

In this paper, a metaphysics is proposed that includes everything that can be represented by a well-founded multiset. It is shown that this metaphysics, apart from being self-explanatory, is also benevolent. Paradoxically, it turns out that the probability that we were born in another life than our own is zero. More insights are gained by inducing properties from a metaphysics that is not self-explanatory. In particular, digital metaphysics is analyzed, which claims that only computable things exist. First of all, it (...) is shown that digital metaphysics contradicts itself by leading to the conclusion that the shortest computer program that computes the world is infinitely long. This means that the Church-Turing conjecture must be false. Secondly, the applicability of Occam’s razor is explained by evolution: in an evolving physics it can appear at each moment as if the world is caused by only finitely many things. Thirdly and most importantly, this metaphysics is benevolent in the sense that it organizes itself to fulfill the deepest wishes of its observers. Fourthly, universal computers with an infinite memory capacity cannot be built in the world. And finally, all the properties of the world, both good and bad, can be explained by evolutionary conservation. (shrink)

Automated reasoning about uncertain knowledge has many applications. One difficulty when developing such systems is the lack of a completely satisfactory integration of logic and probability. We address this problem directly. Expressive languages like higher-order logic are ideally suited for representing and reasoning about structured knowledge. Uncertain knowledge can be modeled by using graded probabilities rather than binary truth-values. The main technical problem studied in this paper is the following: Given a set of sentences, each having some probability of being (...) true, what probability should be ascribed to other (query) sentences? A natural wish-list, among others, is that the probability distribution (i) is consistent with the knowledge base, (ii) allows for a consistent inference procedure and in particular (iii) reduces to deductive logic in the limit of probabilities being 0 and 1, (iv) allows (Bayesian) inductive reasoning and (v) learning in the limit and in particular (vi) allows confirmation of universally quantified hypotheses/sentences. We translate this wish-list into technical requirements for a prior probability and show that probabilities satisfying all our criteria exist. We also give explicit constructions and several general characterizations of probabilities that satisfy some or all of the criteria and various (counter) examples. We also derive necessary and sufficient conditions for extending beliefs about finitely many sentences to suitable probabilities over all sentences, and in particular least dogmatic or least biased ones. We conclude with a brief outlook on how the developed theory might be used and approximated in autonomous reasoning agents. Our theory is a step towards a globally consistent and empirically satisfactory unification of probability and logic. (shrink)

Tight correlations between mental states and brain states have been observed time and again within the ethology of biologically ingrained animal behaviors, the comparative psychology of animal minds, the evolutionary psychology of mental adaptations, the behavioral genetics of inherited mental traits, the developmental psychology of the maturing mind, the psychopharmacology of mind-altering substances, and cognitive neuroscience more generally. They imply that our mental lives are only made possible because of brain activity—that having a functioning brain is a necessary condition for (...) having conscious experiences. This mind-brain dependence thesis yields predictions that have been spectacularly confirmed by the evidence, whereas its antithesis has been massively disconfirmed—at least so long as it is not formulated so vaguely as to yield no specific empirical consequences at all. Dualists are thus forced to make a difficult choice: either retain a belief in personal survival at the expense of ignoring or dismissing the implications of our best evidence, or accept those implications at the expense of acknowledging that the prospects for personal survival are extremely dim. -/- 1. Introduction: A Probabilistic Argument -- 2. Scientific Reasoning I: Mill’s Methods and Modern Epidemiological Principles - 2.1 When Correlation Indicates Causation -- 3. Scientific Reasoning II: Inference to the Best Explanation, Testability, and Predictive Success - 3.1 Parsimony: Tempering Explanations to Minimize False Beliefs - 3.2 Plausibility: Harmonizing Explanations with Background Knowledge - 3.3 Testability and Confirmation: Checking Hypotheses Against the Data - 3.3.1 Confirmation of the Dependence Thesis from Failed Attempts to Falsify It - 3.3.2 Confirmation of the Dependence Thesis from the Falsification of its Rival - 3.4 Scope: Bringing Disparate Facts Together - 3.5 Fertility: Generating New Lines of Research - 3.6 Overview -- 4. Scientific Reasoning III: Bayesian and Information-Theoretic Approaches - 4.1 Bayesian Probability - 4.2 Bayesian Confirmation Theory - 4.3 A Bayesian Approach to the Dependence and Independence Theses - 4.4 The Probabilistic Price of Ad Hoc Accommodation: Simplicity vs. Goodness of Fit - 4.4.1 Simplicity Reflected in the Likelihood - 4.4.2 Simplicity Reflected in the Prior Probability - 4.5 Prediction vs. Accommodation - 4.6 Bayesian Analysis of the Dependence and Independence Theses -- 5. The Dualist’s Dilemma: Reject Science, or Reject Personal Survival? - 5.1 What’s Left for a Soul to Do? -- 6. Conclusion. (shrink)

Early work on the frequency theory of probability made extensive use of the notion of randomness, conceived of as a property possessed by disorderly collections of outcomes. Growing out of this work, a rich mathematical literature on algorithmic randomness and Kolmogorov complexity developed through the twentieth century, but largely lost contact with the philosophical literature on physical probability. The present chapter begins with a clarification of the notions of randomness and probability, conceiving of the former as a property of a (...) sequence of outcomes, and the latter as a property of the process generating those outcomes. A discussion follows of the nature and limits of the relationship between the two notions, with largely negative verdicts on the prospects for any reduction of one to the other, although the existence of an apparently random sequence of outcomes is good evidence for the involvement of a genuinely chancy process. (shrink)

Independence of Conditionals (IC) has recently been proposed as a basic rule for causal structure learning. If a Bayesian network represents the causal structure, its Conditional Probability Distributions (CPDs) should be algorithmically independent. In this paper we compare IC with causal faithfulness (FF), stating that only those conditional independences that are implied by the causal Markov condition hold true. The latter is a basic postulate in common approaches to causal structure learning. The common spirit of FF and IC is to (...) reject causal graphs for which the joint distribution looks ‘non-generic’. The difference lies in the notion of genericity: FF sometimes rejects models just because one of the CPDs is simple, for instance if the CPD describes a deterministic relation. IC does not behave in this undesirable way. It only rejects a model when there is a non-generic relation between different CPDs although each CPD looks generic when considered separately. Moreover, it detects relations between CPDs that cannot be captured by conditional independences. IC therefore helps in distinguishing causal graphs that induce the same conditional independences (i.e., they belong to the same Markov equivalence class). The usual justification for FF implicitly assumes a prior that is a probability density on the parameter space. IC can be justified by Solomonoff’s universal prior, assigning non-zero probability to those points in parameter space that have a finite description. In this way, it favours simple CPDs, and therefore respects Occam’s razor. Since Kolmogorov complexity is uncomputable, IC is not directly applicable in practice. We argue that it is nevertheless helpful, since it has already served as inspiration and justification for novel causal inference algorithms. (shrink)

Putnam construed the aim of Carnap’s program of inductive logic as the specification of a “universal learning machine,” and presented a diagonal proof against the very possibility of such a thing. Yet the ideas of Solomonoff and Levin lead to a mathematical foundation of precisely those aspects of Carnap’s program that Putnam took issue with, and in particular, resurrect the notion of a universal mechanical rule for induction. In this paper, I take up the question whether the Solomonoff–Levin proposal is (...) successful in this respect. I expose the general strategy to evade Putnam’s argument, leading to a broader discussion of the outer limits of mechanized induction. I argue that this strategy ultimately still succumbs to diagonalization, reinforcing Putnam’s impossibility claim. (shrink)

Developing new ways to estimate probabilities can be valuable for science, statistics, engineering, and other fields. By considering the information content of different output patterns, recent work invoking algorithmic information theory inspired arguments has shown that a priori probability predictions based on pattern complexities can be made in a broad class of input-output maps. These algorithmic probability predictions do not depend on a detailed knowledge of how output patterns were produced, or historical statistical data. Although quantitatively fairly accurate, a main (...) weakness of these predictions is that they are given as an upper bound on the probability of a pattern, but many low complexity, low probability patterns occur, for which the upper bound has little predictive value. Here, we study this low complexity, low probability phenomenon by looking at example maps, namely a finite state transducer, natural time series data, RNA molecule structures, and polynomial curves. Some mechanisms causing low complexity, low probability behaviour are identified, and we argue this behaviour should be assumed as a default in the real-world algorithmic probability studies. Additionally, we examine some applications of algorithmic probability and discuss some implications of low complexity, low probability patterns for several research areas including simplicity in physics and biology, a priori probability predictions, Solomonoff induction and Occam’s razor, machine learning, and password guessing. (shrink)

Of all the sub-disciplines of philosophy, the philosophy of science has perhaps the most privileged relationship to information theory. This relationship has been forged through a common interest in themes like induction, probability, confirmation, simplicity, non-ad hocness, unification and, more generally, ontology. It also has historical roots. One of the founders of algorithmic information theory, Ray Solomonoff, produced his seminal work on inductive inference as a direct result of grappling with problems first encountered as a student of the influential philosopher (...) of science Rudolf Carnap. There are other such historical connections between the two fields. Alas, there is no space to explore them here. Instead this essay will restrict its attention to a broad and accessible overview of the aforementioned common themes, which, given their nature, mandate an emphasis on AIT as opposed to general information theory. (shrink)