Logical Prior Probability

Demski, Abram

doi:10.1007/978-3-642-35506-6_6

Cited by 10 publications

(14 citation statements)

References 6 publications

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“…The first formal statement of Desideratum 7 that we know of is given by Demski [9], though it is implicitly assumed whenever asking for a set of beliefs that can reason accurately about arbitrary arithmetical claims (as is done by, e.g., Savage [44] and Hacking [26] Proposed by Hintikka [30], Desideratum 10 is popular among epistemic logicians. This desideratum has been formalized in many different ways; see [8,5] for a sample.…”

Section: Desideratum 7 (Uniform Non-dogmatism) a Good Reasoner Shoulmentioning

confidence: 99%

A Formal Approach to the Problem of Logical Non-Omniscience

Garrabrant

Benson-Tilsen

Critch

et al. 2017

Electron. Proc. Theor. Comput. Sci.

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We present the logical induction criterion for computable algorithms that assign probabilities to every logical statement in a given formal language, and refine those probabilities over time. The criterion is motivated by a series of stock trading analogies. Roughly speaking, each logical sentence phi is associated with a stock that is worth $1 per share if phi is true and nothing otherwise, and we interpret the belief-state of a logically uncertain reasoner as a set of market prices, where pt_N(phi)=50% means that on day N, shares of phi may be bought or sold from the reasoner for 50%. A market is then called a logical inductor if (very roughly) there is no polynomial-time computable trading strategy with finite risk tolerance that earns unbounded profits in that market over time. We then describe how this single criterion implies a number of desirable properties of bounded reasoners; for example, logical inductors outpace their underlying deductive process, perform universal empirical induction given enough time to think, and place strong trust in their own reasoning process.Comment: In Proceedings TARK 2017, arXiv:1707.0825

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Section: Desideratum 7 (Uniform Non-dogmatism) a Good Reasoner Shoulmentioning

confidence: 99%

A Formal Approach to the Problem of Logical Non-Omniscience

Garrabrant

Benson-Tilsen

Critch

et al. 2017

Electron. Proc. Theor. Comput. Sci.

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“…Outside the field of online learning, our work has interesting parallels in the field of mathematical logic. Hutter et al [21] and Demski [22] study the problem of assigning probabilities to sentences in logic while respecting certain relationships between them, a practice that dates back to Gaifman [23]. Because sentences in mathematical logic are expressive enough to make claims about the behavior of computations (such as "this computation will use less memory than that one"), their work can be seen as a different approach to the problems we discuss in this paper.…”

Section: Related Workmentioning

confidence: 99%

Asymptotic Convergence in Online Learning with Unbounded Delays

Garrabrant,

Soares,

Taylor

2016

Preprint

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We study the problem of predicting the results of computations that are too expensive to run, via the observation of the results of smaller computations. We model this as an online learning problem with delayed feedback, where the length of the delay is unbounded, which we study mainly in a stochastic setting. We show that in this setting, consistency is not possible in general, and that optimal forecasters might not have average regret going to zero. However, it is still possible to give algorithms that converge asymptotically to Bayesoptimal predictions, by evaluating forecasters on specific sparse independent subsequences of their predictions. We give an algorithm that does this, which converges asymptotically on good behavior, and give very weak bounds on how long it takes to converge. We then relate our results back to the problem of predicting large computations in a deterministic setting.

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“…Gibbs sampling [CG92,Ber04], are directly built for the purpose of sampling from a posterior distribution. A recent article, Logical Prior Probability [Dem12], samples sentences from a first order theory for the purpose of approximating a universal distribution over sequences.…”

Section: Overview Of the Frameworkmentioning

confidence: 99%

Learning Agents with Evolving Hypothesis Classes

Sunehag

Hutter

2013

Artificial General Intelligence

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It has recently been shown that a Bayesian agent with a universal hypothesis class resolves most induction problems discussed in the philosophy of science. These ideal agents are, however, neither practical nor a good model for how real science works. We here introduce a framework for learning based on implicit beliefs over all possible hypotheses and limited sets of explicit theories sampled from an implicit distribution represented only by the process by which it generates new hypotheses. We address the questions of how to act based on a limited set of theories as well as what an ideal sampling process should be like. Finally, we discuss topics in philosophy of science and cognitive science from the perspective of this framework.

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Logical Prior Probability

Cited by 10 publications

References 6 publications

A Formal Approach to the Problem of Logical Non-Omniscience

A Formal Approach to the Problem of Logical Non-Omniscience

Asymptotic Convergence in Online Learning with Unbounded Delays

Learning Agents with Evolving Hypothesis Classes

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