Probabilities on Sentences in an Expressive Logic

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.Comment: 52 LaTeX pages, 64 definiton/theorems/etc, presented at conference Progic 2011 in New Yor

Single-qubit lasing and cooling at the Rabi frequency

For a superconducting qubit driven to perform Rabi oscillations and coupled to a slow electromagnetic or nano-mechanical oscillator we describe previously unexplored quantum optics effects. When the Rabi frequency is tuned to resonance with the oscillator the latter can be driven far from equilibrium. Blue detuned driving leads to a population inversion in the qubit and a bi-stability with lasing behavior of the oscillator; for red detuning the qubit cools the oscillator. This behavior persists at the symmetry point where the qubit-oscillator coupling is quadratic and decoherence effects are minimized. There the system realizes a "single-atom-two-photon laser".Comment: Replaced with final published version, fig. 2 compresse

Relativistic jet models for the BL Lacertae object Mrk 421 during three epochs of observation

Coordinated observation of the nearby BL Lacertae object Mrk 421 obtained during May 1980, January 1984, and March 1984 are described. These observations give a time-frozen picture of the continuous spectrum of Mrk 421 at X-ray, ultraviolet, optical, and radio wavelengths. The observed spectra have been fitted to an inhomogeneous relativistic jet model. In general, the models reproduce the data well. Many of the observed differences during the three epochs can be attributed to variations in the opening angle of the jet and in the angle that the jet makes to the line of sight. The jet models obtained here are compared with the homogeneous, spherically symmetric, synchrotron self-Compton models for this source. The models are also compared with the relativistic jet models obtained for other active galactic nuclei

Indefinitely Oscillating Martingales

We construct a class of nonnegative martingale processes that oscillate indefinitely with high probability. For these processes, we state a uniform rate of the number of oscillations and show that this rate is asymptotically close to the theoretical upper bound. These bounds on probability and expectation of the number of upcrossings are compared to classical bounds from the martingale literature. We discuss two applications. First, our results imply that the limit of the minimum description length operator may not exist. Second, we give bounds on how often one can change one's belief in a given hypothesis when observing a stream of data.Comment: ALT 2014, extended technical repor

Accurate Hartree-Fock energy of extended systems using large Gaussian basis sets

Calculating highly accurate thermochemical properties of condensed matter via wave function-based approaches (such as e.g. Hartree-Fock or hybrid functionals) has recently attracted much interest. We here present two strategies providing accurate Hartree-Fock energies for solid LiH in a large Gaussian basis set and applying periodic boundary conditions. The total energies were obtained using two different approaches, namely a supercell evaluation of Hartree-Fock exchange using a truncated Coulomb operator and an extrapolation toward the full-range Hartree-Fock limit of a Pad\'e fit to a series of short-range screened Hartree-Fock calculations. These two techniques agreed to significant precision. We also present the Hartree-Fock cohesive energy of LiH (converged to within sub-meV) at the experimental equilibrium volume as well as the Hartree-Fock equilibrium lattice constant and bulk modulus.Comment: 7.5 pages, 2 figures, submitted to Phys. Rev. B; v2: typos removed, References adde

Heat shock factor 1 regulates lifespan as distinct from disease onset in prion disease

Prion diseases are fatal, transmissible, neurodegenerative diseases caused by the misfolding of the prion protein (PrP). At present, the molecular pathways underlying prion-mediated neurotoxicity are largely unknown. We hypothesized that the transcriptional regulator of the stress response, heat shock factor 1 (HSF1), would play an important role in prion disease. Uninoculated HSF1 knockout (KO) mice used in our study do not show signs of neurodegeneration as assessed by survival, motor performance, or histopathology. When inoculated with Rocky Mountain Laboratory (RML) prions HSF1 KO mice had a dramatically shortened lifespan, succumbing to disease ≈20% faster than controls. Surprisingly, both the onset of home-cage behavioral symptoms and pathological alterations occurred at a similar time in HSF1 KO and control mice. The accumulation of proteinase K (PK)-resistant PrP also occurred with similar kinetics and prion infectivity accrued at an equal or slower rate. Thus, HSF1 provides an important protective function that is specifically manifest after the onset of behavioral symptoms of prion disease

Robust Inference of Trees

This paper is concerned with the reliable inference of optimal tree-approximations to the dependency structure of an unknown distribution generating data. The traditional approach to the problem measures the dependency strength between random variables by the index called mutual information. In this paper reliability is achieved by Walley's imprecise Dirichlet model, which generalizes Bayesian learning with Dirichlet priors. Adopting the imprecise Dirichlet model results in posterior interval expectation for mutual information, and in a set of plausible trees consistent with the data. Reliable inference about the actual tree is achieved by focusing on the substructure common to all the plausible trees. We develop an exact algorithm that infers the substructure in time O(m^4), m being the number of random variables. The new algorithm is applied to a set of data sampled from a known distribution. The method is shown to reliably infer edges of the actual tree even when the data are very scarce, unlike the traditional approach. Finally, we provide lower and upper credibility limits for mutual information under the imprecise Dirichlet model. These enable the previous developments to be extended to a full inferential method for trees.Comment: 26 pages, 7 figure