51,053 research outputs found
Minimum Description Length Induction, Bayesianism, and Kolmogorov Complexity
The relationship between the Bayesian approach and the minimum description
length approach is established. We sharpen and clarify the general modeling
principles MDL and MML, abstracted as the ideal MDL principle and defined from
Bayes's rule by means of Kolmogorov complexity. The basic condition under which
the ideal principle should be applied is encapsulated as the Fundamental
Inequality, which in broad terms states that the principle is valid when the
data are random, relative to every contemplated hypothesis and also these
hypotheses are random relative to the (universal) prior. Basically, the ideal
principle states that the prior probability associated with the hypothesis
should be given by the algorithmic universal probability, and the sum of the
log universal probability of the model plus the log of the probability of the
data given the model should be minimized. If we restrict the model class to the
finite sets then application of the ideal principle turns into Kolmogorov's
minimal sufficient statistic. In general we show that data compression is
almost always the best strategy, both in hypothesis identification and
prediction.Comment: 35 pages, Latex. Submitted IEEE Trans. Inform. Theor
On the Asymptotic Efficiency of Approximate Bayesian Computation Estimators
Many statistical applications involve models for which it is difficult to
evaluate the likelihood, but from which it is relatively easy to sample.
Approximate Bayesian computation is a likelihood-free method for implementing
Bayesian inference in such cases. We present results on the asymptotic variance
of estimators obtained using approximate Bayesian computation in a large-data
limit. Our key assumption is that the data are summarized by a
fixed-dimensional summary statistic that obeys a central limit theorem. We
prove asymptotic normality of the mean of the approximate Bayesian computation
posterior. This result also shows that, in terms of asymptotic variance, we
should use a summary statistic that is the same dimension as the parameter
vector, p; and that any summary statistic of higher dimension can be reduced,
through a linear transformation, to dimension p in a way that can only reduce
the asymptotic variance of the posterior mean. We look at how the Monte Carlo
error of an importance sampling algorithm that samples from the approximate
Bayesian computation posterior affects the accuracy of estimators. We give
conditions on the importance sampling proposal distribution such that the
variance of the estimator will be the same order as that of the maximum
likelihood estimator based on the summary statistics used. This suggests an
iterative importance sampling algorithm, which we evaluate empirically on a
stochastic volatility model.Comment: Main text shortened and proof revised. To appear in Biometrik
Reversibility and Adiabatic Computation: Trading Time and Space for Energy
Future miniaturization and mobilization of computing devices requires energy
parsimonious `adiabatic' computation. This is contingent on logical
reversibility of computation. An example is the idea of quantum computations
which are reversible except for the irreversible observation steps. We propose
to study quantitatively the exchange of computational resources like time and
space for irreversibility in computations. Reversible simulations of
irreversible computations are memory intensive. Such (polynomial time)
simulations are analysed here in terms of `reversible' pebble games. We show
that Bennett's pebbling strategy uses least additional space for the greatest
number of simulated steps. We derive a trade-off for storage space versus
irreversible erasure. Next we consider reversible computation itself. An
alternative proof is provided for the precise expression of the ultimate
irreversibility cost of an otherwise reversible computation without
restrictions on time and space use. A time-irreversibility trade-off hierarchy
in the exponential time region is exhibited. Finally, extreme
time-irreversibility trade-offs for reversible computations in the thoroughly
unrealistic range of computable versus noncomputable time-bounds are given.Comment: 30 pages, Latex. Lemma 2.3 should be replaced by the slightly better
``There is a winning strategy with pebbles and erasures for
pebble games with , for all '' with appropriate
further changes (as pointed out by Wim van Dam). This and further work on
reversible simulations as in Section 2 appears in quant-ph/970300
Three-Dimensional MHD Simulation of Caltech Plasma Jet Experiment: First Results
Magnetic fields are believed to play an essential role in astrophysical jets
with observations suggesting the presence of helical magnetic fields. Here, we
present three-dimensional (3D) ideal MHD simulationsof the Caltech plasma jet
experiment using a magnetic tower scenario as the baseline model. Magnetic
fields consist of an initially localized dipole-like poloidal component and a
toroidal component that is continuously being injected into the domain. This
flux injection mimics the poloidal currents driven by the anode-cathode voltage
drop in the experiment. The injected toroidal field stretches the poloidal
fields to large distances, while forming a collimated jet along with several
other key features. Detailed comparisons between 3D MHD simulations and
experimental measurements provide a comprehensive description of the interplay
among magnetic force, pressure and flow effects. In particular, we delineate
both the jet structure and the transition process that converts the injected
magnetic energy to other forms. With suitably chosen parameters that are
derived from experiments, the jet in the simulation agrees quantitatively with
the experimental jet in terms of magnetic/kinetic/inertial energy, total
poloidal current, voltage, jet radius, and jet propagation velocity.
Specifically, the jet velocity in the simulation is proportional to the
poloidal current divided by the square root of the jet density, in agreement
with both the experiment and analytical theory. This work provides a new and
quantitative method for relating experiments, numerical simulations and
astrophysical observation, and demonstrates the possibility of using
terrestrial laboratory experiments to study astrophysical jets.Comment: accepted by ApJ 37 pages, 15 figures, 2 table
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