Harmonical mean

Definitions

  • Webster's Revised Unabridged Dictionary
    • Harmonical mean (Arith. & Alg) certain relations of numbers and quantities, which bear an analogy to musical consonances.
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Usage

In literature:

When you told me that you 'adored and worshipped' Doctor Harmon, did you mean it, or was that the delirium of fever?
"The Harvester" by Gene Stratton Porter
The only means of harmonizing your two wills is to arrange from the first that there shall be but one; and that will must be yours.
"The Marriage Contract" by Honore de Balzac
The new vibration by no means harmonizes.
"Fantasia of the Unconscious" by D. H. Lawrence
The formation of the harmonic basis, which is essentially in four parts, does not by any means devolve upon the wood-wind alone.
"Principles of Orchestration" by Nikolay Rimsky-Korsakov
But Preparation here means Friday, and noon is not the hour needed to harmonize with Mark.
"A Harmony of the Gospels for Students of the Life of Christ" by Archibald Thomas Robertson
It proves that, to produce its highest effects, sacred music must harmonize with the meaning of the words and with the environment.
"Increasing Personal Efficiency" by Russell H. Conwell
The attempt to harmonize different ideas means that in themselves they are discrepant.
"Essays in Experimental Logic" by John Dewey
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In news:

The expression "one man's trash is another man's treasure" finds new meaning in the documentary "Landfill Harmonic".
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In science:

In order to test whether the primordial temperature anisotropies have zero mean (which follows from (cid:104)φ(cid:105) = 0) we need additional information about the distribution of the harmonic coefficients a(cid:96)m .
Do Cosmological Perturbations Have Zero Mean?
Keywords and Phrases: evidence; marginal likelihood; Markov chain Monte Carlo; harmonic mean estimator; power posteriors; annealed importance sampling; nested sampling.
Estimating the evidence -- a review
The harmonic mean estimator (Newton and Raftery 1994) is an easy to implement estimator of the evidence, based on draws from the target distribution.
Estimating the evidence -- a review
Hence in such situations, the harmonic mean estimator will not change much as the prior changes.
Estimating the evidence -- a review
This approach to evaluating the harmonic mean should be more numerically stable than a direct implementation in cases where the log likelihoods are large negative numbers.
Estimating the evidence -- a review
Our interest was to see how the harmonic mean estimator performed for varying values of τ0 .
Estimating the evidence -- a review
In Table 1, the harmonic mean estimate of the logarithm of the evidence, log ˆπ(y), based on 106 iterations of a Gibbs sampler, is presented for varying values of τ0 .
Estimating the evidence -- a review
Table 1: The true value of log π(y) is presented for various values of τ0 , together with estimates of the logarithm of π(y), based on the harmonic mean estimator.
Estimating the evidence -- a review
As above, the explanation for the poor performance of the harmonic mean estimator lies in the fact that the posterior distribution does not change very much, as the prior distribution for µ becomes more diffuse, from τ0 = 1 through to τ0 = 0.0001.
Estimating the evidence -- a review
Therefore, samples from the four different posterior distributions in this example will be very similar, as reflected in the similar harmonic mean estimates.
Estimating the evidence -- a review
For the Laplace approximation at the MAP and the harmonic mean estimator, results were based on a Gibbs sampling run of 505,000, taking 20% of this as a burn-in.
Estimating the evidence -- a review
Issues with the harmonic mean estimator have been mentioned earlier.
Estimating the evidence -- a review
Keywords: Markov operators, harmonic functions, Martin boundary, mean value property, finite variation measures.
Strong and weak mean value properties on trees
In this paper we study a mean value property (see Definition 3), which is actually the dual property of the one commonly known about harmonic functions.
Strong and weak mean value properties on trees
Besides we consider the mean walue property with respect to a set of (harmonic) functions F = {k(·, ξ ) : ξ ∈ F } where F is a suitable subset of the martin boundary of harmonic measure 1 (see the next section for the definitions).
Strong and weak mean value properties on trees
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