28 research outputs found
On mutual information, likelihood-ratios and estimation error for the additive Gaussian channel
This paper considers the model of an arbitrary distributed signal x observed
through an added independent white Gaussian noise w, y=x+w. New relations
between the minimal mean square error of the non-causal estimator and the
likelihood ratio between y and \omega are derived. This is followed by an
extended version of a recently derived relation between the mutual information
I(x;y) and the minimal mean square error. These results are applied to derive
infinite dimensional versions of the Fisher information and the de Bruijn
identity. The derivation of the results is based on the Malliavin calculus.Comment: 21 pages, to appear in the IEEE Transactions on Information Theor
Band-limited functions and the sampling theorem
The definition of band-limited functions (and random processes) is extended to include functions and processes which do not possess a Fourier integral representation. This definition allows a unified approach to band-limited functions and band-limited (but not necessarily stationary) processes. The sampling theorem for functions and processes which are band-limited under the extended definition is derived
The Distribution Route from Ancestors to Descendants
We study the distribution of descendants of a known personality, or of
anybody else, as it propagates along generations from father or mother through
any of their children. We ask for the ratio of the descendants to the total
population and construct a model for the route of Distribution from Ancestors
to Descendants (DAD). The population ratio is found to be given by the
recursive equation that provides the
transition from the th to the th generation. and
is the total relevant population at the first generation. The number of
generations it takes to make half the population descendants is and additional generations make everyone a descendent (=the full
descendant spreading time). These results are independent of the population
growth factor even if it changes along generations. As a running example we
consider the offspring of King David. Assuming a population between and of Israelites at King David's time ( BC),
it took 24 to 26 generations (about 600-650 years, when taking 25 years for a
generation) to make every Israelite a King David descendent. The conclusion is
that practically every Israelite living today (and in fact already at 350-400
BC), and probably also many others beyond them, are descendants of King David.
We note that this work doesn't deal with any genetical aspect. We also didn't
take into account here any geo-social-demographic factor. Nevertheless, along
tens of generations, about 120 from King David's time till today, the DAD route
is likely to govern the distribution in communities that are not very isolated.Comment: 16 pages, 2 figures. appears in BDD (Bar-Ilan University Press) 23_71
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