Gambling and information theory Statistical inference might be thought of as gambling theory applied to the world around. The myriad applications for logarithmic information measures tell us precisely how to take the best guess in the face of partial information.[1] In that sense, information theory might be considered a formal expression of the theory of gambling. It is no surprise, therefore, that information theory has applications to games of chance.[2] Kelly Betting[edit] Kelly betting or proportional betting is an application of information theory to investing and gambling. Part of Kelly's insight was to have the gambler maximize the expectation of the logarithm of his capital, rather than the expected profit from each bet. Side information[edit] where Y is the side information, X is the outcome of the betable event, and I is the state of the bookmaker's knowledge. The nature of side information is extremely finicky. Doubling rate[edit] Doubling rate in gambling on a horse race is [3] where there are (e.g if the where
Entropy where T is the absolute temperature of the system, dividing an incremental reversible transfer of heat into that system (dQ). (If heat is transferred out the sign would be reversed giving a decrease in entropy of the system.) The above definition is sometimes called the macroscopic definition of entropy because it can be used without regard to any microscopic description of the contents of a system. The concept of entropy has been found to be generally useful and has several other formulations. Entropy was discovered when it was noticed to be a quantity that behaves as a function of state, as a consequence of the second law of thermodynamics. The absolute entropy (S rather than ΔS) was defined later, using either statistical mechanics or the third law of thermodynamics. In the modern microscopic interpretation of entropy in statistical mechanics, entropy is the amount of additional information needed to specify the exact physical state of a system, given its thermodynamic specification.
Information theory Overview[edit] The main concepts of information theory can be grasped by considering the most widespread means of human communication: language. Two important aspects of a concise language are as follows: First, the most common words (e.g., "a", "the", "I") should be shorter than less common words (e.g., "roundabout", "generation", "mediocre"), so that sentences will not be too long. Note that these concerns have nothing to do with the importance of messages. Information theory is generally considered to have been founded in 1948 by Claude Shannon in his seminal work, "A Mathematical Theory of Communication". Historical background[edit] The landmark event that established the discipline of information theory, and brought it to immediate worldwide attention, was the publication of Claude E. Prior to this paper, limited information-theoretic ideas had been developed at Bell Labs, all implicitly assuming events of equal probability. With it came the ideas of Quantities of information[edit] .
Entropy Figure 1: In a naive analogy, energy in a physical system may be compared to water in lakes, rivers and the sea. Only the water that is above the sea level can be used to do work (e.g. propagate a turbine). Entropy represents the water contained in the sea. In classical physics, the entropy of a physical system is proportional to the quantity of energy no longer available to do physical work. Entropy is central to the second law of thermodynamics, which states that in an isolated system any activity increases the entropy. History The term entropy was coined in 1865 [Cl] by the German physicist Rudolf Clausius from Greek en- = in + trope = a turning (point). The Austrian physicist Ludwig Boltzmann [B] and the American scientist Willard Gibbs [G] put entropy into the probabilistic setup of statistical mechanics (around 1875). The formulation of Maxwell's paradox by James C. Entropy in physics Thermodynamical entropy - macroscopic approach Entropy in quantum mechanics Black hole entropy
Binomial options pricing model Use of the model[edit] The Binomial options pricing model approach is widely used as it is able to handle a variety of conditions for which other models cannot easily be applied. This is largely because the BOPM is based on the description of an underlying instrument over a period of time rather than a single point. As a consequence, it is used to value American options that are exercisable at any time in a given interval as well as Bermudan options that are exercisable at specific instances of time. Being relatively simple, the model is readily implementable in computer software (including a spreadsheet). Although computationally slower than the Black–Scholes formula, it is more accurate, particularly for longer-dated options on securities with dividend payments. Method[edit] The binomial pricing model traces the evolution of the option's key underlying variables in discrete-time. Option valuation using this method is, as described, a three-step process: or and ). , we have: Where Max [ (
8. Reductio ad Absurdum – A Concise Introduction to Logic 8.1 A historical example In his book, The Two New Sciences,[10] Galileo Galilea (1564-1642) gives several arguments meant to demonstrate that there can be no such thing as actual infinities or actual infinitesimals. One of his arguments can be reconstructed in the following way. Galileo proposes that we take as a premise that there is an actual infinity of natural numbers (the natural numbers are the positive whole numbers from 1 on): He also proposes that we take as a premise that there is an actual infinity of the squares of the natural numbers. Now, Galileo reasons, note that these two groups (today we would call them “sets”) have the same size. If we can associate every natural number with one and only one square number, and if we can associate every square number with one and only one natural number, then these sets must be the same size. But wait a moment, Galileo says. Galileo argues that the reason we reached a contradiction is because we assumed that there are actual infinities.
Latent Dirichlet allocation In natural language processing, latent Dirichlet allocation (LDA) is a generative model that allows sets of observations to be explained by unobserved groups that explain why some parts of the data are similar. For example, if observations are words collected into documents, it posits that each document is a mixture of a small number of topics and that each word's creation is attributable to one of the document's topics. LDA is an example of a topic model and was first presented as a graphical model for topic discovery by David Blei, Andrew Ng, and Michael Jordan in 2003.[1] Topics in LDA[edit] In LDA, each document may be viewed as a mixture of various topics. For example, an LDA model might have topics that can be classified as CAT_related and DOG_related. Each document is assumed to be characterized by a particular set of topics. Model[edit] With plate notation, the dependencies among the many variables can be captured concisely. is the topic distribution for document i, The 1. , where .
untitled Chapter 4: Music I preface the following by the admission that I have developed little to no musical aptitude as yet and have never studied music theory. However, that does not seem to have stopped me from uncovering what looks to be some very interesting observations having applied Mod 9 to the frequencies generated by the black and white keys of the musical scale. 1955 saw the introduction of the International Standard Tuning of 440 Hz on the A of Middle C Octave. The C's were all 3 & 6, same for C sharp, then D's are all 9's and on until I came to F which revealed the 1 2 4 8 7 5 sequence, in order. You will notice that using this tuning at 440 Hz we see that: 5 sections of the octave are 1 2 4 8 7 5 4 sections are 3 & 6 3 sections are 9 Immediately the Pythagorean 3 4 5 triangle springs to mind. Above, we can clearly see that, as with the numbers, the octaves are paired up symmetrically and reflected, separated by 3 octaves. 2 sections that are 3 6, versus 4, using the 440 Hz tuning.
Value: The Third Factor Of Investing A stock's valuation is the final factor of the Fama-French three-factor model of investment returns. A stock's valuation is measured on a continuum from "value" to "growth." In broad strokes, value stocks are cheap and growth stocks are expensive. But there are compelling reasons why an investor might be willing to pay more for a growth stock than a value stock. Consider a local utility company whose stock is selling for $10 a share. This company has a price per earnings (P/E) ratio of 10. In contrast, consider a technology startup company that has shown meteoric growth in the past three years. Investors might rightly decide that the growing technology company is worth more than the static regional utility. The P/E ratio is one common measurement used to place stocks on the value to growth continuum. Some measurements use the past four quarters of earnings, which is often called the trailing P/E ratio. Is this author on the ball? Follow and be the first to know when they publish.
The Zero Point Field: How Thoughts Become Matter? | HuffPost Life Since I have mentioned the zero point field (ZPF) so much in my past HuffPost articles, and seeing as how it is a vital component to what is going on, it only makes sense to provide a more detailed analysis for all those Quantum buffs who struggle with my theory that thoughts equal matter. So, let's start with the basics and show what is known about the ZPF, and how its discovery come about? ZPF Basics In quantum field theory, the vacuum state is the quantum state with the lowest possible energy; it contains no physical particles, and is the energy of the ground state. Liquid helium-4 is a great example: Under atmospheric pressure, even at absolute zero, it does not freeze solid and will remain a liquid. This would seem to imply that a vacuum state -- or simply vacuum -- is not empty at all, but the ground state energy of all fields in space, and may collectively be called the zero point field. In physics, there is something called the Casmir effect.
Learn to Trade Forex (Currencies), Stocks, & CFDs | InformedTrades The “Sound” of Weyl Fermions Binghai Yan, Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot, Israel A prediction of a new heat-transport mechanism—called chiral zero sound—may explain recent observations of a “giant” thermal conductivity in Weyl semimetals. Heat in a solid is mainly carried by lattice vibrations (phonons) and conducting electrons. These mechanisms of heat conduction are so dominant over other forms that in an ordinary material they are often assumed to be the only ones that matter. But as reported in a pair of papers, an exotic “vibration” of electrons might provide a third, significant way to conduct heat at low temperatures [1, 2]. This effect occurs in a family of materials called Weyl semimetals [3, 4], and it might be harnessed to guide heat through a material. A Weyl semimetal is a material that hosts particles known as Weyl fermions. Because of this intrinsic chirality, Weyl quasiparticles behave differently than electrons in ordinary metals or semiconductors. Z.