Bayes' Original Paper

I was looking for some historical information about Thomas Bayes (1702-1761) this week in order to write a little text and found his biography in Wikipedia. In the end of the article, there is a link to the notorious paper that is credit for starting it all, I mean, the celebrated Bayesian interpretation of probability theory. The name of the paper is An essay towards solving a Problem in the Doctrine of Chances and it was published postumously in 1763 being presented to the Royal Society of London by a friend named Richard Price. Indeed, it seems that Bayes published only three papers: two in life and one in death.

Doctrine of Chances was the name given to Probability Theory at those days. I started to read the paper and discovered that it was written with a strange point of view about probabilities. As Kolmogorov didn´t put everything in rigorous ground in that time, the paper talks about games, expectations of receiving and losing amounts of (probably) money and is full of complicated definitions and propositions that could be written and deduced in simpler ways today. There is even a comment in the beggining by Price saying that "His solution he has applied to a very important purpose, and thereby shewn that those a remuch mistaken who have insinuated that the Doctrine of Chances in mathematics is of trivial consequence, and cannot have a place in any serious enquiry." Can anyone think something like this about Probability Theory today?

The problem that Bayes solves in the paper is: "Given the number of times on which an unknown event has happened and failed: Required the chance that the probability of its happening in a single trial lies somewhere between any two degrees of probability that can be named." Note that Bayes ask how to estimate a probability after just one try of a random event. Here is the cern of the Bayesian interpretation, where estimates can be made without reference to frequencies, but the modern shape of Bayes Inference started to appear only with Laplace, which used the ideas in Bayes' paper to estimate the mass of Saturn.

Being written in such a different mathematical language, the paper is difficult to read, but it's a funny exercise and sheds light on the foundations of Bayesian theory. Bayes inference is an interesting matter by itself and I intend (did I promise it already?) to te a bigger post about that. It's beign prepared and I hope to finish it someday...

On my desk:

• The physics of forgetting: Landauer's erasure principle and information theory, M.B. Plenio, V. Vitelli (quant-ph/0103108)
• Deriving Landauer's erasure principle from statistical mechanics, Kurt Jacobs (quant-ph/0512105)
• Relational Quantum Mechanics, Carlo Rovelli (quant-ph/9609002)

Cartoons

These ones are funny science cartoons... I put them in my site...

Official Dilbert Website

Official Calvin and Hobbes Website

Cat and Girl

Reading...

I've been busy in the last weeks with a lot of things. Some are related to Physics but most are not (like my wedding). As I have not the time to write a post, I decided to list the papers I've been reading. So, here it goes:

• Uncertainty relations on the slit and Fisher information, Z. Hradil & J. Rehacek (quant-ph/0309184)


• Replica analysis of a preferential urn model, Jun Ohkubo, Muneki Yasuda, Kazuyuki Tanaka (cond-mat/0601193)


• Introduction to Modern Canonical Quantum General Relativity, Thomas Thiemann (gr-qc/0110034)


• Quantum Mechanics as Quantum Information (and only a little more), Christopher A. Fuchs


• Quantum Cryptography, Miloslav Dusek, Norbert Lutkenhaus, Martin Hendrych (quant-ph/0601207)


• An introduction to quantum filtering, Luc Bouten, Ramon van Handel, Matthew James (math.OC/0601741)

Picture taken from: www.ibiblio.org/ eldritch/js/a18.htm

MaxEnt 2006

This year MaxEnt 2006 will be in Paris from 8th to 13th, July. MaxEnt is an annual conference on Maximum Entropy methods and their applications in science and this year it will be the 26th time it happens. The conference-chair is Ali Mohammad-Djafari and it has some notorious people in the organization: G. L. Bretthorst, A. Caticha, C. Rodriquez and J. Skilling among others.

Maximum Entropy methods are extensively used not only in physics but also in statistics and engineering since the landmark paper of Shannon, named A Mathematical Theory of Communication, where he found that the same quantity called entropy in statistical physics could be used as a measure of information.

One of the most interesting applications of MaxEnt is related to Bayesian inference. It is often said that the Bayesian interpretation is a "subjective" interpretation of probability theory and a lot of criticism come from this, some even say that due to this Bayesian probabilities are not even scientific. I'm in the course of writing a post about Bayesian inference, but while I cannot finish it, let me say that MaxEnt can be used as a method to generate Bayesian prior distributions in a totally objective way. The idea, roughly, is that as entropy is a measure of "uncertainty" or "disinformation", the correct prior distribution is obtained by maximizing the entropy of the distribution subject to constraints given by the avaiable information. This means that when using this principle we are trying to use just the avaiable inforamtion and assuming nothing about everything else, i.e., we assume that our disinformation about the rest is maximal.

An interesting and simple example of this method is related to the well-known Gaussian distribution. It can be shown that the probability distribution obtained by MaxEnt when the only information we have is the mean and the covariance is given by a Gaussian, i.e., if the only thing you have is a mean and a covariance matrix, your best guess is a Gaussian one.

Picture: Entropy Trails, by Tetragrammaton Productions.

Renewed Homepage

After some months, I finally finished the "restauration" of my homepage. It does not have too much information yet, mainly because it has been renegated to second place after I started this blog, but I have some ideas and the new format is simpler for me to modify and include new things.

Any suggestion or critique is wellcomed, and I would like very much to receive a lot of both.

On my desk:

• Vacuum Fluctuations, Geometric Modular Action and Relativistic Quantum Information Theory, Verch (gr-qc/0512053)


• The Meaning of the Interaction-Free Measurements, Vaidman (quant-ph/0103081)


• Relative entropy in quantum information theory, Schumacher & Westmoreland (quant-ph/0004045)


• Quantum interference and Bayes theorem, Neri (quant-ph/0601061)

Space_Exploration@Home

NASA's Stardust spacecraft is comming back to Earth bringing an important load from space: dust from the Comet Wild 2, that is believed to contain information about the conditions of our Solar System in the time of Earth's formation, and interstelar dust collect during the trip.

To analyse the dust, it is important to pinpoint the tiny particles in the aerogel in which they were collected using microscopy photos. It is important to know exactly where the dust is before trying to get it so as not to disturbing unnecessarily the rest of the sample. The odd thing is that NASA will recruit people all over the planet to do that by internet. All you have to do is to subscribe in the site with the name of the project Stardust@Home and as soon as the spacecraft arrive in Earth they will begin to apply a test to evaluate if you have the required ability to help the program.

Those who really find dust particles in the photos will even gain the right to name the dust particle as a reward for the discovery. A good oportunity to become a scientist if even you never thought in being one!

Some news about the project:

Internet users will hunt for Stardust@home

Public to look for dust grains in Stardust detectors

Stardust@home Project Brings Cosmic Dust to Your Desktop

Net Articles

Continuing my efforts to organize my stuff before leaving Brazil, I was organizing my bookmarks and found these links to interesting online articles:

Superstrings
Life in Extreme Environments
Why smart people defend bad ideas
Oklo (A natural nuclear reactor.)
Twistors --- What Are They?
Brief History of Artificial Intelligence
An Introduction to Lagrange Multipliers
A compendium of NP optimization problems
Notes on Functionals
On the Origins of Twistor Theory by Roger Penrose.

As with the links in the last post, finding more articles to post here is just a matter of time as my cleaning proceeds...

On my desk

• Quantum Information can be Negative, Horodecki et al. (quant-ph/0505062)


• Characterizing Entanglement, Bruss (quant-ph/0110078)

Picture taken from: perso.wanadoo.fr/overlord59/

Rusty Links

I took this last week of the year off to organize my things, cause I will probably move to UK next year to begin a work applying Statistical Physics to Cryptography. I was cleaning my wardrobe, which does not have only clothes but also books, old toys and a lot of papers. Then I found a folder and a notebook covered with some dust and with the title "Internet" (time goes by very fast...). I opened then and found a lot of links that I probably found interesting someday and I decided to put them in my del.icio.us page. I begun this task this morning and they are really interesting links. So, I decided to put them here for everyone who read this also have the chance to take a look at them. Let me list them in the exact order that I found them:

Sodaplay: classic. Everyone must know it by now, but I´m glad to have rediscovered it.

Jim Loy´s Homepage: the amount of information about science-related themes is huge. How this guy manage to write so much is a mistery to me.

2d Curves: a collection of mathematical bidimensional curves.

Igor Nikitin´s Homepage: has an interesting document on String Theory.

Kolmogorov: a very complete page about Kolmogorov.

Wilfrid Hodges´ Homepage

Douglas Arnold´s Homepage: with an interesting page about Some disasters attributable to bad numerical computing.

The Online Books Page

O Cerebro Nosso de Cada Dia: a site about the brain originally in Portuguese, but with an English translation as Our Daily Brain.

That´s it. Probably I will find more as my cleaning proceeds and I´ll put them here.

Happy New Year for everyone!

Picture: Rusty Chain by Hilly Wakeford

Phononic Crystals

I found an interesting article in Physics Web by Gorishnyy et al. named Sound ideas (the beautiful picture above was taken from this article). The article talks about a kind of crystal, named a phononic crystal, that can be constructed in such a way to create specific "band gaps" for waves travelling in this solid. This means that you can control which frequency cannot propagate in the crystal, creating, for example, materials that become isolators for particular sounds or mechanical waves.

The band gaps are created by carefull design of the crystals allowing a control of the dispersion relation, the relation between frequency and wave number, in phonons, which are quantized modes of vibration in a solid. This quantization of vibrational modes comes from a treatment using the machinery of quantum mechanics and is a very important mechanism that, among other things, influence the heat condictivity of materials. For a short introduction to the theory of elementary excitations in solids see Elementary Excitations in Solids : Lectures on Phonons, Electrons, and Plasmons by David Pines.

Phononic crystals may have a lot of interesting technological applications described in detail inside the article, in the words of the authors

"Phononic crystals will provide researchers in acoustics and ultrasonics with new components that offer the same level of control over sound that mirrors and lenses provide over light."

Over my desk:

• Quantum Theory from Quantum Gravity, Markopoulou & Smolin (gr-qc/0311059)


• Optical Quantum Cloning – A Review, Cerf & Fiurasek (quant-ph/0512172)


• Quantum Probabilities and Paradoxes of the Quantum Century, Belavkin (math.PR/0512415)


• Basics of M-Theory, Miemiec & Schnakenburg (hep-th/0509137)


• The quantum structure of black holes, Mathur (hep-th/0510180)

Cosmic Collision

A friend send me this link named Cosmic Collision this week. It is a subsite of the official Hubble site where it is described how it will look like the collision of our Milky Way with the Andromeda Galaxy. The site shows the story with a narrated video and have a lot of scientific explanations in a simple but precise way.

The Milky way is indeed colliding already with other minor galaxies of our local neighborhood in our trip in the direction of Virgo Cluster named the Local Group, like the Magellanic Clouds, but the collision with Andromeda will be much more espectacular due to the size of Andromeda. Our planetary system probably will not be affected due to its tiny size relative to interstelar distances, but in the site they show how the night sky will look like during the collision time. In the end, both galaxies will merge into a large elliptical galaxy.

The collision will occur in about 5 billion years from now, what remembered me of a story someone told me once (I don´t remember who...): A scientist was giving a lecture about the death of our sun. At some point, a person raised a shaking arm and asked in a trembling voice Excuse-me, professor, when did you say that will occur?. The professor answered In about 5 billion years.. The guy then took a deep breath and said in relief Oh... I thought you have said 5 MILLION...

As a last comment, the Hubble site has a lot of beautiful pictures and interesting explanations. Don´t be in a hurry when navigating there and you will enjoy every mouse click.

Over my desk:

1. Deriving Landauer’s erasure principle from statistical mechanics, Jacobs (quant-ph/0512105).

2. Spin Glasses: a Perspective, Sherrington (cond-mat/0512425).

3. Projective geometry and special relativity, Delphenich (gr-qc/0512125).

4. The Study of the Pioneer Anomaly: New Data and Objectives for New Investigation, Turyshev (gr-qc/0512121).

5. Quantum information and computation, Bub (quant-ph/0512125).

Picture: Colliding galaxies NGC 2207 and IC 2163, NASA.

Geometric Algebra

The algebra of complex numbers is related to geometry by the Argand plane. Using it, we see that the operation of multiplying by i is equivalent to a 90 degrees rotation in the counterclockwise direction. A little more advanced concept is that of quaternions, that as complex numbers, are a set of numbers that can represent rotations in 3D space. In both these cases, there is a beautiful connection between algebric structures and geometry that can be used to express physical laws in a concise way.

The notorious way to use geometry in physics is by means of Gibbs' vector calculus, which became widespread in physical sciences and engineering. In 1878 Clifford created a structure with the name geometric algebra uniting the dot and the cross products of two vectors into a single entity named the geometric product, which for two vectors a and b is written as
\[ab=a\cdot b +a \wedge b,\]
where the first term is the dot (scalar) product and the second the wedge or exterior product, which generalize the cross product that turns out to be a particular case in 3 dimensions.

Although it has a lot of applications in physics, it was eclipsed by Gibbs' vector calculus and was forgotten untill 1960 when David Hestenes, trying to recover the geometric meaning of the Clifford algebra related to spin discovered that geometric algebra is a "universal language for mathematics, physics and engineering."

There are a complete introductory course as Lecture Notes in the site of the Department of Physics of the University of Cambridge.

The interesting fact, that my former PhD advisor pointed me, is that there is a hope that this structure can lead to a geometric interpretation of the misterious use of complex numbers in Quantum Mechanics. However, I need to read more the lecture notes to talk about that.

Papers over my desk (or in my desktop):

• Vegetation's Red Edge: A Possible Spectroscopic Biosignature of Extraterrestrial Plants - Seager et al. (astro-ph/0503302)
• Causal Sets: Discrete Gravity (Notes for the Valdivia Summer School) - Sorkin (gr-qc/0309009)
• The General Quantum Interference Principle and the Duality Computer - Long (quant-ph/0512120)
• Entropic Priors - Caticha and Preuss (physics/0312131)
• On Math, Matter and Mind - Hut et al. (physics/0510188)

Picture: Quantum Notions - Gerard von Harpe

A Matter of Doubt

In the most fundamental level of nature lies two concepts that are central to physics: energy and matter. Energy is a fundamental entity present everywhere. Even empty space contains energy (what renders the term "empty space" a little inaccurate).

Matter is a concept directly associated with mass. Matter particles are particles with mass. Mass started as two "different" quantities: a measure of inertia, what comes from Newton´s formulaand gravitational charge, again given by Newton aswhere the gravitational constant G is so small that renders gravity the weakest of all forces in nature. Although nothing in principle says that gravitational charge and coefficient of inertia should be the same thing, Newton already confirmed by making experiments that both concepts agree with great precision. This point was late clarified by General Relativity, where we learned that gravity is only a deformation in spacetime and what we see as an atractive force is just a geodesic path, but the detailed explanation can be found in, for example, Robert Wald's General Relativitybook and in Sean Carrol's Website under the title Lecture Notes on General Relativity, so I will postpone it for a future post.

Mass is known to be equivalent to energy since Einstein´s Special Relativity. His famous formulawhich is valid for a body AT REST, means that even objects with no movement and subject to no forces have some energy that can be extracted from its mass. Indeed, the atomic bomb relied on this formula to produce an amazing amount of energy from a relatively small piece of matter.

The making of the atomic bomb shows that to extract energy from matter is (relatively) easy, but the converse is not so. The main problem is that we still does not know what exactly is the mechanism that converts energy into matter. We have clues, both experimental and theoretical, but a complete explanation is still lacking.

In the first place, we expect that energy can be transformed into matter because we believe that in the beginning there was only energy in the universe and, somehow at some point in the far past, this energy gave birth matter particles. Second, we know that it can happen because there are experimental evidence for a phenomenon called pair creation, where a photon acquires sufficient energy and generates a positron and an electron. However, this is totally random and we cannot predict when and how this will happen.

There is a curious theoretical phenomenon called Unruh-Hawking Radiation, sometimes treated separetely as Unruh Effect and Hawking Radiation, which is related to matter creation too. It is theoretical because we can deduce it from quantum mechanics and relativity, but the effect was not observed experimentally at this moment. Hawking discovered that black holes can induce production of pairs of matter particles around its event horizon and the emnission spectrum of these particles is a black body spectrum with temperaturewhere g is the local gravity acceleration. The equivalence principle of general relativity requires that a gravitational field is equivalent to acceleration and this implies that an accelerated observer can see a background of matter particles where an observer at rest see only the vacuum, and this particles obbey the same spectrum distribution of the particles near the black hole with temperaturewhere a is the acceleration.

The only explanation till now about how particles acquire mass comes from the Higgs mechanism, a kind of symmetry breaking involving a particle called the Higgs boson. But the Higgs boson has not been found experimentally at the moment and there is another problem: we must assume that the Higgs has a mass itself, what only puts the problem in another level: from where comes the mass of the Higgs? A self-interaction, you would say, but it´s just a circular argument, does not help too much.

The matter-energy problem has not been in the first plane of research in the last decades, but there are something very fundamental in this problem that must be understood if we want to go on with our aim of understanding how the universe works and how it appeared.

Quantum Limitations

As I already said in another post, quantum mechanics is a wide confirmed and one of the most successful theories about nature we (humans) ever created. The agreement of predictions with experiments is amazing and there are no known experiments that contradict the theory.

However, this is not the end of the story. QM is successful for its mathematics describes nature with tantalizing precision, but the math was tailored from experiments to fit them. This means that QM, unlike Relativity, is not derived from some fundamental principle. The lack of this principle is what is behind the great numbers of alternative interpretations apart from the ortodox one, which leads to strange situations like the Schroedinger Cat.

The lack of a first principles derivation still is responsible for the existence of alternative theories that try to explain qunatum phenomena, like Bohmian Mechanics, where David Bohm tries to explain quantum behavior by a misterious quantum field that permeates spacetime, and Stochastic Electrodynamics, pioneered by Timothy Boyer, which uses classical mechanics plus a random background field of electric particles and is able to find a lot of good results. But no theory yet has been proven to be exactly equal or superior to QM.

When I talk about the success of QM, I´m not talking yet about quantum field theory (QFT). QFT arises from the merging of QM with special relativity. It has a lot of success, but the way these results are extracted from the body of the theory is very trick and most of scientists have the feeling that this should not be the final answer. Although when supplied with some experimental measurements QFT can give results that agree with experiments by one part in a billion (in QED, for example), the calculations come from infinite series expansions that do not converge. The expansions are truncated and a lot of work on renormalizing (take the infinites away) the theory must be made.

The point is that even giving the correct values for several quantities, QFT begins with a very questionable (in my view) procedure: you simply transform equations from classical physics in equations for fields, solve by expanding in Fourier series and then impose quantum commutation relations between the fields and the canonical conjugate momenta. It is a recipe. We don´t know exactly what we are doing, but we borrow the procedure of imposing these relations from plain QM and go on. Another thing: the conjugate momenta comes from a lagrangean density that is constructed in such a way that it gives the correct equations of motion, again without any fuindamental principle. This procedure works with some tricks for Electrodynamics and for Weak and Strong Forces (giving rise to electroweak theory and to QCD), but fails miserably with gravity.

In my view (and my only view, what means that it is not the current view of scientific community), without a clear understanding of what we are really doing, we can´t even be sure if we should quantize gravity. Critics of string theory say that after so much time without success, maybe string theory is a wrong way, but the endeavour of quantizing gravity is much older and we could not do it till this day. I´m not saying that quantum gravity is not worth pursuing, I´m just saying that maybe there is a tiny possibility that nature did not choose this path. However, today the probability that QG exists probably is higher than that it do not. We have to wait more theoretical results or experiments.

Just to cite a tentative for deriving QM from first principles, it is worth looking at the papers of Ariel Caticha, he is trying to show that QM can be obtained by applying principles of information theory and bayesian inference to physics. The main theory is in Insufficient reason and entropy in quantum theory (quant-ph/9810074). He is also trying to show that general relativity can be obtained from the same principles: The Information Geometry of Space and Time (gr-qc/0508108).

Picture: Quantum Foam - taken from http://www.journal-kempten.de/

Avalanches

Avalanches are physical phenomena of great interest, mainly because they represent a big risk for those who live or visit areas where this kind of natural disaster can occur. But avalanches are very complex. They arise from an instability in a pile of granular material like sand or snow. Granular materials can be piled but just until the slope of the sides of this pile is below a critical angle. When the slope is above this angle, any extra grain added to the pile can cause a chain reaction and start an avalanche. The point is that you never know exactly when the avalanche will start.

Avalanches are an example of what is called an emergent behavior. Complex systems, which are composed by a great number of interacting unities, can show exceptional characteristics that are not expected: strange organization phenomena and surprising effects.

Although the problem of modelling granular materials seems to be something easy at first sight, it is a difficult matter and to this date we haven´t a unified theory yet. There are different approaches to attack this problem. One is to try to model granular materials a kind of fluid with special properties. It is a hydrodynamical approach. The other is to build discrete toy models and analyze them mathematically.

The second approach is related to the famous Bak-Tang-Wisenfeld model of a sandpile, where they use a bidimensional cellular automaton to model a pile where at each time step a grain is added at random in some site. When a site has a slope above a critical slope relative to its neighbours, one or more grains is transferred to this neighbour. This model turn out to have a very special behavior called Self-Organized Criticality (SOC). This behavior is rrelated to the distribution of the sizes of the avalanches in the pile and to the fact that the pile has a set of quiescent states, named metastable states, where the pile is momentarily stable.

These models can be complicated or simplified as much as we want and their study is not an easy matter once they are models that should be studied out of equilibrium, and out-of-equilibrium phenomena and, once more, we don´t have a unified theory for them too. An interesting example of a simplified model where you can see "avalanches" was sent to me last week by a friend named Marlo who found it in the internet. It is a game where you have a bidimensional cellular automaton where each site can be in one of four different states. You can change the state of one site and the interaction between states can trigger an avalanche effect. The aim is to trigger the biggest possible avalanche, although it is funny just to look at the dynamics and the metastable states to see how they look like.

My friend becomes excited with the game and said to me that this could have a lot of consequences even in sociology... well, physicists already though of this and he is right. I´ll edit this post another day and will try to put some links to show this.

Picture taken from: Milford Road.

Kung Fu Science

I´m a kung fu fighter. After eigth years and two knee surgeries, last year I finally got my black belt. My style is Ton Long, or Praying Mantis, one of the several kung fu styles that exist. Most of the styles are inspired in the movements of animals, like Tiger (Hung Garr), Crane, Eagle´s Claws and Monkey, but there are others that do not follow the pattern, like Tai Chi Chuan, Drunk Style, Wing Chun (the style of Bruce Lee) or Suai Shiao. In fact, most of the styles are completely different martial arts and kung fu is a common name for all chinese martial arts. Kung fu is not even the correct name, its meaning is "hard work" and in China is used to every kind of art that needs a great effort to learn and master. The chinese name for their martial arts is wushu or kuoshu.

My passion for kung fu is well known among my friends and yesterday one of them send me a link about the physics of kung fu, a site entitled Kung Fu Science. The link is indeed about a study of the physics involved in breaking blocks with bare hands led by a young PhD student of atmosferic physics. The site has beautiful presentation and design and the text is very accessible for those who are not scientists too. There are links to related studies about the physics of other martial arts in the end of the webpage. It is worth to visit.

Picture taken from: International Chinese Kung Fu Association Website.