# Entropy formula dice

The mathematical expressions for thermodynamic entropy in the statistical thermodynamics formulation established by Ludwig Boltzmann and J. Willard Gibbs in the s are similar to the information entropy by Claude Shannon and Ralph Hartleydeveloped in the s.

The defining expression for entropy in the theory of statistical mechanics established by Ludwig Boltzmann and J. Willard Gibbs in the s, is of the form:. The defining expression for entropy in the theory of information established by Claude E. Shannon in is of the form:. If all the microstates are equiprobable a microcanonical ensemblethe statistical thermodynamic entropy reduces to the form, as given by Boltzmann.

Therefore S depends on temperature. If all the messages are equiprobable, the information entropy reduces to the Hartley entropy. The logarithm in the thermodynamic definition is the natural logarithm. It can be shown that the Gibbs entropy formula, with the natural logarithm, reproduces all of the properties of the macroscopic classical thermodynamics of Rudolf Clausius.

See article: Entropy statistical views. The logarithm can also be taken to the natural base in the case of information entropy. This is equivalent to choosing to measure information in nats instead of the usual bits or more formally, shannons.

In practice, information entropy is almost always calculated using base 2 logarithms, but this distinction amounts to nothing other than a change in units. One nat is about 1.

For a simple compressible system that can only perform volume work, the first law of thermodynamics becomes. This equivalence is commonly shown in textbooks.

However, the equivalence between the thermodynamic definition of entropy and the Gibbs entropy is not general but instead an exclusive property of the generalized Boltzmann distribution [2].

Despite the foregoing, there is a difference between the two quantities. The information entropy H can be calculated for any probability distribution if the "message" is taken to be that the event i which had probability p i occurred, out of the space of the events possiblewhile the thermodynamic entropy S refers to thermodynamic probabilities p i specifically.

The difference is more theoretical than actual, however, because any probability distribution can be approximated arbitrarily closely by some thermodynamic system. Moreover, a direct connection can be made between the two. Or, in the words of G. Lewis writing about chemical entropy in"Gain in entropy always means loss of information, and nothing more". To be more concrete, in the discrete case using base two logarithms, the reduced Gibbs entropy is equal to the minimum number of yes—no questions needed to be answered in order to fully specify the microstategiven that we know the macrostate.

Furthermore, the prescription to find the equilibrium distributions of statistical mechanics—such as the Boltzmann distribution—by maximising the Gibbs entropy subject to appropriate constraints the Gibbs algorithm can be seen as something not unique to thermodynamics, but as a principle of general relevance in statistical inference, if it is desired to find a maximally uninformative probability distributionsubject to certain constraints on its averages.

These perspectives are explored further in the article Maximum entropy thermodynamics. The Shannon entropy in information theory is sometimes expressed in units of bits per symbol. The physical entropy may be on a "per quantity" basis h which is called " intensive " entropy instead of the usual total entropy which is called "extensive" entropy.

The "shannons" of a message H are its total "extensive" information entropy and is h times the number of bits in the message. A direct and physically real relationship between h and S can be found by assigning a symbol to each microstate that occurs per mole, kilogram, volume, or particle of a homogeneous substance, then calculating the 'h' of these symbols.

By theory or by observation, the symbols microstates will occur with different probabilities and this will determine h.

If there are N moles, kilograms, volumes, or particles of the unit substance, the relationship between h in bits per unit substance and physical extensive entropy in nats is:. N h is the amount of information in bits needed to describe the state of a physical system with entropy S. Landauer's principle demonstrates the reality of this by stating the minimum energy E required and therefore heat Q generated by an ideally efficient memory change or logic operation by irreversibly erasing or merging N h bits of information will be S times the temperature which is.

This has been experimentally confirmed. This shows that S is a true statistical measure of microstates that does not have a fundamental physical unit other than the units of information, in this case "nats", which is just a statement of which logarithm base was chosen by convention. Consider Maxwell's set-up, but with only a single gas particle in a box.The principle of indifference also called principle of insufficient reason is a rule for assigning epistemic probabilities.

The principle of indifference states that in the absence of any relevant evidence, agents should distribute their credence or 'degrees of belief' equally among all the possible outcomes under consideration. In Bayesian probabilitythis is the simplest non-informative prior. The principle of indifference is meaningless under the frequency interpretation of probability[ citation needed ] in which probabilities are relative frequencies rather than degrees of belief in uncertain propositions, conditional upon state information.

The textbook examples for the application of the principle of indifference are coinsdiceand cards. In a macroscopic system, at least, it must be assumed that the physical laws which govern the system are not known well enough to predict the outcome.

Given enough time and resources, there is no fundamental reason to suppose that suitably precise measurements could not be made, which would enable the prediction of the outcome of coins, dice, and cards with high accuracy: Persi Diaconis 's work with coin-flipping machines is a practical example of this.

A symmetric coin has two sides, arbitrarily labeled heads many coins have the head of a person portrayed on one side and tails. Assuming that the coin must land on one side or the other, the outcomes of a coin toss are mutually exclusive, exhaustive, and interchangeable. It is implicit in this analysis that the forces acting on the coin are not known with any precision. If the momentum imparted to the coin as it is launched were known with sufficient accuracy, the flight of the coin could be predicted according to the laws of mechanics.

Thus the uncertainty in the outcome of a coin toss is derived for the most part from the uncertainty with respect to initial conditions.

This point is discussed at greater length in the article on coin flipping. A symmetric die has n faces, arbitrarily labeled from 1 to n. We assume that the die will land on one face or another upward, and there are no other possible outcomes.

As with coins, it is assumed that the initial conditions of throwing the dice are not known with enough precision to predict the outcome according to the laws of mechanics.

Dice are typically thrown so as to bounce on a table or other surface s. This interaction makes prediction of the outcome much more difficult. The assumption of symmetry is crucial here.

Suppose that we are asked to bet for or against the outcome "6". We might reason that there are two relevant outcomes here "6" or "not 6", and that these are mutually exclusive and exhaustive.

A standard deck contains 52 cards, each given a unique label in an arbitrary fashion, i. This example, more than the others, shows the difficulty of actually applying the principle of indifference in real situations. What we really mean by the phrase "arbitrarily ordered" is simply that we don't have any information that would lead us to favor a particular card. In actual practice, this is rarely the case: a new deck of cards is certainly not in arbitrary order, and neither is a deck immediately after a hand of cards.

In practice, we therefore shuffle the cards; this does not destroy the information we have, but instead hopefully renders our information practically unusable, although it is still usable in principle.

In fact, some expert blackjack players can track aces through the deck; for them, the condition for applying the principle of indifference is not satisfied. Applying the principle of indifference incorrectly can easily lead to nonsensical results, especially in the case of multivariate, continuous variables. A typical case of misuse is the following example:. In this example, mutually contradictory estimates of the length, surface area, and volume of the cube arise because we have assumed three mutually contradictory distributions for these parameters: a uniform distribution for any one of the variables implies a non-uniform distribution for the other two.

In general, the principle of indifference does not indicate which variable e. Another classic example of this kind of misuse is the Bertrand paradox. Edwin T. Jaynes introduced the principle of transformation groupswhich can yield an epistemic probability distribution for this problem. This generalises the principle of indifference, by saying that one is indifferent between equivalent problems rather than indifference between propositions. This still reduces to the ordinary principle of indifference when one considers a permutation of the labels as generating equivalent problems i.

To apply this to the above box example, we have three random variables related by geometric equations. If we have no reason to favour one trio of values over another, then our prior probabilities must be related by the rule for changing variables in continuous distributions.There are three important elements to consider in the StarCraft 2 economy: Minerals, Vespene Gas and Supply. Minerals are the most primitive resource, Vespene Gas is used to build advanced units, and your Supply level limits how many units you can field at any given time.

When you begin a match, you'll have a patch of minerals to harvest and units to start the harvesting process. You are encouraged to keep building a number of workers to ensure the harvesting rate increases over time. Note that certain yellow-coloured Mineral fields grant a greater yield, although you shouldn't worry too much about this in your early days.

The green Vespene Gas geysers allow you to harvest a second mineral, but each race has to build a specific structure on top of the gas patch first. Get extraction of the gas up and running very quickly in each game, as you'll need it sooner rather than later. You can increase the Supply value of your base by building certain structures nearby. Should you find yourself unable to build a new unit because you lack Supply, you are considered Supply-capped.

You should avoid this situation at all costs. It is vital that you grow your economy continuously throughout each match, harvest more and more resources, and ensure you have the necessary Supply infrastructure in place at all times. By managing all three of these elements, you hope to use your resources and production in as efficient a manner as possible. It's a lot to juggle, and our Macro and Micro section below contains more information about building out your fighting force.

Two of the terms you'll hear mentioned a huge amount in discussion of StarCraft 2 are the twin pillars of the game's strategy: Macro and Micro. Put very simply, Macro concerns the maintenance of your economy (keeping workers busy, resources flowing and so on), as well as the growth of your overall fighting force, pursuing whatever pre-determined strategy you've decided to work towards.

Micromanagement, on the other hand, refers to the more fiddly business of controlling individual elements of your army, or groups of fighting units. It is absolutely vital that you pay close attention to both of these elements, and continue doing so at every stage of any given match. Fail to do so and you might find yourself starved of resources to rebuild after a fiddly, losing battle.

Spend too much time playing SimCity, on the other hand, and you'll lack both the intel and means to defend against your opponent's mounting forces. Take comfort from the fact this process comes naturally to nobody, and in each match you should simply focus on trying to do one thing better than you did in the previous match. If you keep practising in this way, then you'll eventually find yourself passively improving at monitoring both elements of the game. It's a painful learning process, but a necessary one and you will get better if you persevere.

To make things a little easier on yourself, it's important to go into each game with something called a build order in mind. The following section contains a little more information on this point, as well as a beginner's example for each race.

Build orders are quite simply your pre-determined production plans for the earliest stages of the game - what you're going to build and when you're going to build it. This ensures you have a goal to work towards, and helps you work efficiently towards achieving it. There are more build order options out there than we could possibly summarise in any one guide such as this, but we wanted to highlight a solid starting list for each race that you can use to get going with.

In each video you'll find a versatile build order that should serve you well in all of your early match-ups. This is one area where you can take a little time to play against the AI, getting the flow of your hotkey and control groups together (see further on in this guide), while starting the process of building a super-efficient and robust economy.

Again, just don't get addicted to stomping easy AI opponents in this way, and get back into real multiplayer at your earliest opportunity.

If you commit your chosen build to memory, you will find yourself at a huge advantage in the early stages of the ladder climb.

Having a solid plan in place will also free you up to focus on honing your other skills in the early days. If you do not make use of hotkeys, you are always going to be at a permanent disadvantage to anyone who does - even if (all things being equal) they're an inferior player to you.

It takes time to move between multiple UI elements, after all, moving your mouse and then clicking.There is no perfect system. We strive to substantially increase your tennis predictions betting expert chances of winning. Since zulubet today sure bet prediction win high odd football betting tips champions league match.

Softmax and Cross Entropy Gradients for Backpropagation

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### Entropy in thermodynamics and information theory

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The Yellow Submarine aim to make amends for back-to-back defeats to the likes of Sevilla and Leganes and a win over Barcelona would see them retain their spot in top six. The Mussi Volanti have been having their ups and downs in the Serie A this season and they sit in the middle of the table ahead of Matchday 16.

Chievo Verona are a real force to be reckoned with at home and you should bear in mind that they are ey. The Neapolitans dropped to second place in the standings, but a win over the Viola would see them regain first place in the table. The Partenopei need to get back on course as soon as possible and we predict. As expected, SPAL suffered a 3-1 loss to Roma at Stadio Olimpico, but it has to be noted that the newcomers were with ten men on the field for more than 80 minutes.

The Biancazzurri extended their winless run in the Serie A to four games, but a win over Verona could see them escape the relegat. The Zebrette sit in the danger zone in the standings ahead of the visit of Benevento, but they are widely expected to pick up all three points from the match. The Neroverdi do not seem to be on top of their game at the moment and they suffered five defeats in previous six league fixtures.

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The rock-bottom team are desperate to put an end to their 14-game winless run. Die Roten extended their winless run in Bundesliga to four games, but they are seven points clear of the relegation zone ahead of Matchday 15 and they are on the right track to secure the top flight survival.

The Bavarians have been playing above expectations in the German top flight this term and it is no. Les Aiglons have been climbing the Ligue 1 standings week by week and a win over Nantes would see them regain their spot in top ten.

Mario Balotelli scored his eighth goal of the season in the. OM are nine points behind the league leaders in the Ligue 1 standings and we predict the star-studded team will go all-out attack against Saint-Etienne. The Acca Smacker supplies predictions for each outcome of a betting market, with the best prediction highlighted.

Form Streaks and Form Stats tabs filter predictions using streak data and detailed form comparison. Form Streaks: Highlight predictions with results streaks of three or more consecutive games for a specific betting market (KickOff Pro Members Only).Nagel knows Sunday's game comes down to the Seahawks' defense (No.

If the Seahawks can pressure Rodgers and force turnovers, they'll win on the road. But will that happen. Nagel knows the Seahawks were 2-5 against the spread on the road last season, while the Packers are 4-0 against the spread in their last four home games. The Packers used their first four draft picks to bolster a defense that was No. In that game, Matt Ryan had a field day, throwing for 338 yards and three TDs.

But will they be able to slow down Wilson, who looked red-hot in the preseason. Nagel thinks Seahawks vs. Packers will probably be a low-scoring game that stays Under, but there's one team he thinks is a very strong play to cover. He's sharing which one it is over at SportsLine.

So which side of Seahawks-Packers should you take. Visit SportsLine now to see what huge x-factor determines Seahawks-Packers, and what team is a very strong play to cover, all from the handicapper who's nailed four of his last five Seahawks picks. Firefox Up next: Packers vs. Seahawks odds, picks from red-hot Seahawks betting expert Packers vs.

Seahawks odds, picks from red-hot Seahawks betting expert Josh Nagel went 4-1 on Seahawks games last season and makes a pick for Sunday's game vs. Brady formally apologizes to McDaniels 'He knows how much I love him,' Brady said of McDaniels Week 14 injuries: Tyrod questionable Here's every official injury designation for each NFL team for Week 14 Steelers to honor Shazier with cleats Shazier, who suffered a brutal back injury on Monday, underwent surgery on Wednesday Peters suspended for spat with coach It turns out, Peters' suspension wasn't just about his flag toss Watt on broken leg: 'Easier mentally' The Texans' star opens up about his recovery from a season-ending injury window.

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## Principle of indifference

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