# Shannon entropy example

Entropy forms the basis of the universe and everything in it. Why should deep learning be any different? In layman terms, you describe entropy as:.

The most basic example you get is of a fair coin: when you toss it, what will you get? Heads 1 or Tails 0. There is no way to tell. So, if you are playing this coin-tossing game on the phone.

Then, you must tell the other a single outcome or you need a single bit 0 or 1 to convey the information about this.

Would you need to tell the other person the result? The simple answer is NO. Hence, you need not say anything. You are not using even a single bit. And its mathematical definition:. What is probability? The likelihood of an event to occur.

The definition of probability talks about a single event, not the whole system. Thus, what probability can give us is a local picture, a limited picture of the whole system. Probability gives a local picture of the whole system. In order for us to get a sense of the whole system, we need to come up with a way that tells us a global picture of the whole system.

We need to evaluate the parts of the system and see their effect in summation. How do you evaluate a part of the system for the randomness it will contribute to the system?Shannon entropy is defined for a given discrete probability distribution; it measures how much information is required, on average, to identify random samples from that distribution.

Consider a coin with probability B for bias of flipping heads. We flip coins in a sequence known as the Bernoulli process and transmit each outcome to a reciever.

We can represent the outcome of each flip with a binary 1 heads or 0 tailstherefore on average it takes one bit of information to transmit one coin flip. Note that this method works regardless of the value of B, and therefore B does not need to be known to the sender or receiver.

If B is known and is exactly one half i. If B is exactly 0 or 1 then no bits need to be transmitted at all; i. For values of B other than 0, 1 and 0. The probability of any given coin flip sequence S consisting of h head and t tail flips, and for a coin with bias B, is given by the following equation from Estimating a Biased Coin :. The probability of each sequence is:. Huffman coding assigns a code to each sequence such that more probable frequent sequences are assigned shorter codes, in an attempt to reduce the number of bits we need to send on average.

## Entropy (information theory)

Applying the standard Huffman coding scheme we obtain these code allocations:. We can now calculate how many bits we need to send, on average, per coin flip, by multiplying each sequence's code length with the probability of that sequence occurring, and summing over all four possible sequences:.

Therefore this coding scheme requires 1. Huffman coding captures some of the possible gains in efficiency but is not optimal in the general case, i. However, our example demonstrates a key aspect of the Shannon entropy equation; that multiplying each possible sequence's code length by its probability gives the code length we would need to send on average.

This averaging over a coding scheme is precisely what the Shannon entropy equation describes:. We can clarify the equation further by applying the following logarithm law :.

Note that we have a sum over the product of each outcome's probability and a log term. The only difference between this and the above Huffman coding example is that the code length has been replaced with a log term; why?

Logarithms are inherently a measure of information quantity. Consider transmitting long numbers, e. Each value can take one of out of a million possible states, and yet we can transmit each number with only 6 decimal digits.

Noting that:. Note that the log base matches the number of symbols 0 to 9and that the result is the number of decimal digits needed to encode a number with one million possible states.

Hence we can obtain the number of bits needed to encode each number by changing the log base to two:. By taking the reciprocal we obtain the number of possible states that could have that probability; we then take the log of that number of states to obtain how many bits, on average, it would take to distinguish between that many states. Multiplying by the probability of the state, and summing over all states, gives the Shannon entropy equation. The Shannon entropy equation is agnostic with regard to logarithm base.

Base 2 will give a measure of information stated in bits, which is a well understood metric and a natural choice when dealing with digital communications. However we can choose any base, e.All three of them have just completed a medical test which, after some processing, yields one of two possible results: the disease is either present or absent.

I would like to focus on a simple question. All other things being equal, which of the three patients is confronted with the greatest degree of uncertainty?

I think the answer is clear: patient C. What he is going through is the greatest degree of uncertainty possible under the circumstances: a dramatic medical version of a flip with a fair coin. Compare this with patient A. Sure, the overall situation looks quite grim, but at least this patient is experiencing little uncertainty with regard to his medical prospects.

Intuitively speaking, what can we say about patient B? This is where entropy comes in. Entropy, in other words, is a measure of uncertainty. It is also a measure of information, but, personally, I prefer the uncertainty interpretation. It might just be me, but things seemed a lot clearer when I no longer attempted to impose my preconceived notion of information on the equations. Given certain assumptions and foreshadowing an important result mentioned belowentropy is the measure of uncertainty.

If you are anything like me when I first looked at this formula, you might be asking yourself questions such as: Why the logarithm? Why is this a good measure of uncertainty at all? And, of course, why the letter H?

Apparently, the use of the English letter H evolved from the the Greek capital letter Etaalthough the history appears to be quite complicated. And 2 Are they any competing constructs that have all of the these desirable properties? In short, the answers for Shannon entropy as a measure of uncertainty are: 1 many and 2 no. If your goal is to minimize uncertainty, stay away from uniform probability distributions. Quick reminder: A probability distribution is a function that assigns a probability to every possible outcome such that the probabilities add up to 1.

A distribution is uniform when all of the outcomes have the same probability. A good measure of uncertainty achieves its highest values for uniform distributions. Entropy satisfies the criterion. Given n possible outcomes, maximum entropy is maximized by equiprobable outcomes:. Here is the plot of the Entropy function as applied to Bernoulli trials events with two possible outcomes and probabilities p and 1-p :. Let A and B be independent events.

In other words, knowing the outcome of event A does not tell us anything about the outcome of event B. The uncertainty associated with both events â€” this is another item on our wish list â€” should be the sum of the individual uncertainties:. We can either flip both coins simultaneously or first flip one coin and then flip the other one.The idea of entropy was presented by Claude E. Historically, numerous musings of entropy have been proposed. The etymology of the word entropy goes back to Clausius, inwho named this term from the Greek troops, which means change, and a prefix en-to recalls the indissociable in his work connection to the possibility of vitality by Jaynes[43].

A statistical idea of entropy was presented by Shannon in the hypothesis of correspondence and transmission of data [50]. It is formally like Boltzmann entropy related with the statistical portrayal of the infinitesimal setups of many-body frameworks and how it represents their plainly visible direct.

Working up the connections between statistical entropy, statistical mechanics, and thermodynamic entropy was started by Jaynes [43]. In an initially totally alternate perspective, an idea of entropy rate was produced in dynamical frameworks hypothesis and representative gathering analysis. The issue of pressure is sometimes established in data hypothesis and Shannon entropy, while in different cases it is established in algorithmic Flexibility.

As a result of this decent variety of employment and ideas, we may request whether the utilization from the term entropy has any importance. Is there really something connecting this decent variety, or is the utilization of a similar term in such huge numbers of implications basically deceptive? A short historical record of the diverse ideas of entropy was given by Jaynes thirty years earlier [43]. I here propose a more detailed review of the connections between the distinctive considerations of entropy, not in a historical perspective yet rather as they show up today, featuring spans between likelihood, data hypothesis, dynamical frameworks hypothesis and statistical material science.

I will develop my argumentation in light of mathematical outcomes identified with Shannon entropy, operational framework, and demand change. They offer a strong both qualitative and quantitative guide for the correct utilize and interpretation of these ideas.

Specifically, they give a rationale, and also several stipulations, to the most extreme entropy standard.

### Report on Shannon Entropy

In this examination, a method to quantify the operational flexibility is produced. The scope of the volume to create for every item blend which is the idea of operational flexibility-can be calculated using information from the working expenses and all the technical information of the manufacturing framework and of the item to be created. Rudolf Clausius, a German physicist, formulated the second law of thermodynamics in by stating that heat flowed spontaneously from hot bodies to cool ones, never the reverse.

He conjectured that matter must have a previously unrecognized property which he called entropy. He further showed that total entropy always increased for all changes in any natural process.

This observation led him to formulate the second law as the entropy of the universe tends to a maximum. Empirically, Clausius defined entropy, S, in a differential frame:. This definition, however, fails to provide much understanding as to how the concept can be used concretely. From the perspective of statistical mechanics, entropy is viewed as the probability that certain events may happen inside the framework of all possible events.

By observing the behavior of large numbers of particles, statistical mechanics has succeeded in giving equations to the calculation of entropy as well as justification for equating entropy with a degree of disorder. Shannon[50] looked at information as a component of a priori probability of a given state or outcome among the universe of physically possible states.

He considered entropy as equivalent to uncertainty. Along these lines, information theory parallels the second law of thermodynamics as expressed by Clasius in claiming that the uncertainty on the planet always tends to increase. Indeed, as our perception of the world becomes increasingly complex, the number of phenomena about which we are uncertain increases and the uncertainty about each phenomenon also increases. To decrease this uncertainty, one collects an ever increasing amount of information describe by Kapur, [45] A system facing uncertainty uses flexibility as an adaptive response to cope with change.

The flexibility in the action of the system depends on the decision alternatives or the choices available and on the freedom with which various choices can be made by R.

Caprihan [11]. A greater number of choices leads to more uncertainty of outcomes, and hence, increased flexibility. This inference has been the main driver to apply entropy as a measure of flexibility by different researchers.

Jaynes [43] Demonstrated consistency between the definition of entropy in statistical mechanics and the definition in information theory. He showed that the measure of uncertainty defined by Shannon could be taken as a primitive one and be used to derive state probabilities. Jaynes also introduced a formal entropy-maximization principle, Tribus [44] which subsequently used to demonstrate that all the laws of classical thermodynamics could also be derived from the uncertainty measure.The Shannon entropy equation provides a way to estimate the average minimum number of bits needed to encode a string of symbols, based on the frequency of the symbols.

Such an optimal encoding would allocate fewer bits for the frequency occuring symbols e. Note that the frequence of the symbols also happens to match the frequence in the string. This will not usually be the case and it seems to me that there are two ways to apply the Shannon entropy equation:. The symbol set has a known frequency, which does not necessarily correspond to the frequency in the message string.

For example, characters in a natural language, like english, have a particular average frequency. The number of bits per character can be calculated from this frequency set using the Shannon entropy equation.

A constant number of bits per character is used for any string in the natural language. Symbol frequency can be calculated for a particular message.

The Shannon entropy equation can be used calculate the number of bits per symbol for that particular message. Shannon entropy provides a lower bound for the compression that can be achieved by the data representation coding compression step. Shannon entropy makes no statement about the compression efficiency that can be achieved by predictive compression. Algorithmic complexity Kolmogorov complexity theory deals with this area.

Given an infinite data set something that only mathematicians possessthe data set can be examined for randomness. If the data set is not random, then there is some program that will generate or approximate it and the data set can, in theory, be compressed. Note that without an infinite data set, this determination is not always possible.

A finite set of digits generated for a pi expansion satisify tests for randomness. However, these digits must be pseudo-random, since they are generated from a deterministic process. Algorithmic complexity theory views a pi expansion of any number of digits as compressible to the function that generated the sequence a relatively small number of bits.

Back to Lossless Wavelet Compression. Back to Wavelets and Signal Processing.In information theorythe entropy of a random variable is the average level of "information", "surprise", or "uncertainty" inherent in the variable's possible outcomes.

The concept of information entropy was introduced by Claude Shannon in his paper " A Mathematical Theory of Communication ", [1] [2] and is sometimes called Shannon entropy in his honour. As an example, consider a biased coin with probability p of landing on heads and probability 1- p of landing on tails.

Other values of p give different entropies between zero and one bits. Base 2 gives the unit of bits or " shannons "while base e gives the "natural units" natand base 10 gives a unit called "dits", "bans", or " hartleys ".

An equivalent definition of entropy is the expected value of the self-information of a variable. The entropy was originally created by Shannon as part of his theory of communication, in which a data communication system is composed of three elements: a source of data, a communication channeland a receiver.

In Shannon's theory, the "fundamental problem of communication" â€” as expressed by Shannon â€” is for the receiver to be able to identify what data was generated by the source, based on the signal it receives through the channel.

Shannon strengthened this result considerably for noisy channels in his noisy-channel coding theorem. Entropy in information theory is directly analogous to the entropy in statistical thermodynamics. Entropy has relevance to other areas of mathematics such as combinatorics.

The definition can be derived from a set of axioms establishing that entropy should be a measure of how "surprising" the average outcome of a variable is. For a continuous random variable, differential entropy is analogous to entropy.

The basic idea of information theory is that the "informational value" of a communicated message depends on the degree to which the content of the message is surprising. If an event is very probable, it is no surprise and generally uninteresting when that event happens as expected; hence transmission of such a message carries very little new information.

However, if an event is unlikely to occur, it is much more informative to learn that the event happened or will happen. For instance, the knowledge that some particular number will not be the winning number of a lottery provides very little information, because any particular chosen number will almost certainly not win. However, knowledge that a particular number will win a lottery has high value because it communicates the outcome of a very low probability event.

Entropy measures the expected i. Consider the example of a coin toss. If the probability of heads is the same as the probability of tails, then the entropy of the coin toss is as high as it could be for a two-outcome trial.

## Entropy is a measure of uncertainty

English text, treated as a string of characters, has fairly low entropy, i. If we do not know exactly what is going to come next, we can be fairly certain that, for example, 'e' will be far more common than 'z', that the combination 'qu' will be much more common than any other combination with a 'q' in it, and that the combination 'th' will be more common than 'z', 'q', or 'qu'.

The Misunderstood Nature of Entropy

After the first few letters one can often guess the rest of the word. English text has between 0. If a compression scheme is lossless â€” one in which you can always recover the entire original message by decompression â€” then a compressed message has the same quantity of information as the original but communicated in fewer characters. It has more information higher entropy per character.

A compressed message has less redundancy. Shannon's source coding theorem states a lossless compression scheme cannot compress messages, on average, to have more than one bit of information per bit of message, but that any value less than one bit of information per bit of message can be attained by employing a suitable coding scheme.

The entropy of a message per bit multiplied by the length of that message is a measure of how much total information the message contains. Information theory gives a way of calculating the smallest possible amount of information that will convey this.

In this case, 'A' would be coded as '0' one bit'B' as '10', and 'C' and 'D' as '' and '', respectively. The calculation of the sum of probability-weighted log probabilities measures and captures this effect.

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### Do You Know What is Shannonâ€™s Entropy?

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