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Entropy (information theory)

In information theory, the entropy of a random variable is the average level of "information", "surprise", or "uncertainty" inherent to the variable's possible outcomes. Given a discrete random variable $$X$$, which takes values in the set $$\mathcal{X}$$ and is distributed according to $$p\colon \mathcal{X}\to[0, 1]$$, the entropy is $$\Eta(X) := -\sum_{x \in \mathcal{X}} p(x) \log p(x),$$ where $$\Sigma$$ denotes the sum over the variable's possible values. The choice of base for $$\log$$, the logarithm, varies for different applications. Base 2 gives the unit of bits (or "shannons"), while base e gives "natural units" nat, and base 10 gives units of "dits", "bans", or "hartleys". An equivalent definition of entropy is the expected value of the self-information of a variable.

The concept of information entropy was introduced by Claude Shannon in his 1948 paper "A Mathematical Theory of Communication", and is also referred to as Shannon entropy. Shannon's theory defines a data communication system composed of three elements: a source of data, a communication channel, and a receiver. The "fundamental problem of communication" – as expressed by Shannon – 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 considered various ways to encode, compress, and transmit messages from a data source, and proved in his source coding theorem that the entropy represents an absolute mathematical limit on how well data from the source can be losslessly compressed onto a perfectly noiseless 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. The analogy results when the values of the random variable designate energies of microstates, so Gibbs's formula for the entropy is formally identical to Shannon's formula. Entropy has relevance to other areas of mathematics such as combinatorics and machine learning. The definition can be derived from a set of axioms establishing that entropy should be a measure of how informative the average outcome of a variable is. For a continuous random variable, differential entropy is analogous to entropy. The definition $$\mathbb{E}[-\log p(X)] $$ generalizes the above.

Introduction
The core 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 a highly likely event occurs, the message carries very little information. On the other hand, if a highly unlikely event occurs, the message is much more informative. 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 informational value because it communicates the occurrence of a very low probability event.

The information content, also called the surprisal or self-information, of an event $$E$$ is a function which increases as the probability $$p(E)$$ of an event decreases. When $$p(E)$$ is close to 1, the surprisal of the event is low, but if $$p(E)$$ is close to 0, the surprisal of the event is high. This relationship is described by the function $$\log\left(\frac{1}{p(E)}\right) ,$$ where $$\log$$ is the logarithm, which gives 0 surprise when the probability of the event is 1. In fact, $log$ is the only function that satisfies а specific set of conditions defined in section .

Hence, we can define the information, or surprisal, of an event $$E$$ by $$I(E) = -\log_2(p(E)) ,$$ or equivalently, $$I(E) = \log_2\left(\frac{1}{p(E)}\right) .$$

Entropy measures the expected (i.e., average) amount of information conveyed by identifying the outcome of a random trial. This implies that rolling a die has higher entropy than tossing a coin because each outcome of a die toss has smaller probability ($$p=1/6$$) than each outcome of a coin toss ($$p=1/2$$).

Consider a coin with probability $p$ of landing on heads and probability $1 − p$ of landing on tails. The maximum surprise is when $p = 1/2$, for which one outcome is not expected over the other. In this case a coin flip has an entropy of one bit. (Similarly, one trit with equiprobable values contains $$\log_2 3$$ (about 1.58496) bits of information because it can have one of three values.) The minimum surprise is when $p = 0$ or $p = 1$, when the event outcome is known ahead of time, and the entropy is zero bits. When the entropy is zero bits, this is sometimes referred to as unity, where there is no uncertainty at all – no freedom of choice – no information. Other values of p give entropies between zero and one bits.

Example
Information theory is useful to calculate the smallest amount of information required to convey a message, as in data compression. For example, consider the transmission of sequences comprising the 4 characters 'A', 'B', 'C', and 'D' over a binary channel. If all 4 letters are equally likely (25%), one cannot do better than using two bits to encode each letter. 'A' might code as '00', 'B' as '01', 'C' as '10', and 'D' as '11'. However, if the probabilities of each letter are unequal, say 'A' occurs with 70% probability, 'B' with 26%, and 'C' and 'D' with 2% each, one could assign variable length codes. In this case, 'A' would be coded as '0', 'B' as '10', 'C' as '110', and 'D' as '111'. With this representation, 70% of the time only one bit needs to be sent, 26% of the time two bits, and only 4% of the time 3 bits. On average, fewer than 2 bits are required since the entropy is lower (owing to the high prevalence of 'A' followed by 'B' – together 96% of characters). The calculation of the sum of probability-weighted log probabilities measures and captures this effect.

English text, treated as a string of characters, has fairly low entropy; i.e. it is fairly predictable. 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'. After the first few letters one can often guess the rest of the word. English text has between 0.6 and 1.3 bits of entropy per character of the message.

Definition
Named after Boltzmann's Η-theorem, Shannon defined the entropy $Η$ (Greek capital letter eta) of a discrete random variable $X$, which takes values in the set $$\mathcal{X}$$ and is distributed according to $$p: \mathcal{X} \to [0, 1]$$ such that $$p(x) := \mathbb{P}[X = x]$$:

$$\Eta(X) = \mathbb{E}[\operatorname{I}(X)] = \mathbb{E}[-\log p(X)].$$

Here $$\mathbb{E}$$ is the expected value operator, and $I$ is the information content of $X$. $$\operatorname{I}(X)$$ is itself a random variable.

The entropy can explicitly be written as: $$\Eta(X) = -\sum_{x \in \mathcal{X}} p(x)\log_b p(x) ,$$ where $b$ is the base of the logarithm used. Common values of $b$ are 2, Euler's number $e$, and 10, and the corresponding units of entropy are the bits for $b = 2$, nats for $b = e$, and bans for $b = 10$.

In the case of $$p(x) = 0$$ for some $$x \in \mathcal{X}$$, the value of the corresponding summand $0 log_{b}(0)$ is taken to be $0$, which is consistent with the limit: $$\lim_{p\to0^+}p\log (p) = 0.$$

One may also define the conditional entropy of two variables $$X$$ and $$Y$$ taking values from sets $$\mathcal{X}$$ and $$\mathcal{Y}$$ respectively, as: $$ \Eta(X|Y)=-\sum_{x,y \in \mathcal{X} \times \mathcal{Y}} p_{X,Y}(x,y)\log\frac{p_{X,Y}(x,y)}{p_Y(y)} ,$$ where $$p_{X,Y}(x,y) := \mathbb{P}[X=x,Y=y]$$ and $$p_Y(y) = \mathbb{P}[Y = y]$$. This quantity should be understood as the remaining randomness in the random variable $$X$$ given the random variable $$Y$$.

Measure theory
Entropy can be formally defined in the language of measure theory as follows: Let $$(X, \Sigma, \mu)$$ be a probability space. Let $$A \in \Sigma$$ be an event. The surprisal of $$A$$ is $$ \sigma_\mu(A) = -\ln \mu(A) .$$

The expected surprisal of $$A$$ is $$ h_\mu(A) = \mu(A) \sigma_\mu(A) .$$

A $$\mu$$-almost partition is a set family $$P \subseteq \mathcal{P}(X)$$ such that $$\mu(\mathop{\cup} P) = 1$$ and $$\mu(A \cap B) = 0$$ for all distinct $$A, B \in P$$. (This is a relaxation of the usual conditions for a partition.) The entropy of $$P$$ is $$ \Eta_\mu(P) = \sum_{A \in P} h_\mu(A) .$$

Let $$M$$ be a sigma-algebra on $$X$$. The entropy of $$M$$ is $$ \Eta_\mu(M) = \sup_{P \subseteq M} \Eta_\mu(P) .$$ Finally, the entropy of the probability space is $$\Eta_\mu(\Sigma)$$, that is, the entropy with respect to $$\mu$$ of the sigma-algebra of all measurable subsets of $$X$$.

Example
Binary entropy plot.svg surprisal) of a coin flip, measured in bits, graphed versus the bias of the coin $Η(X)$, where $Pr(X = 1)$ represents a result of heads.

Here, the entropy is at most 1 bit, and to communicate the outcome of a coin flip (2 possible values) will require an average of at most 1 bit (exactly 1 bit for a fair coin). The result of a fair die (6 possible values) would have entropy log26 bits.]]

Consider tossing a coin with known, not necessarily fair, probabilities of coming up heads or tails; this can be modelled as a Bernoulli process.

The entropy of the unknown result of the next toss of the coin is maximized if the coin is fair (that is, if heads and tails both have equal probability 1/2). This is the situation of maximum uncertainty as it is most difficult to predict the outcome of the next toss; the result of each toss of the coin delivers one full bit of information. This is because $$\begin{align} \Eta(X) &= -\sum_{i=1}^n {p(x_i) \log_b p(x_i)} \\ &= -\sum_{i=1}^2 {\frac{1}{2}\log_2{\frac{1}{2}}} \\ &= -\sum_{i=1}^2 {\frac{1}{2} \cdot (-1)} = 1. \end{align}$$

However, if we know the coin is not fair, but comes up heads or tails with probabilities $X = 1$ and $p$, where $q$, then there is less uncertainty. Every time it is tossed, one side is more likely to come up than the other. The reduced uncertainty is quantified in a lower entropy: on average each toss of the coin delivers less than one full bit of information. For example, if $p ≠ q$ = 0.7, then $$\begin{align} \Eta(X) &= - p \log_2 (p) - q \log_2 (q) \\ &= - 0.7 \log_2 (0.7) - 0.3 \log_2 (0.3) \\ &\approx - 0.7 \cdot (-0.515) - 0.3 \cdot (-1.737) \\ &= 0.8816 < 1. \end{align}$$

Uniform probability yields maximum uncertainty and therefore maximum entropy. Entropy, then, can only decrease from the value associated with uniform probability. The extreme case is that of a double-headed coin that never comes up tails, or a double-tailed coin that never results in a head. Then there is no uncertainty. The entropy is zero: each toss of the coin delivers no new information as the outcome of each coin toss is always certain.

Characterization
To understand the meaning of $p$, first define an information function $−Σ p_{i} log(p_{i})$ in terms of an event $I$ with probability $i$. The amount of information acquired due to the observation of event $p_{i}$ follows from Shannon's solution of the fundamental properties of information:
 * 1) $i$ is monotonically decreasing in $I(p)$: an increase in the probability of an event decreases the information from an observed event, and vice versa.
 * 2) $p$: events that always occur do not communicate information.
 * 3) $I(1) = 0$: the information learned from independent events is the sum of the information learned from each event.

Given two independent events, if the first event can yield one of $I(p_{1}·p_{2}) = I(p_{1}) + I(p_{2})$ equiprobable outcomes and another has one of $n$ equiprobable outcomes then there are $m$ equiprobable outcomes of the joint event. This means that if $mn$ bits are needed to encode the first value and $log_{2}(n)$ to encode the second, one needs $log_{2}(m)$ to encode both.

Shannon discovered that a suitable choice of $$\operatorname{I}$$ is given by: $$\operatorname{I}(p) = \log\left(\tfrac{1}{p}\right) = -\log(p).$$

In fact, the only possible values of $$\operatorname{I}$$ are $$\operatorname{I}(u) = k \log u$$ for $$k<0$$. Additionally, choosing a value for $log_{2}(mn) = log_{2}(m) + log_{2}(n)$ is equivalent to choosing a value $$x>1$$ for $$k = - 1/\log x$$, so that $k$ corresponds to the base for the logarithm. Thus, entropy is characterized by the above four properties.

 * {| class="toccolours collapsible collapsed" width="80%" style="text-align:left"

!Proof
 * Let $\operatorname{I}$ be the information function which one assumes to be twice continuously differentiable, one has:
 * Let $\operatorname{I}$ be the information function which one assumes to be twice continuously differentiable, one has:

 * $$\begin{align}

& \operatorname{I}(p_1 p_2) &=\ & \operatorname{I}(p_1) + \operatorname{I}(p_2) && \quad \text{Starting from property 3} \\ & p_2 \operatorname{I}'(p_1 p_2) &=\ & \operatorname{I}'(p_1) && \quad \text{taking the derivative w.r.t}\ p_1 \\ & \operatorname{I}'(p_1 p_2) + p_1 p_2 \operatorname{I}''(p_1 p_2) &=\ & 0 && \quad \text{taking the derivative w.r.t}\ p_2 \\ & \operatorname{I}'(u) + u \operatorname{I}''(u) &=\ & 0 && \quad \text{introducing}\, u = p_1 p_2 \\ & (u \mapsto u \operatorname{I}'(u))' &=\ & 0 \end{align}$$

This differential equation leads to the solution $$\operatorname{I}(u) = k \log u + c$$ for some $$k, c \in \mathbb{R}$$. Property 2 gives $$c = 0$$. Property 1 and 2 give that $$\operatorname{I}(p)\ge 0$$ for all $$p\in [0,1]$$, so that $$k < 0$$.
 * }

The different units of information (bits for the binary logarithm $x$, nats for the natural logarithm $log_{2}$, bans for the decimal logarithm $ln$ and so on) are constant multiples of each other. For instance, in case of a fair coin toss, heads provides $log_{10}$ bit of information, which is approximately 0.693 nats or 0.301 decimal digits. Because of additivity, $log_{2}(2) = 1$ tosses provide $n$ bits of information, which is approximately $n$ nats or $0.693n$ decimal digits.

The meaning of the events observed (the meaning of messages) does not matter in the definition of entropy. Entropy only takes into account the probability of observing a specific event, so the information it encapsulates is information about the underlying probability distribution, not the meaning of the events themselves.

Alternative characterization
Another characterization of entropy uses the following properties. We denote $0.301n$ and $p_{i} = Pr(X = x_{i})$.

 * 1) Continuity: $Η_{n}(p_{1}, ..., p_{n}) = Η(X)$ should be continuous, so that changing the values of the probabilities by a very small amount should only change the entropy by a small amount.
 * 2) Symmetry: $H$ should be unchanged if the outcomes $H$ are re-ordered. That is, $$\Eta_n\left(p_1, p_2, \ldots p_n \right) = \Eta_n\left(p_{i_1}, p_{i_2}, \ldots, p_{i_n} \right)$$ for any permutation $$\{i_1, ..., i_n\}$$ of $$\{1, ..., n\}$$.
 * 3) Maximum: $$\Eta_n$$ should be maximal if all the outcomes are equally likely i.e. $$\Eta_n(p_1,\ldots,p_n) \le \Eta_n\left(\frac{1}{n}, \ldots, \frac{1}{n}\right)$$.
 * 4) Increasing number of outcomes: for equiprobable events, the entropy should increase with the number of outcomes i.e. $$\Eta_n\bigg(\underbrace{\frac{1}{n}, \ldots, \frac{1}{n}}_{n}\bigg) < \Eta_{n+1}\bigg(\underbrace{\frac{1}{n+1}, \ldots, \frac{1}{n+1}}_{n+1}\bigg).$$
 * 5) Additivity: given an ensemble of $x_{i}$ uniformly distributed elements that are partitioned into $n$ boxes (sub-systems) with $k$ elements each, the entropy of the whole ensemble should be equal to the sum of the entropy of the system of boxes and the individual entropies of the boxes, each weighted with the probability of being in that particular box.

Discussion
The rule of additivity has the following consequences: for positive integers $b_{1}, ..., b_{k}$ where $b_{i}$,
 * $$\Eta_n\left(\frac{1}{n}, \ldots, \frac{1}{n}\right) = \Eta_k\left(\frac{b_1}{n}, \ldots, \frac{b_k}{n}\right) + \sum_{i=1}^k \frac{b_i}{n} \, \Eta_{b_i}\left(\frac{1}{b_i}, \ldots, \frac{1}{b_i}\right).$$

Choosing $b_{1} + ... + b_{k} = n$, $k = n$ this implies that the entropy of a certain outcome is zero: $b_{1} = ... = b_{n} = 1$. This implies that the efficiency of a source set with $Η_{1}(1) = 0$ symbols can be defined simply as being equal to its $n$-ary entropy. See also Redundancy (information theory).

The characterization here imposes an additive property with respect to a partition of a set. Meanwhile, the conditional probability is defined in terms of a multiplicative property, $$P(A\mid B)\cdot P(B)=P(A\cap B)$$. Observe that a logarithm mediates between these two operations. The conditional entropy and related quantities inherit simple relation, in turn. The measure theoretic definition in the previous section defined the entropy as a sum over expected surprisals $$\mu(A)\cdot \ln\mu(A)$$ for an extremal partition. Here the logarithm is ad hoc and the entropy is not a measure in itself. At least in the information theory of a binary string, $$\log_2$$ lends itself to practical interpretations.

Motivated by such relations, a plethora of related and competing quantities have been defined. For example, David Ellerman's analysis of a "logic of partitions" defines a competing measure in structures dual to that of subsets of a universal set. Information is quantified as "dits" (distinctions), a measure on partitions. "Dits" can be converted into Shannon's bits, to get the formulas for conditional entropy, and so on.

Alternative characterization via additivity and subadditivity
Another succinct axiomatic characterization of Shannon entropy was given by Aczél, Forte and Ng, via the following properties:

 * 1) Subadditivity:  $$\Eta(X,Y) \le \Eta(X)+\Eta(Y)$$   for jointly distributed random variables $$X,Y$$.
 * 2) Additivity:  $$\Eta(X,Y) = \Eta(X)+\Eta(Y)$$ when the random variables $$X,Y$$ are independent.
 * 3) Expansibility:  $$\Eta_{n+1}(p_1, \ldots, p_n, 0) = \Eta_n(p_1, \ldots, p_n)$$, i.e., adding an outcome with probability zero does not change the entropy.
 * 4) Symmetry: $$\Eta_n(p_1, \ldots, p_n)$$ is invariant under permutation of $$p_1, \ldots, p_n$$.
 * 5) Small for small probabilities:  $$\lim_{q \to 0^+} \Eta_2(1-q, q) = 0$$.

Discussion
It was shown that any function $$\Eta$$ satisfying the above properties must be a constant multiple of Shannon entropy, with a non-negative constant. Compared to the previously mentioned characterizations of entropy, this characterization focuses on the properties of entropy as a function of random variables (subadditivity and additivity), rather than the properties of entropy as a function of the probability vector $$p_1,\ldots ,p_n$$.

It is worth noting that if we drop the "small for small probabilities" property, then $$\Eta$$ must be a non-negative linear combination of the Shannon entropy and the Hartley entropy.

Further properties
The Shannon entropy satisfies the following properties, for some of which it is useful to interpret entropy as the expected amount of information learned (or uncertainty eliminated) by revealing the value of a random variable $n$:

 * Adding or removing an event with probability zero does not contribute to the entropy:
 * $$\Eta_{n+1}(p_1,\ldots,p_n,0) = \Eta_n(p_1,\ldots,p_n)$$.

 * The maximal entropy of an event with n different outcomes is $X$: it is attained by the uniform probability distribution. That is, uncertainty is maximal when all possible events are equiprobable:
 * $$\Eta(p_1,\dots,p_n) \leq \log_b n$$.

 * The entropy or the amount of information revealed by evaluating $log_{b}(n)$ (that is, evaluating $(X,Y)$ and $X$ simultaneously) is equal to the information revealed by conducting two consecutive experiments: first evaluating the value of $Y$, then revealing the value of $Y$ given that you know the value of $X$. This may be written as:
 * $$ \Eta(X,Y)=\Eta(X|Y)+\Eta(Y)=\Eta(Y|X)+\Eta(X).$$

 * If $$Y=f(X)$$ where $$f$$ is a function, then $$\Eta(f(X)|X) = 0$$. Applying the previous formula to $$\Eta(X,f(X))$$ yields
 * $$ \Eta(X)+\Eta(f(X)|X)=\Eta(f(X))+\Eta(X|f(X)),$$
 * so $$\Eta(f(X)) \le \Eta(X)$$, the entropy of a variable can only decrease when the latter is passed through a function.

 * If $Y$ and $X$ are two independent random variables, then knowing the value of $Y$ doesn't influence our knowledge of the value of $Y$ (since the two don't influence each other by independence):
 * $$ \Eta(X|Y)=\Eta(X).$$

 * More generally, for any random variables $X$ and $X$, we have
 * $$ \Eta(X|Y)\leq \Eta(X)$$.

 * The entropy of two simultaneous events is no more than the sum of the entropies of each individual event i.e., $$ \Eta(X,Y)\leq \Eta(X)+\Eta(Y)$$, with equality if and only if the two events are independent.
 * The entropy $$\Eta(p)$$ is concave in the probability mass function $$p$$, i.e.
 * $$\Eta(\lambda p_1 + (1-\lambda) p_2) \ge \lambda \Eta(p_1) + (1-\lambda) \Eta(p_2)$$
 * for all probability mass functions $$p_1,p_2$$ and $$ 0 \le \lambda \le 1$$.
 * Accordingly, the negative entropy (negentropy) function is convex, and its convex conjugate is LogSumExp.

Relationship to thermodynamic entropy
The inspiration for adopting the word entropy in information theory came from the close resemblance between Shannon's formula and very similar known formulae from statistical mechanics.

In statistical thermodynamics the most general formula for the thermodynamic entropy $Y$ of a thermodynamic system is the Gibbs entropy
 * $$S = - k_\text{B} \sum p_i \ln p_i \,,$$

where $S$ is the Boltzmann constant, and $k_{B}$ is the probability of a microstate. The Gibbs entropy was defined by J. Willard Gibbs in 1878 after earlier work by Boltzmann (1872).

The Gibbs entropy translates over almost unchanged into the world of quantum physics to give the von Neumann entropy introduced by John von Neumann in 1927:
 * $$S = - k_\text{B} \,{\rm Tr}(\rho \ln \rho) \,,$$

where ρ is the density matrix of the quantum mechanical system and Tr is the trace.

At an everyday practical level, the links between information entropy and thermodynamic entropy are not evident. Physicists and chemists are apt to be more interested in changes in entropy as a system spontaneously evolves away from its initial conditions, in accordance with the second law of thermodynamics, rather than an unchanging probability distribution. As the minuteness of the Boltzmann constant $p_{i}$ indicates, the changes in $k_{B}$ for even tiny amounts of substances in chemical and physical processes represent amounts of entropy that are extremely large compared to anything in data compression or signal processing. In classical thermodynamics, entropy is defined in terms of macroscopic measurements and makes no reference to any probability distribution, which is central to the definition of information entropy.

The connection between thermodynamics and what is now known as information theory was first made by Ludwig Boltzmann and expressed by his equation:

 * $$S=k_\text{B} \ln W,$$

where $$S$$ is the thermodynamic entropy of a particular macrostate (defined by thermodynamic parameters such as temperature, volume, energy, etc.), $S / k_{B}$ is the number of microstates (various combinations of particles in various energy states) that can yield the given macrostate, and $W$ is the Boltzmann constant. It is assumed that each microstate is equally likely, so that the probability of a given microstate is $k_{B}$. When these probabilities are substituted into the above expression for the Gibbs entropy (or equivalently $p_{i} = 1/W$ times the Shannon entropy), Boltzmann's equation results. In information theoretic terms, the information entropy of a system is the amount of "missing" information needed to determine a microstate, given the macrostate.

In the view of Jaynes (1957), thermodynamic entropy, as explained by statistical mechanics, should be seen as an application of Shannon's information theory: the thermodynamic entropy is interpreted as being proportional to the amount of further Shannon information needed to define the detailed microscopic state of the system, that remains uncommunicated by a description solely in terms of the macroscopic variables of classical thermodynamics, with the constant of proportionality being just the Boltzmann constant. Adding heat to a system increases its thermodynamic entropy because it increases the number of possible microscopic states of the system that are consistent with the measurable values of its macroscopic variables, making any complete state description longer. (See article: maximum entropy thermodynamics). Maxwell's demon can (hypothetically) reduce the thermodynamic entropy of a system by using information about the states of individual molecules; but, as Landauer (from 1961) and co-workers have shown, to function the demon himself must increase thermodynamic entropy in the process, by at least the amount of Shannon information he proposes to first acquire and store; and so the total thermodynamic entropy does not decrease (which resolves the paradox). Landauer's principle imposes a lower bound on the amount of heat a computer must generate to process a given amount of information, though modern computers are far less efficient.

Data compression
Shannon's definition of entropy, when applied to an information source, can determine the minimum channel capacity required to reliably transmit the source as encoded binary digits. Shannon's entropy measures the information contained in a message as opposed to the portion of the message that is determined (or predictable). Examples of the latter include redundancy in language structure or statistical properties relating to the occurrence frequencies of letter or word pairs, triplets etc. The minimum channel capacity can be realized in theory by using the typical set or in practice using Huffman, Lempel–Ziv or arithmetic coding. (See also Kolmogorov complexity.) In practice, compression algorithms deliberately include some judicious redundancy in the form of checksums to protect against errors. The entropy rate of a data source is the average number of bits per symbol needed to encode it. Shannon's experiments with human predictors show an information rate between 0.6 and 1.3 bits per character in English; the PPM compression algorithm can achieve a compression ratio of 1.5 bits per character in English text.

If a compression scheme is lossless – one in which you can always recover the entire original message by decompression – 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. Shannon's theorem also implies that no lossless compression scheme can shorten all messages. If some messages come out shorter, at least one must come out longer due to the pigeonhole principle. In practical use, this is generally not a problem, because one is usually only interested in compressing certain types of messages, such as a document in English, as opposed to gibberish text, or digital photographs rather than noise, and it is unimportant if a compression algorithm makes some unlikely or uninteresting sequences larger.

A 2011 study in Science estimates the world's technological capacity to store and communicate optimally compressed information normalized on the most effective compression algorithms available in the year 2007, therefore estimating the entropy of the technologically available sources. The authors estimate humankind technological capacity to store information (fully entropically compressed) in 1986 and again in 2007. They break the information into three categories—to store information on a medium, to receive information through one-way broadcast networks, or to exchange information through two-way telecommunication networks.

Entropy as a measure of diversity
Entropy is one of several ways to measure biodiversity, and is applied in the form of the Shannon index. A diversity index is a quantitative statistical measure of how many different types exist in a dataset, such as species in a community, accounting for ecological richness, evenness, and dominance. Specifically, Shannon entropy is the logarithm of $k_{B}$, the true diversity index with parameter equal to 1. The Shannon index is related to the proportional abundances of types.

Entropy of a sequence
There are a number of entropy-related concepts that mathematically quantify information content of a sequence or message: (The "rate of self-information" can also be defined for a particular sequence of messages or symbols generated by a given stochastic process: this will always be equal to the entropy rate in the case of a stationary process.) Other quantities of information are also used to compare or relate different sources of information.
 * the self-information of an individual message or symbol taken from a given probability distribution (message or sequence seen as an individual event),
 * the joint entropy of the symbols forming the message or sequence (seen as a set of events),
 * the entropy rate of a stochastic process (message or sequence is seen as a succession of events).

It is important not to confuse the above concepts. Often it is only clear from context which one is meant. For example, when someone says that the "entropy" of the English language is about 1 bit per character, they are actually modeling the English language as a stochastic process and talking about its entropy rate. Shannon himself used the term in this way.

If very large blocks are used, the estimate of per-character entropy rate may become artificially low because the probability distribution of the sequence is not known exactly; it is only an estimate. If one considers the text of every book ever published as a sequence, with each symbol being the text of a complete book, and if there are $^{1}D$ published books, and each book is only published once, the estimate of the probability of each book is $N$, and the entropy (in bits) is $1/N$. As a practical code, this corresponds to assigning each book a unique identifier and using it in place of the text of the book whenever one wants to refer to the book. This is enormously useful for talking about books, but it is not so useful for characterizing the information content of an individual book, or of language in general: it is not possible to reconstruct the book from its identifier without knowing the probability distribution, that is, the complete text of all the books. The key idea is that the complexity of the probabilistic model must be considered. Kolmogorov complexity is a theoretical generalization of this idea that allows the consideration of the information content of a sequence independent of any particular probability model; it considers the shortest program for a universal computer that outputs the sequence. A code that achieves the entropy rate of a sequence for a given model, plus the codebook (i.e. the probabilistic model), is one such program, but it may not be the shortest.

The Fibonacci sequence is 1, 1, 2, 3, 5, 8, 13, .... treating the sequence as a message and each number as a symbol, there are almost as many symbols as there are characters in the message, giving an entropy of approximately $−log_{2}(1/N) = log_{2}(N)$. The first 128 symbols of the Fibonacci sequence has an entropy of approximately 7 bits/symbol, but the sequence can be expressed using a formula [$log_{2}(n)$ for $F(n) = F(n−1) + F(n−2)$, $n = 3, 4, 5, ...$, $F(1) =1$] and this formula has a much lower entropy and applies to any length of the Fibonacci sequence.

Limitations of entropy in cryptography
In cryptanalysis, entropy is often roughly used as a measure of the unpredictability of a cryptographic key, though its real uncertainty is unmeasurable. For example, a 128-bit key that is uniformly and randomly generated has 128 bits of entropy. It also takes (on average) $$2^{127}$$ guesses to break by brute force. Entropy fails to capture the number of guesses required if the possible keys are not chosen uniformly. Instead, a measure called guesswork can be used to measure the effort required for a brute force attack.

Other problems may arise from non-uniform distributions used in cryptography. For example, a 1,000,000-digit binary one-time pad using exclusive or. If the pad has 1,000,000 bits of entropy, it is perfect. If the pad has 999,999 bits of entropy, evenly distributed (each individual bit of the pad having 0.999999 bits of entropy) it may provide good security. But if the pad has 999,999 bits of entropy, where the first bit is fixed and the remaining 999,999 bits are perfectly random, the first bit of the ciphertext will not be encrypted at all.

Data as a Markov process
A common way to define entropy for text is based on the Markov model of text. For an order-0 source (each character is selected independent of the last characters), the binary entropy is:

 * $$\Eta(\mathcal{S}) = - \sum p_i \log p_i ,$$

where $F(2) = 1$ is the probability of $p_{i}$. For a first-order Markov source (one in which the probability of selecting a character is dependent only on the immediately preceding character), the entropy rate is:

 * $$\Eta(\mathcal{S}) = - \sum_i p_i \sum_j \  p_i (j) \log p_i (j) ,$$

where $i$ is a state (certain preceding characters) and $$p_i(j)$$ is the probability of $i$ given $j$ as the previous character.

For a second order Markov source, the entropy rate is

 * $$\Eta(\mathcal{S}) = -\sum_i p_i \sum_j p_i(j) \sum_k p_{i,j}(k)\ \log \ p_{i,j}(k) .$$

Efficiency (normalized entropy)
A source set $$\mathcal{X}$$ with a non-uniform distribution will have less entropy than the same set with a uniform distribution (i.e. the "optimized alphabet"). This deficiency in entropy can be expressed as a ratio called efficiency:

 * $$\eta(X) = \frac{H}{H_{max}} = -\sum_{i=1}^n \frac{p(x_i) \log_b (p(x_i))}{\log_b (n)}.

$$ Applying the basic properties of the logarithm, this quantity can also be expressed as:
 * $$\eta(X) = -\sum_{i=1}^n \frac{p(x_i) \log_b (p(x_i))}{\log_b (n)} = \sum_{i=1}^n \frac{\log_b(p(x_i)^{-p(x_i)})}{\log_b(n)} =

\sum_{i=1}^n \log_n(p(x_i)^{-p(x_i)}) = \log_n (\prod_{i=1}^n p(x_i)^{-p(x_i)}). $$

Efficiency has utility in quantifying the effective use of a communication channel. This formulation is also referred to as the normalized entropy, as the entropy is divided by the maximum entropy $${\log_b (n)}$$. Furthermore, the efficiency is indifferent to choice of (positive) base $i$, as indicated by the insensitivity within the final logarithm above thereto.

Differential entropy
The Shannon entropy is restricted to random variables taking discrete values. The corresponding formula for a continuous random variable with probability density function $b$ with finite or infinite support $$\mathbb X$$ on the real line is defined by analogy, using the above form of the entropy as an expectation:

 * $$\Eta(X) = \mathbb{E}[-\log f(X)] = -\int_\mathbb X f(x) \log f(x)\, \mathrm{d}x.$$

This is the differential entropy (or continuous entropy). A precursor of the continuous entropy $f(x)$ is the expression for the functional $h[f]$ in the H-theorem of Boltzmann.

Although the analogy between both functions is suggestive, the following question must be set: is the differential entropy a valid extension of the Shannon discrete entropy? Differential entropy lacks a number of properties that the Shannon discrete entropy has – it can even be negative – and corrections have been suggested, notably limiting density of discrete points.

To answer this question, a connection must be established between the two functions:

In order to obtain a generally finite measure as the bin size goes to zero. In the discrete case, the bin size is the (implicit) width of each of the $Η$ (finite or infinite) bins whose probabilities are denoted by $n$. As the continuous domain is generalized, the width must be made explicit.

To do this, start with a continuous function $p_{n}$ discretized into bins of size $$\Delta$$.

By the mean-value theorem there exists a value $f$ in each bin such that $$f(x_i) \Delta = \int_{i\Delta}^{(i+1)\Delta} f(x)\, dx$$ the integral of the function $x_{i}$ can be approximated (in the Riemannian sense) by $$\int_{-\infty}^{\infty} f(x)\, dx = \lim_{\Delta \to 0} \sum_{i = -\infty}^{\infty} f(x_i) \Delta ,$$ where this limit and "bin size goes to zero" are equivalent.

We will denote $$\Eta^{\Delta} := - \sum_{i=-\infty}^{\infty} f(x_i) \Delta \log \left(  f(x_i)  \Delta \right)$$ and expanding the logarithm, we have $$\Eta^{\Delta} = - \sum_{i=-\infty}^{\infty} f(x_i)  \Delta \log (f(x_i)) -\sum_{i=-\infty}^{\infty} f(x_i) \Delta \log (\Delta).$$

As $f$, we have

 * $$\begin{align}

\sum_{i=-\infty}^{\infty} f(x_i) \Delta &\to \int_{-\infty}^{\infty} f(x)\, dx = 1 \\ \sum_{i=-\infty}^{\infty} f(x_i) \Delta \log (f(x_i)) &\to \int_{-\infty}^{\infty} f(x) \log f(x)\, dx. \end{align}$$

Note; $Δ → 0$ as $log(Δ) → −∞$, requires a special definition of the differential or continuous entropy:

 * $$h[f] = \lim_{\Delta \to 0} \left(\Eta^{\Delta} + \log \Delta\right) = -\int_{-\infty}^{\infty} f(x) \log f(x)\,dx,$$

which is, as said before, referred to as the differential entropy. This means that the differential entropy is not a limit of the Shannon entropy for $Δ → 0$. Rather, it differs from the limit of the Shannon entropy by an infinite offset (see also the article on information dimension).

Limiting density of discrete points
It turns out as a result that, unlike the Shannon entropy, the differential entropy is not in general a good measure of uncertainty or information. For example, the differential entropy can be negative; also it is not invariant under continuous co-ordinate transformations. This problem may be illustrated by a change of units when $n → ∞$ is a dimensioned variable. $x$ will then have the units of $f(x)$. The argument of the logarithm must be dimensionless, otherwise it is improper, so that the differential entropy as given above will be improper. If $1/x$ is some "standard" value of $Δ$ (i.e. "bin size") and therefore has the same units, then a modified differential entropy may be written in proper form as:
 * $$\Eta=\int_{-\infty}^\infty f(x) \log(f(x)\,\Delta)\,dx ,$$

and the result will be the same for any choice of units for $x$. In fact, the limit of discrete entropy as $$ N \rightarrow \infty $$ would also include a term of $$ \log(N)$$, which would in general be infinite. This is expected: continuous variables would typically have infinite entropy when discretized. The limiting density of discrete points is really a measure of how much easier a distribution is to describe than a distribution that is uniform over its quantization scheme.

Relative entropy
Another useful measure of entropy that works equally well in the discrete and the continuous case is the relative entropy of a distribution. It is defined as the Kullback–Leibler divergence from the distribution to a reference measure $x$ as follows. Assume that a probability distribution $m$ is absolutely continuous with respect to a measure $p$, i.e. is of the form $m$ for some non-negative $p(dx) = f(x)m(dx)$-integrable function $m$ with $f$-integral 1, then the relative entropy can be defined as
 * $$D_{\mathrm{KL}}(p \| m ) = \int \log (f(x)) p(dx) = \int f(x)\log (f(x)) m(dx) .$$

In this form the relative entropy generalizes (up to change in sign) both the discrete entropy, where the measure $m$ is the counting measure, and the differential entropy, where the measure $m$ is the Lebesgue measure. If the measure $m$ is itself a probability distribution, the relative entropy is non-negative, and zero if $m$ as measures. It is defined for any measure space, hence coordinate independent and invariant under co-ordinate reparameterizations if one properly takes into account the transformation of the measure $p = m$. The relative entropy, and (implicitly) entropy and differential entropy, do depend on the "reference" measure $m$.

Use in number theory
Terence Tao used entropy to make a useful connection trying to solve the Erdős discrepancy problem.

Intuitively the idea behind the proof was if there is low information in terms of the Shannon entropy between consecutive random variables (here the random variable is defined using the Liouville function (which is a useful mathematical function for studying distribution of primes) $m$ = $$\lambda(n+H)$$. And in an interval [n, n+H] the sum over that interval could become arbitrary large.  For example, a sequence of +1's (which are values of $X_{H}$ could take) have trivially low entropy and their sum would become big. But the key insight was showing a reduction in entropy by non negligible amounts as one expands H leading inturn to unbounded growth of a mathematical object over this random variable is equivalent to showing the unbounded growth per the Erdős discrepancy problem.

The proof is quite involved and it bought together breakthroughs not just in novel use of Shannon Entropy, but also its used the Liouville function along with averages of modulated multiplicative functions in short intervals. Proving it also broke the "parity barrier" for this specific problem.

While the use of Shannon Entropy in the proof is novel it is likely to open new research in this direction.

Use in combinatorics
Entropy has become a useful quantity in combinatorics.

Loomis–Whitney inequality
A simple example of this is an alternative proof of the Loomis–Whitney inequality: for every subset $X_{H'}$, we have
 * $$ |A|^{d-1}\leq \prod_{i=1}^{d} |P_{i}(A)|$$

where $A ⊆ Z^{d}$ is the orthogonal projection in the $P_{i}$th coordinate:
 * $$ P_{i}(A)=\{(x_{1}, \ldots, x_{i-1}, x_{i+1}, \ldots, x_{d}) : (x_{1}, \ldots, x_{d})\in A\}.$$

The proof follows as a simple corollary of Shearer's inequality: if $i$ are random variables and $X_{1}, ..., X_{d}$ are subsets of $S_{1}, ..., S_{n}$ such that every integer between 1 and ${1, ..., d}$ lies in exactly $d$ of these subsets, then
 * $$ \Eta[(X_{1}, \ldots ,X_{d})]\leq \frac{1}{r}\sum_{i=1}^{n}\Eta[(X_{j})_{j\in S_{i}}]$$

where $$ (X_{j})_{j\in S_{i}}$$ is the Cartesian product of random variables $r$ with indexes $X_{j}$ in $j$ (so the dimension of this vector is equal to the size of $S_{i}$).

We sketch how Loomis–Whitney follows from this: Indeed, let $S_{i}$ be a uniformly distributed random variable with values in $X$ and so that each point in $A$ occurs with equal probability. Then (by the further properties of entropy mentioned above) $A$, where $Η(X) = log|A|$ denotes the cardinality of $|A|$. Let $A$. The range of $$(X_{j})_{j\in S_{i}}$$ is contained in $S_{i} = {1, 2, ..., i−1, i+1, ..., d}$ and hence $$ \Eta[(X_{j})_{j\in S_{i}}]\leq \log |P_{i}(A)|$$. Now use this to bound the right side of Shearer's inequality and exponentiate the opposite sides of the resulting inequality you obtain.

Approximation to binomial coefficient
For integers $P_{i}(A)$ let $0 < k < n$. Then
 * $$\frac{2^{n\Eta(q)}}{n+1} \leq \tbinom nk \leq 2^{n\Eta(q)},$$

where
 * $$\Eta(q) = -q \log_2(q) - (1-q) \log_2(1-q).$$

 * {| class="toccolours collapsible collapsed" width="80%" style="text-align:left"

!Proof (sketch)
 * Note that $$\tbinom nk q^{qn}(1-q)^{n-nq}$$ is one term of the expression
 * $$\sum_{i=0}^n \tbinom ni q^i(1-q)^{n-i} = (q + (1-q))^n = 1.$$
 * $$\sum_{i=0}^n \tbinom ni q^i(1-q)^{n-i} = (q + (1-q))^n = 1.$$

Rearranging gives the upper bound. For the lower bound one first shows, using some algebra, that it is the largest term in the summation. But then,
 * $$\binom nk q^{qn}(1-q)^{n-nq} \geq \frac{1}{n+1}$$

since there are $q = k/n$ terms in the summation. Rearranging gives the lower bound.
 * }

A nice interpretation of this is that the number of binary strings of length $n + 1$ with exactly $n$ many 1's is approximately $$2^{n\Eta(k/n)}$$.

Use in machine learning
Machine learning techniques arise largely from statistics and also information theory. In general, entropy is a measure of uncertainty and the objective of machine learning is to minimize uncertainty.

Decision tree learning algorithms use relative entropy to determine the decision rules that govern the data at each node. The information gain in decision trees $$IG(Y,X)$$, which is equal to the difference between the entropy of $$Y$$ and the conditional entropy of $$Y$$ given $$X$$, quantifies the expected information, or the reduction in entropy, from additionally knowing the value of an attribute $$X$$. The information gain is used to identify which attributes of the dataset provide the most information and should be used to split the nodes of the tree optimally.

Bayesian inference models often apply the principle of maximum entropy to obtain prior probability distributions. The idea is that the distribution that best represents the current state of knowledge of a system is the one with the largest entropy, and is therefore suitable to be the prior.

Classification in machine learning performed by logistic regression or artificial neural networks often employs a standard loss function, called cross-entropy loss, that minimizes the average cross entropy between ground truth and predicted distributions. In general, cross entropy is a measure of the differences between two datasets similar to the KL divergence (also known as relative entropy).

Textbooks on information theory

 * Cover, T.M., Thomas, J.A. (2006), Elements of Information Theory – 2nd Ed., Wiley-Interscience, ISBN 978-0-471-24195-9
 * MacKay, D.J.C. (2003), Information Theory, Inference and Learning Algorithms, Cambridge University Press, ISBN 978-0-521-64298-9
 * Arndt, C. (2004), Information Measures: Information and its Description in Science and Engineering, Springer, ISBN 978-3-540-40855-0
 * Gray, R. M. (2011), Entropy and Information Theory, Springer.
 * Shannon, C.E., Weaver, W. (1949) The Mathematical Theory of Communication, Univ of Illinois Press. ISBN 0-252-72548-4
 * Stone, J. V. (2014), Chapter 1 of Information Theory: A Tutorial Introduction, University of Sheffield, England. ISBN 978-0956372857.
 * Stone, J. V. (2014), Chapter 1 of Information Theory: A Tutorial Introduction, University of Sheffield, England. ISBN 978-0956372857.

title: Entropy (information theory)


text:
Ithaca College

Ithaca College is a private college in Ithaca, New York. It was founded by William Egbert in 1892 as a conservatory of music. Ithaca College is known for its media-related programs and entertainment programs within the Roy H. Park School of Communications and the Ithaca College School of Music, Theatre, and Dance. The college has a liberal arts focus, and offers several pre-professional programs, along with some graduate programs.

Beginnings
Ithaca College was founded as the Ithaca Conservatory of Music in 1892 when a local violin teacher, William Grant Egbert, rented four rooms and arranged for the instruction of eight students. For nearly seven decades the institution flourished in the city of Ithaca, adding to its music curriculum the study of elocution, dance, physical education, speech correction, radio, business, and the liberal arts. In 1931 the conservatory was chartered as a private college under its current name, Ithaca College. The college was originally housed in the Boardman House, that later became the Ithaca College Museum of Art, and it was listed on the National Register of Historic Places in 1971.

Modern era
By 1960, some 2,000 students were in attendance. A modern campus was built on South Hill in the 1960s, and students were shuttled between the old and new during the construction. The hillside campus continued to grow in the ensuing 30 years to accommodate more than 6,000 students.

As the campus expanded, the college also began to expand its curriculum. By the 1990s, some 2,000 courses in more than 100 programs of study were available in the college's five schools. The school attracts a multicultural student body with representatives from almost every state and from 78 foreign countries. In October 2020, the college announced that 130 out of 547 faculty positions would be cut due to a need to cut $30 million from the school's budget. This in turn was said to be a result of declining enrollment. 4,957 undergraduate students enrolled for Fall 2020 versus 5,852 undergraduates in Fall 2019 and 6,101 in Fall 2018.

Presidents
Ithaca's current president is Dr. LaJerne Terry Cornish. She was named the school's 10th President in March 2022 after having served in as interim President since August 30, 2021.

She replaced Shirley M. Collado who departed Ithaca College to become the president and CEO of College Track, a comprehensive college completion program. She was named the ninth president of Ithaca College on February 22, 2017, and assumed the presidency on July 1, 2017. She was previously executive vice chancellor and chief operating officer at Rutgers University–Newark and vice president of student affairs and dean of the college at Middlebury College. She is the first Dominican American to be named president of a college in the United States. During Collado's time as president she was the center of multiple controversies. Collado faced backlash when students and faculty discovered she was accused of sexually abusing a female patient while working as a psychologist in Washington, D.C., in 2000 and was convicted of sexual abuse in 2001. Students further questioned her transparency when she announced plans to cut 116 full-time faculty members, some of whom had worked at the school for decades, after receiving a $172,769 payment. Collado eventually announced in July 2021 that she will step down in January to become president and CEO of College Track.

Collado succeeded Thomas Rochon, who was named eighth president of Ithaca College on April 11, 2008. Rochon took over as president of the college following Peggy Williams, who had announced on July 12, 2007, that she would retire from the presidency post effective May 31, 2009, following a one-year sabbatical. During the fall 2015 semester, multiple protests focusing on campus climate and Rochon's leadership were led by students and faculty. After multiple racially charged events including student house party themes and racially tinged comments at administration led-programs, students, faculty and staff all decided to hold votes of "no confidence" in Rochon. Students voted "no confidence" by a count of 72% no confidence, 27% confidence, and 1% abstaining. The faculty voted 77.8% no confidence to 22.2% confidence. Rochon retired on July 1, 2017.

Campus
Ithaca College's current campus was built in the 1960s on South Hill. The college's final academic department moved from downtown to the South Hill campus in 1968, making the move complete.

Satellite campuses
Besides its Ithaca campus, Ithaca College has also operated satellite campuses in other cities. The Ithaca College London Center has been in existence since 1972. Ithaca runs the Ithaca College Los Angeles Program at the James B. Pendleton Center.

Former programs include the Ithaca College Antigua Program and the Ithaca College Walkabout Down Under Program in Australia.

Ithaca College also operates direct enrollment exchange programs with several universities, including Griffith University, La Trobe University, Murdoch University, and University of Tasmania (Australia); Chengdu Sport University and Beijing Sport University (China); University of Hong Kong (Hong Kong); Masaryk University (Czech Republic); Akita International University and University of Tsukuba (Japan); Hanyang University (Korea); Nanyang Technological University (Singapore); University of Valencia (Spain); and Jönköping University (Sweden). Ithaca College is also affiliated with study abroad programs such as IES Abroad and offers dozens of exchange or study abroad options to students.

Academics

The college offers a curriculum with more than 100 degree programs in its five schools:

 * School of Business
 * Roy H. Park School of Communications
 * School of Health Sciences & Human Performance
 * School of Humanities & Sciences
 * School of Music, Theatre, and Dance

Until the spring of 2011, several cross-disciplinary degree programs, along with the Center for the Study of Culture, Race, and Ethnicity, were housed in the Division of Interdisciplinary and International Studies; in 2011, the division was eliminated and its programs, centers and institutes were absorbed into other schools.

, the most popular majors included visual and performing arts, health professions and related programs, business, management, marketing, and related support services and biological and biomedical Sciences.

Media and publications

 * The Ithacan is Ithaca College's official weekly newspaper that is written, edited and published by students. The Ithacan and its staff have won over 200 major collegiate journalism awards and is generally recognized as one of the top student-run newspapers in the country. Most notably, the newspaper is a consistent recipient of the Associated Collegiate Press' National Pacemaker Award; it has received the National Newspaper Pacemaker Award six times and the Online Pacemaker Award nine times (both most recently in 2015). The Pacemaker has been widely considered the "Pulitzer Prize of collegiate journalism." The Ithacan is also a five-time recipient of the Gold Crown Award from the Columbia Scholastic Press Association, most recently receiving the award in 2016. The Ithacan was also ranked #3 on the 2018 The Princeton Review Best College Newspaper list.
 * Ithaca College Television (ICTV) is the world's oldest student-operated college television channel. Broadcasting since 1958, ICTV is available to 26,000 cable households. It is also one of the most awarded student-run television stations, with its news program, Newswatch, receiving best news telecast accolades from organizations including the New York State Associated Press Broadcasters Association, Society of Professional Journalists and Collegiate Broadcasters Inc. The show also received ICTV's first College Emmy Award from the Academy of Television Arts & Sciences. ICTV is housed and operated in the Roy H. Park School of Communications. Approximately 15 to 20 production teams operate simultaneously, utilizing around 400 volunteers each semester. Programming varies by semester, but typically includes news, sports, entertainment, scripted, and podcast programs.
 * WICB is a student-operated, 4,100 Watt FM station that serves Tompkins County and beyond, reaching from northern Pennsylvania to Lake Ontario, with a potential audience of over 250,000. The majority of programming on WICB – which broadcasts from 91.7 on the FM band – is modern rock, but the station also airs a number of specialty shows, which includes a number of genres (including blues, Broadway, jam band music and "homeless" music) that is not normally heard on public airwaves. Recently, readers of the Ithaca Times voted 92 WICB "Best Radio Station." WICB has also won the MTV U's Woodie Award for Best College Radio, while the Princeton Review ranks WICB the number one college radio station in the country.
 * VIC Radio is Ithaca College's second student-run radio station. Previously available on 105.9 FM, VIC Radio is now an online-only radio station. It is most well known for its annual 50 Hour Marathon, in which four DJs broadcast for 50 hours straight to raise money for local community organizations.
 * Buzzsaw Magazine, formerly Buzzsaw Haircut, was founded in 1999 and is an independent monthly alternative magazine written, produced and distributed by Ithaca College students. It is a progressive publication with a goal to "publish original creative journalism, commentary and satire that works to deconstruct society, pop culture, politics, college life and dominant Western beliefs." The faculty adviser is media critic Jeff Cohen, who is also the founder of the college's Park Center for Independent Media. In 2011, the organization added a new multimedia section to Buzzsaw, titled Seesaw, dedicated to creating documentaries, radio pieces, interactive graphics, and other multimedia pieces to complement the print and online magazine. Buzzsaw has also won a number of national awards, including the Campus Alternative Journalism Project's award for "Best Sense of Humor" and the Independent Press Association's Campus Independent Journalism Awards for "Best Campus Publication with a Budget Under $10,000" and "Best Political Commentary."
 * Park Productions is a professional production unit within the Roy H. Park School of Communications which allows students to collaborate with faculty and industry professionals to create interdisciplinary media projects. Park Productions partners with community organizations, government agencies, and higher education institutions and has produced over 200 titles including documentaries, feature films, shorts, commercials, museum exhibits, television programs, educational, corporate, and web-based media. Awards and juried screenings include LA Webfest, Mexico International Film Festival, CINE Awards, Chicago International Film Festival, Official selections at Miami, University Film and Video Festival, Cinema in Industry Awards, Multiple International Communicator Awards, Oberhausen, Montreal, Palm Springs, and Hudson Valley Film Festivals.
 * Distinct Magazine is self described as Ithaca College's "fashion magazine devoted to the style and culture of the students on campus...[it aims] to break gender and social class stereotypes in the fashion world, and to build a safe space for people to express themselves." The first issue was released online in 2016. The magazine is separated into five content sections: Fashion, Beauty, Life, Culture, and Health and Fitness. Distinct is released in print twice a semester (Fall, Winter, Spring and Summer editions).
 * Embrace is an IC magazine that aims "to create a platform in which underrepresented students are able to see a representation of themselves on campus and within society." The magazine is separated into content sections: LGBTQ+, Fashion, Politics and News, Mind Body Spirit, Personal Narratives and Alumni Highlight. It was first published in February 2016.

Greek life
Historically, various independent and national fraternities and sororities had active chapters at Ithaca College. However, due to a series of highly publicized hazing incidents in the 1980s, including one that was responsible for the death of a student, the college administration reevaluated their Greek life policy and only professional music fraternities were allowed to remain affiliated with the school.

, three recognized Greek organizations remain on campus, all of which are music-oriented:

 * Phi Mu Alpha Sinfonia (Delta Chapter)
 * Sigma Alpha Iota (Epsilon Chapter)
 * Mu Phi Epsilon (Lambda Chapter)

A fourth house, performing arts fraternity Kappa Gamma Psi (Iota Chapter) became inactive in 2008. Although there are potentially plans to reactivate the chapter, it is unclear whether this will be permitted or not due to the college's policy on Greek Life.

However, there are various Greek letter organizations at Ithaca College that are unaffiliated with the school, and therefore not subject to the same housing privileges or rules that contribute to the safety of their members such as non-hazing and non-drinking policies. Additionally, while not particularly common, Ithaca College students may rush for Greek houses affiliated with nearby Ivy institution Cornell University, subject to the rules of each individual fraternity or sorority. Some Cornell-affiliated Greek organizations actively recruit Ithaca College students.

There are a few unaffiliated fraternities that some Ithaca College students join - ΔΚΕ (Delta Kappa Epsilon), ΑΕΠ (Alpha Epsilon Pi), ΦΚΣ (Phi Kappa Sigma), ΦΙΑ (Phi Iota Alpha), ΛΥΛ (Lambda Upsilon Lambda), and ΚΣ (Kappa Sigma). There are also unaffiliated sororities including - ΓΔΠ (Gamma Delta Pi), ΠΛΧ (Pi Lambda Chi), ΦΜΖ (Phi Mu Zeta),.

Athletics
Ithaca competes in athletics at the NCAA Division III level as a members of the Liberty League and the Eastern College Athletic Conference (ECAC). Ithaca has one of Division III's strongest athletic programs, with the Bombers winning a total of 14 national titles in seven team sports and five individual sports. Ithaca was previously a member of the Empire 8.

The Ithaca athletics nickname "Bombers" is unique in NCAA athletics, and the origins of the nickname are obscure. Ithaca College's sports teams were originally named the Cayugas, but the name was changed to the Bombers sometime in the 1930s. Some other names that have been used for Ithaca College's teams include: Blue Team, Blues, Blue and Gold, Collegians, and the Seneca Streeters. Several possibilities for the change to the "Bombers" have been posited. The most common explanation is that the school's baseball uniforms—white with navy blue pinstripes and an interlocking "IC" on the left chest—bear a striking resemblance to the distinctive home uniforms of the New York Yankees, who are known as the Bronx Bombers. It may also have referred to the Ithaca basketball team of that era and its propensity for half-court "bombs". Grumman Aircraft also manufactured airplanes including bombers in Ithaca for many years. The first "Bombers" reference on record was in the December 17, 1938 issue of the Rochester Times-Union in a men's basketball article.

The name has at times sparked controversy for its perceived violent connotations. It is an occasional source of umbrage from Ithaca's prominent pacifist community, but the athletics department has consistently stated it has no interest in changing the name. The athletics logo has in the past incorporated World War II era fighter planes, but currently does not, and the school does not currently have a physical mascot to personify the name. In 2010 the school launched a contest to choose one. It received over 250 suggestions and narrowed the field down to three: a phoenix, a flying squirrel, and a Lake Beast. In June 2011, President Rochon announced that the school would discontinue the search due to opposition in the alumni community.

Ithaca College remodeled the Hill Center in 2013. The building features hardwood floors (Ben Light Gymnasium) as well as coaches offices. The building is home to Ithaca's men's and women's basketball teams, women's volleyball team, wrestling, and gymnastics. Ithaca also opened the Athletics & Events Center in 2011, a $65.5 million facility funded by donors. The facility is mainly used by the school's varsity athletes. It has a 47,000 square foot, 9-lane 50 meter Olympic-size pool. The building also has Glazer Arena, a 130,000 square foot event space. It is a track and field center that doubles as a practice facility for lacrosse, field hockey, soccer, baseball, tennis, and football. The facility was designed by the architectural firm Moody Nolan and began construction in June 2009.

Coached by Jim Butterfield for 27 years, the football team has won three NCAA Division III Football Championships in 1979, 1988 and 1991 (a total surpassed only by Augustana College, Mount Union and the Wisconsin–Whitewater). Bomber football teams made a record seven appearances in the Division III national championship game, the Amos Alonzo Stagg Bowl, which has since been surpassed by Mount Union in 2003. The Bombers play the SUNY Cortland Red Dragons for the Cortaca Jug, which was added in 1959 to an already competitive rivalry. The match-up is one of the most prominent in Division III college football.

Gymnastics won the NCAA Division III national championships in 1998.

Women's field hockey won the 1982 NCAA Division III Field Hockey Championship.

The Men's and Women's Crew programs are housed in the Robert B. Tallman Rowing Center, a $2.6 million boathouse dedicated in 2012. The new boathouse replaced the Haskell Davidson Boathouse, which was constructed in 1974 on Cayuga Inlet. The old boathouse was razed to make room for the new facility. At 8,500 square feet, the Tallman boathouse is almost twice the size of the previous structure.

Intramurals
Along with Intercollegiate athletics, Ithaca College has a large intramural sport program. This extracurricular program serves approximately 25% of the undergraduate population yearly. Fourteen traditional team activities are offered throughout the year and include basketball, flag football, kickball, soccer, softball, ultimate, ski racing, and volleyball.

For most activities, divisions are offered for men's, women's, and co-recreational teams. Throughout the year usually two or more activities run concurrently and participants are able to play on a single sex team and co-recreational team for each activity.

Sustainability
Ithaca's School of Business was the first college or university business school in the world to achieve LEED Platinum Certification alongside Yale University, which had the second. Ithaca's Peggy Ryan Williams Center is also LEED Platinum certified. It makes extensive use of day light in occupied spaces. There are sensors that regulate lighting and ventilation based on occupancy and natural light. Over 50% of the building energy comes from renewable sources such as wind power. The college also has a LEED Gold Certified building, the Athletics & Events Center. The college composts its dining hall waste, runs a "Take It or Leave It" Green move-out program, and offers a sustainable living option. It also operates an office supply collection and reuse program, as well as a sustainability education program during new student orientation. Ithaca received a B− grade on the Sustainable Endowments Institute's 2009 College Sustainability Report Card and an A− for 2010.

In 2017, Ithaca College was listed as one of Princeton Review's top "green colleges" for being environmentally responsible.

Commitments to action on climate change
In the spring of 2007, then-President Peggy R. Williams signed the American College & University President's Climate Commitment (ACUPCC), pledging Ithaca College to the task of developing a strategy and long-range plan to achieve "carbon neutrality" at some point in the future. In 2009 the Ithaca College Board of Trustees approved the Ithaca College Climate Action Plan, which calls for 100% carbon neutrality by 2050. In 2009, the Ithaca College Board of Trustees approved the Ithaca College Climate Action Plan, which calls for 100% carbon neutrality by 2050 and offers a 40-year action plan to work toward that ambitious goal.

Energy profile
The college purchases 100 percent of its electricity from renewable sources. Including offsets from a solar farm, the college's overall energy usage is 45 percent carbon neutral.

Energy investments
The college aims to optimize investment returns and does not invest the endowment in on-campus sustainability projects, renewable energy funds, or community development loan funds. The college's investment policy reserves the right of the investment committee to restrict investments for any reason, which could include environmental and sustainability factors.

Community impact
While the Ithaca College Natural Lands has issued a statement that Ithaca College should join efforts calling for a moratorium on horizontal drilling and high volume ("slick water") hydraulic fracturing, or fracking, the college as a whole has refused to issue a statement regarding the issue.

Notable alumni
Ithaca College has over 70,000 alumni, with clubs in Boston, Chicago, Connecticut, Los Angeles, Metro New York, National Capital, North and South Carolina, Philadelphia, Rochester (NY), San Diego, and Southern Florida. Alumni events are hosted in cooperation with city-specific clubs and through a program called "IC on the Road".

Notable faculty
Notable current and former Ithaca College faculty include:
 * Asma Barlas, politics, director of the Center for the Study of Culture, Race, and Ethnicity. Specializes in comparative and international politics; women, gender, and Islam; Islam and Qur'anic hermeneutics
 * Rick Beato, music educator and producer
 * Jeff Cohen, journalism, and founding director, Park Center for Independent Media; media critic and commentator, author, founder of Fairness & Accuracy in Reporting (FAIR) media watchdog group
 * Patrick Conway, famous early 20th-century bandleader
 * Andrew Ezergailis, history; expert in 20th-century history of Latvia
 * Robert Allen Iger, an American media business executive who serves as the chief executive officer (CEO) of The Walt Disney Company.
 * A. Van Jordan, poet
 * Marisa Kelly, political scientist, President of Suffolk University in Boston, Massachusetts
 * Nicholas Muellner, media arts, sciences and studies; photographer and writer whose work has been acclaimed by Time magazine and the Paris Photo–Aperture Foundation
 * Alex Perialas, sound recording technology; acclaimed audio engineer and record producer known for his extensive work during the "golden age" of thrash metal
 * Rod Serling, communications; creator, producer and host of The Twilight Zone
 * Saviana Stănescu, theater; poet, playwright, script analyst, journalist
 * Gordon Stout, music; percussionist, composer, specializes in marimba
 * Fred A. Wilcox, writing; nonfiction writer, fiction writer, expert on Vietnam War and effects of U.S. use of Agent Orange on U.S. military members and Vietnamese people (retired 2014)
 * Dana Wilson, the Charles A. Dana Professor of Music; composer, jazz pianist

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title: Ithaca College


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Differential psychology

Differential psychology studies the ways in which individuals differ in their behavior and the processes that underlie it. This is a discipline that develops classifications (taxonomies) of psychological individual differences. This is distinguished from other aspects of psychology in that although psychology is ostensibly a study of individuals, modern psychologists often study groups, or attempt to discover general psychological processes that apply to all individuals. This particular area of psychology was first named and still retains the name of "differential psychology" by William Stern in his book (1900).

While prominent psychologists, including Stern, have been widely credited for the concept of individual differences, historical records show that it was Charles Darwin (1859) who first spurred the scientific interest in the study of individual differences. His interest was further pursued by his half-cousin Francis Galton in his attempt to quantify individual differences among people.

For example, in evaluating the effectiveness of a new therapy, the mean performance of the therapy in one treatment group might be compared to the mean effectiveness of a placebo (or a well-known therapy) in a second, control group. In this context, differences between individuals in their reaction to the experimental and control manipulations are actually treated as errors rather than as interesting phenomena to study. This approach is applied because psychological research depends upon statistical controls that are only defined upon groups of people.

Importance of individual differences
Importantly, individuals can also differ not only in their current state, but in the magnitude or even direction of response to a given stimulus. Such phenomena, often explained in terms of inverted-U response curves, place differential psychology at an important location in such endeavours as personalized medicine, in which diagnoses are customised for an individual's response profile.

Areas of study
Individual differences research typically includes personality, temperament (neuro-chemically based behavioural traits), motivation, intelligence, ability, IQ, interests, values, self-concept, self-efficacy, and self-esteem (to name just a few). Although the United States has seen a decrease in individual differences research since the 1960s, researchers are found in a variety of applied and experimental fields. These fields include clinical psychology, psychophysiology, educational psychology, Industrial and organizational psychology, personality psychology, social psychology, behavioral genetics, and developmental psychology programs, in the neo-Piagetian theories of cognitive development in particular.

Methods of research
To study individual differences, psychologists use a variety of methods. The method is to compare and analyze the psychology and behaviour of individuals or groups under different environmental conditions. By correlating observed psychological and behavioral differences with known accompanying environments, the relative roles of different variables in psychological and behavioral development can be probed. Psychophysiological experiments on both humans and other mammals include EEG and ERPs, PET-scans, MRI, functional MRI, neurochemistry experiments with neurotransmitter and hormonal systems, caffeine and controlled drug challenges. These methods can be used for a search of biomarkers of consistent, biologically based behavioural patterns (temperament traits and symptoms of psychiatric disorders). Other sets of methods include behavioural experiments, to see how different people behave in similar settings. Behavioural experiments are often used in personality and social psychology, and include lexical and self-report methods where people are asked to complete paper-based and computer-based forms prepared by psychologists.

title: Differential psychology