Tossing 24 coins a few times might give you a feel for the surprisal of getting all heads on the first try. The additive nature of this measure also comes in handy when weighing alternatives. For example, imagine that the surprisal of harm from a vaccination is 20 bits. If the surprisal of catching a disease without it is 16 bits, but the surprisal of harm from the disease if you catch it is 2 bits, then the surprisal of harm from NOT getting the vaccination is only 16+2=18 bits.
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Later, surprisal analysis was extended to mesoscopic systems, bulk systems and to dynamical processes.
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However, Jaynes pointed out that with true-false assertions one can also define bits of evidence ebits as the surprisal against minus the surprisal for. This evidence in bits relates simply to the odds ratio = p/(1-p) = 2ebits, and has advantages similar to those of self-information itself.
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Maximum entropy methods are at the core of a new view of scientific inference, allowing analysis and interpretation of large and sometimes noisy data. Surprisal analysis extends principles of maximal entropy and of thermodynamics, where both equilibrium thermodynamics and statistical mechanics are assumed to be inferences processes. This enables surprisal analysis to be an effective method of information quantification and compaction and of providing an unbiased characterization of systems. Surprisal analysis is particularly useful to characterize and understand dynamics in small systems, where energy fluxes otherwise negligible in large systems, heavily influence system behavior.
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Surprisal and evidence in bits, as logarithmic measures of probability and odds respectively. The logarithmic probability measure self-information or surprisal,Tribus, Myron (1961) Thermodynamics and Thermostatics: An Introduction to Energy, Information and States of Matter, with Engineering Applications (D. Van Nostrand Company Inc., 24 West 40 Street, New York 18, New York, U.S.A) ASIN: B000ARSH5S.
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The surprisal theory is a theory of sentence processing based on information theory.Levy, R. (2008). Expectation-based syntactic comprehension. Cognition, 106(3), 1126-1177.
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Surprisal analysis was extended to better characterize and understand cellular processes, see figure, biological phenomena and human disease with reference to personalized diagnostics. Surprisal analysis was first utilized to identify genes implicated in the balance state of cells in vitro; the genes mostly present in the balance state were genes directly responsible for the maintenance of cellular homeostasis. Similarly, it has been used to discern two distinct phenotypes during the EMT of cancer cells.
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The processing of extraposed structures in English. Cognition, 122(1), 12-36.Levy, R. (2011). Integrating surprisal and uncertain-input models in online sentence comprehension: formal techniques and empirical results.
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Surprisal analysis is an information-theoretical analysis technique that integrates and applies principles of thermodynamics and maximal entropy. Surprisal analysis is capable of relating the underlying microscopic properties to the macroscopic bulk properties of a system. It has already been applied to a spectrum of disciplines including engineering, physics, chemistry and biomedical engineering. Recently, it has been extended to characterize the state of living cells, specifically monitoring and characterizing biological processes in real time using transcriptional data.
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Surprisal analysis was initially applied to characterize a small three molecule system that did not seemingly conform to principles of thermodynamics and a single dominant constraint was identified that was sufficient to describe the dynamic behavior of the three molecule system. Similar results were then observed in nuclear reactions, where differential states with varying energy partitioning are possible. Often chemical reactions require energy to overcome an activation barrier. Surprisal analysis is applicable to such applications as well.
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Another account is that the N400 reflects prediction error or surprisal. Word-based surprisal was a strong predictor of N400 amplitude in an ERP corpus. In addition, connectionist models make use of prediction error for learning and linguistic adaptation, and these models can explain several N400/P600 results in terms of prediction error propagation for learning. As research in the field of electrophysiology continues to progress, these theories will likely be refined to include a complete account of just what the N400 represents.
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This perspective is also used in regression analysis, where least squares finds the solution that minimizes the distances from it, and analogously in logistic regression, a maximum likelihood estimate minimizes the surprisal (information distance).
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Foremost, surprisal analysis identifies the state of a system when it reaches its maximal entropy, or thermodynamic equilibrium. This is known as balance state of the system because once a system reaches its maximal entropy, it can no longer initiate or participate in spontaneous processes. Following the determination of the balanced state, surprisal analysis then characterizes all the states in which the system deviates away from the balance state. These deviations are caused by constraints; these constraints on the system prevent the system from reaching its maximal entropy.
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Here the quantity that's measured in bits is the logarithmic information measure mentioned above. Hence there are N bits of surprisal in landing all heads on one's first toss of N coins. The additive nature of surprisals, and one's ability to get a feel for their meaning with a handful of coins, can help one put improbable events (like winning the lottery, or having an accident) into context. For example if one out of 17 million tickets is a winner, then the surprisal of winning from a single random selection is about 24 bits.
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Quantitative uses of the terms uncertainty and risk are fairly consistent from fields such as probability theory, actuarial science, and information theory. Some also create new terms without substantially changing the definitions of uncertainty or risk. For example, surprisal is a variation on uncertainty sometimes used in information theory. But outside of the more mathematical uses of the term, usage may vary widely.
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Beauclerk describes Buckhurst: "Cultured, witty, satirical, dissolute, and utterly charming".Beaclerk, p. 103. He was one of a handful of court wits, the "merry gang" as named by Andrew Marvell. Sometime after the end of April and her last recorded role that season (in Robert Howard's The Surprisal), Gwyn and Buckhurst left London for a country holiday in Epsom, accompanied by Charles Sedley, another wit in the merry gang.
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The notion of a "center" as minimizing variation can be generalized in information geometry as a distribution that minimizes divergence (a generalized distance) from a data set. The most common case is maximum likelihood estimation, where the maximum likelihood estimate (MLE) maximizes likelihood (minimizes expected surprisal), which can be interpreted geometrically by using entropy to measure variation: the MLE minimizes cross entropy (equivalently, relative entropy, Kullback–Leibler divergence). A simple example of this is for the center of nominal data: instead of using the mode (the only single-valued "center"), one often uses the empirical measure (the frequency distribution divided by the sample size) as a "center". For example, given binary data, say heads or tails, if a data set consists of 2 heads and 1 tails, then the mode is "heads", but the empirical measure is 2/3 heads, 1/3 tails, which minimizes the cross-entropy (total surprisal) from the data set.
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Best-guess states (e.g. for atoms in a gas) are inferred by maximizing the average surprisal S (entropy) for a given set of control parameters (like pressure P or volume V). This constrained entropy maximization, both classically and quantum mechanically, minimizes Gibbs availability in entropy unitsJ.W. Gibbs (1873), "A method of geometrical representation of thermodynamic properties of substances by means of surfaces", reprinted in The Collected Works of J. W. Gibbs, Volume I Thermodynamics, ed.
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Whether or not you decide to get the vaccination (e.g. the monetary cost of paying for it is not included in this discussion), you can in that way at least take responsibility for a decision informed to the fact that not getting the vaccination involves more than one bit of additional risk. More generally, one can relate probability p to bits of surprisal sbits as probability = 1/2sbits. As suggested above, this is mainly useful with small probabilities.
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This measure has also been called surprisal, as it represents the "surprise" of seeing the outcome (a highly improbable outcome is very surprising). This term (as a log-probability measure) was coined by Myron Tribus in his 1961 book Thermostatics and Thermodynamics.R. B. Bernstein and R. D. Levine (1972) "Entropy and Chemical Change. I. Characterization of Product (and Reactant) Energy Distributions in Reactive Molecular Collisions: Information and Entropy Deficiency", The Journal of Chemical Physics 57, 434-449 link.
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4 (October 2019), pp. 62–67. (p. 67.) Some scientists prefer to use Bayesian methods, a more direct statistical approach which takes initial beliefs, adds in new evidence, and updates the beliefs. Another alternative procedure is to use the surprisal, a mathematical quantity that adjust p values to produce bits – as in computer bits – of information; in that perspective, 0.05 is a weak standard. When Ronald Fisher embraced the concept of "significance" in the early 20th century, it meant "signifying" but not "important".
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In the surprisal theory, the cost of processing a word is determined by its self-information, or how predictable the word is, given its context. A highly probable word carries a small amount of self-information and would therefore be processed easily, as measured by reduced reaction time, a smaller N400 response, or reduced fixation times in an eyetracking reading study. Empirical tests of this theory have shown a high degree of match between processing cost measures and the self-information values assigned to words.Levy, R., Fedorenko, E., Breen, M. and Gibson, T. (2011).
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Surprisal (a term coined in this context by Myron TribusMyron Tribus (1961) Thermodynamics and Thermostatics: An Introduction to Energy, Information and States of Matter, with Engineering Applications (D. Van Nostrand, 24 West 40 Street, New York 18, New York, U.S.A) Tribus, Myron (1961), pp. 64-66 borrow.) was first introduced to better understand the specificity of energy release and selectivity of energy requirements of elementary chemical reactions. This gave rise to a series of new experiments which demonstrated that in elementary reactions, the nascent products could be probed and that the energy is preferentially released and not statistically distributed.
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In information theory, the information content, self-information, surprisal, or Shannon information is a basic quantity derived from the probability of a particular event occurring from a random variable. It can be thought of as an alternative way of expressing probability, much like odds or log-odds, but which has particular mathematical advantages in the setting of information theory. The Shannon information can be interpreted as quantifying the level of "surprise" of a particular outcome. As it is such a basic quantity, it also appears in several other settings, such as the length of a message needed to transmit the event given an optimal source coding of the random variable.
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Differential entropy (also referred to as continuous entropy) is a concept in information theory that began as an attempt by Shannon to extend the idea of (Shannon) entropy, a measure of average surprisal of a random variable, to continuous probability distributions. Unfortunately, Shannon did not derive this formula, and rather just assumed it was the correct continuous analogue of discrete entropy, but it is not . The actual continuous version of discrete entropy is the limiting density of discrete points (LDDP). Differential entropy (described here) is commonly encountered in the literature, but it is a limiting case of the LDDP, and one that loses its fundamental association with discrete entropy.
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However, knowledge that a particular number will win a lottery has high value because it communicates the outcome of a very low probability event. The information content (also called the surprisal) of an event E is a function which decreases as the probability p(E) of an event increases, defined by I(E) = -\log_2(p(E)) or equivalently I(E) = \log_2(1/p(E)), where \log is the logarithm. Entropy measures the expected (i.e., average) amount of information conveyed by identifying the outcome of a random trial. This implies that casting a die has higher entropy than tossing a coin because each outcome of a die toss has smaller probability (about p=1/6) than each outcome of a coin toss (p=1/2).
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On the other hand, other models suggest that the P600 may not reflect these processes in particular, but just the amount of time and effort in general it takes to build up coherent structure in a sentence, or the general processes of creating or destroying syntactic structure (not specifically because of repair). Another proposal is that the P600 does not necessarily reflect any linguistic processes per se, but is similar to the P300 in that it is triggered when a subject encounters "improbable" stimuli-- since ungrammatical sentences are relatively rare in natural speech, a P600 may not be a linguistic response but simply an effect of the subject's "surprise" upon encountering an unexpected stimulus. Another account is that the P600 reflects error/surprisal propagation due to learning processes that take place during linguistic adaptation and this account has been implemented in a connectionist model that explains several P600/N400 results.
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