Spike train entropy-rate estimation using
... where s0 denotes the context s with the earliest symbol removed. This choice gives the prior distribution of gs mean gs0 , as desired. We continue constructing the prior with gs00 |gs0 ∼ Beta(α|s0 | gs00 , α|s0 | (1 − gs00 )) and so on until g[] ∼ Beta(α0 p∅ , α0 (1 − p∅ )) where g[] is the probabil ...
... where s0 denotes the context s with the earliest symbol removed. This choice gives the prior distribution of gs mean gs0 , as desired. We continue constructing the prior with gs00 |gs0 ∼ Beta(α|s0 | gs00 , α|s0 | (1 − gs00 )) and so on until g[] ∼ Beta(α0 p∅ , α0 (1 − p∅ )) where g[] is the probabil ...
On the Foundations of Quantitative Information Flow
... H(H) seems appropriate. For the remaining uncertainty about H, the conditional entropy H(H|L) seems appropriate. Finally, for the information leaked to L, the entropy H(L) might appear appropriate as well, but this cannot be correct in the case where c is probabilistic. For in that case, L might get ...
... H(H) seems appropriate. For the remaining uncertainty about H, the conditional entropy H(H|L) seems appropriate. Finally, for the information leaked to L, the entropy H(L) might appear appropriate as well, but this cannot be correct in the case where c is probabilistic. For in that case, L might get ...
Entropy Measures vs. Kolmogorov Complexity
... The Kolmogorov complexity K(x) measures the amount of information contained in an individual object (usually a string) x, by the size of the smallest program that generates it. It naturally characterizes a probability distribution over Σ∗ (the set of all finite binary strings), assigning a probabili ...
... The Kolmogorov complexity K(x) measures the amount of information contained in an individual object (usually a string) x, by the size of the smallest program that generates it. It naturally characterizes a probability distribution over Σ∗ (the set of all finite binary strings), assigning a probabili ...
Markov Lie Monoid Entropies as Network Metrics
... I + λC by ignoring higher order infinitesimals. Here one sees that the value or weight of the connection matrix between two nodes, gives the M matrix element as the relative infinitesimal transition rate between those two components of the vector. Thus it follows that given a probability distributio ...
... I + λC by ignoring higher order infinitesimals. Here one sees that the value or weight of the connection matrix between two nodes, gives the M matrix element as the relative infinitesimal transition rate between those two components of the vector. Thus it follows that given a probability distributio ...
ENTROPY
... states, Ω, in the way S=klnΩ, so that the equilibrium state is the one with most chances (really he said proportional to the statistic weight of possible states; states can only be numbered with quantum theory, and, only if states can be numbered, can absolute entropy be defined). In the 1930s, A. E ...
... states, Ω, in the way S=klnΩ, so that the equilibrium state is the one with most chances (really he said proportional to the statistic weight of possible states; states can only be numbered with quantum theory, and, only if states can be numbered, can absolute entropy be defined). In the 1930s, A. E ...
Time-reversed dynamical entropy and irreversibility in Markovian
... probability distribution, although the outgoing particles have a probability distribution which depends on their interaction inside the system and are therefore finely correlated. The time-reversed steady state is in principle possible but highly improbable because it would require the incoming part ...
... probability distribution, although the outgoing particles have a probability distribution which depends on their interaction inside the system and are therefore finely correlated. The time-reversed steady state is in principle possible but highly improbable because it would require the incoming part ...
On fuzzy information theory
... hartley, which is based on the common logarithm. In what follows, an expression of the form is considered by convention to be equal to zero whenever p = 0. This is justified because for any logarithmic base (Jaynes 1957, Mackay 2003). ...
... hartley, which is based on the common logarithm. In what follows, an expression of the form is considered by convention to be equal to zero whenever p = 0. This is justified because for any logarithmic base (Jaynes 1957, Mackay 2003). ...
Entropy and Uncertainty
... subadditivity as it reduces to the subadditive condition (15) if β is the trivial process with only one outcome. Example There are two cities, for example Melbourne and Canberra, and the citizens of one always tells the truth but the citizens of the other never tell the truth. An absent-minded mathe ...
... subadditivity as it reduces to the subadditive condition (15) if β is the trivial process with only one outcome. Example There are two cities, for example Melbourne and Canberra, and the citizens of one always tells the truth but the citizens of the other never tell the truth. An absent-minded mathe ...
Which Processes Satisfy the Second Law?
... entropy state. Thus, the time asymmetry comes from the asymmetry between the initial and final conditions. However, if the process is in equilibrium, then the entropy is constant. Nothing could be more time symmetric than that. However, the conditional entropy H(X, IXo} of a state at time 11 given t ...
... entropy state. Thus, the time asymmetry comes from the asymmetry between the initial and final conditions. However, if the process is in equilibrium, then the entropy is constant. Nothing could be more time symmetric than that. However, the conditional entropy H(X, IXo} of a state at time 11 given t ...
printer version
... The entropy of the joint random variables defines a point in the r positive orthant of R2 , with one coordinate for each subset of {1, 2, . . . , r}. Since the entropy of f∅ is 0, we can omit the first r coordinate, and regard the point as lying in R2 −1 . Let Γr be the set of all such points (arisi ...
... The entropy of the joint random variables defines a point in the r positive orthant of R2 , with one coordinate for each subset of {1, 2, . . . , r}. Since the entropy of f∅ is 0, we can omit the first r coordinate, and regard the point as lying in R2 −1 . Let Γr be the set of all such points (arisi ...
On measures of entropy and information.
... indicated). Notice that the quantity (3.2) is defined for two finite discrete probability distributions aP = (pi, * *, pn) and Q = (ql, * * *, q.) only if Pk > 0 for k = 1, 2, -- , n (among the qk there may be zeros) and if there is given a oneto-one correspondence between the elements of the distri ...
... indicated). Notice that the quantity (3.2) is defined for two finite discrete probability distributions aP = (pi, * *, pn) and Q = (ql, * * *, q.) only if Pk > 0 for k = 1, 2, -- , n (among the qk there may be zeros) and if there is given a oneto-one correspondence between the elements of the distri ...
A Characterization of Entropy in Terms of Information Loss
... where one has objects and morphisms between them. However, the reader need only know the definition of ‘category’ to understand this paper. Our main result is that Shannon entropy has a very simple characterization in terms of information loss. To state it, we consider a category where a morphism f ...
... where one has objects and morphisms between them. However, the reader need only know the definition of ‘category’ to understand this paper. Our main result is that Shannon entropy has a very simple characterization in terms of information loss. To state it, we consider a category where a morphism f ...
Entropic Inference
... The uncertainty about a variable x ∈ X (whether discrete or continuous, in one or several dimensions) is described by a probability distribution q(x). Our goal is to design a method to update from a prior distribution q(x) to a posterior distribution P (x) when new information in the form of constra ...
... The uncertainty about a variable x ∈ X (whether discrete or continuous, in one or several dimensions) is described by a probability distribution q(x). Our goal is to design a method to update from a prior distribution q(x) to a posterior distribution P (x) when new information in the form of constra ...
Mutual Information and Channel Capacity
... The conditional entropy is a measure of how much information loss occurs in the encoding process, and if it is equal to zero, then the encoder is information lossless. w.l.o.g., the encoder can be viewed as a channel in which the Source alphabet is the same as the codeword alphabet, and the encoding ...
... The conditional entropy is a measure of how much information loss occurs in the encoding process, and if it is equal to zero, then the encoder is information lossless. w.l.o.g., the encoder can be viewed as a channel in which the Source alphabet is the same as the codeword alphabet, and the encoding ...
Mutual Information and Channel Capacity
... The channel capacity is the maximum average information that can be sent per channel use. Notice that the mutual information is a function of the probability distribution of A. By changing Pa, we get different I(A;B). ...
... The channel capacity is the maximum average information that can be sent per channel use. Notice that the mutual information is a function of the probability distribution of A. By changing Pa, we get different I(A;B). ...
Lecture3
... Assume now that we choose a random ordering of the variables by choosing xi,j ∼ U [0, 1] independently for each (i, j), and ordering the variables in order of decreasing xi,j . Let x = (xi,j )i,j , and observe that this is, of course, nothing but a uniformly random ordering, but it is convenient to ...
... Assume now that we choose a random ordering of the variables by choosing xi,j ∼ U [0, 1] independently for each (i, j), and ordering the variables in order of decreasing xi,j . Let x = (xi,j )i,j , and observe that this is, of course, nothing but a uniformly random ordering, but it is convenient to ...
Entropy Rate Constancy in Text - ACL Anthology Reference Corpus
... We have proposed a fundamental principle of language generation, namely the entropy rate constancy principle. We have shown that entropy of the sentences taken without context increases with the sentence number, which is in agreement with the above principle. We have also examined the causes of this ...
... We have proposed a fundamental principle of language generation, namely the entropy rate constancy principle. We have shown that entropy of the sentences taken without context increases with the sentence number, which is in agreement with the above principle. We have also examined the causes of this ...
Information Theory and Predictability. Lecture 3: Stochastic Processes
... showing that the initial condition probability and the conditional probability functions p(xj |xj−1 ) are all that are required to describe a Markov process. Intuitively a Markov process is one in which the probability at a given step depends only on the previous step and not on earlier steps. In mo ...
... showing that the initial condition probability and the conditional probability functions p(xj |xj−1 ) are all that are required to describe a Markov process. Intuitively a Markov process is one in which the probability at a given step depends only on the previous step and not on earlier steps. In mo ...
Full text in PDF form
... Equality obtains on the left in (10) if and only if p(Ci ) = 1 for some i: if you already know in advance that ξ will come up with state Ci , then you gain no knowledge by performing the experiment. Equality obtains on the right in (10) if and only if p(Ci ) = r−1 for each i: the most informative ex ...
... Equality obtains on the left in (10) if and only if p(Ci ) = 1 for some i: if you already know in advance that ξ will come up with state Ci , then you gain no knowledge by performing the experiment. Equality obtains on the right in (10) if and only if p(Ci ) = r−1 for each i: the most informative ex ...
Principle of Maximum Entropy: Simple Form
... The Principle of Maximum Entropy is a technique that can be used to estimate input probabilities more generally. The result is a probability distribution that is consistent with known constraints expressed in terms of averages, or expected values, of one or more quantities, but is otherwise as unbia ...
... The Principle of Maximum Entropy is a technique that can be used to estimate input probabilities more generally. The result is a probability distribution that is consistent with known constraints expressed in terms of averages, or expected values, of one or more quantities, but is otherwise as unbia ...
Full text in PDF form
... includes also Shannon’s entropy H. Considerations of choice of the value of α imply that exp(H) appears to be the most appropriate measure of Ess. Entropy and Ess can be viewed thanks to their log / exp relationship as two aspects of the same thing. In Probability and Statistics the Ess aspect could ...
... includes also Shannon’s entropy H. Considerations of choice of the value of α imply that exp(H) appears to be the most appropriate measure of Ess. Entropy and Ess can be viewed thanks to their log / exp relationship as two aspects of the same thing. In Probability and Statistics the Ess aspect could ...
Lecture1
... 1. Source Coding in ML: In ML, the source is essentially a model (e.g. p(X1 , ..., Xn )) that generates data points X1 , ..., Xn , and the least number of bits needed to encode these data reflect the complexity of the source or model. Thus, source coding can be used to pick a descriptive model with ...
... 1. Source Coding in ML: In ML, the source is essentially a model (e.g. p(X1 , ..., Xn )) that generates data points X1 , ..., Xn , and the least number of bits needed to encode these data reflect the complexity of the source or model. Thus, source coding can be used to pick a descriptive model with ...
Lecture Notes
... The concept of entropy originated in thermodynamics in the 19th century where it was intimately related to heat flow and central to the second law of the subject. Later the concept played a central role in the physical illumination of thermodynamics by statistical mechanics. This was due to the effo ...
... The concept of entropy originated in thermodynamics in the 19th century where it was intimately related to heat flow and central to the second law of the subject. Later the concept played a central role in the physical illumination of thermodynamics by statistical mechanics. This was due to the effo ...
Using Curve Fitting as an Example to Discuss Major Issues in ML
... Experiment: Given a function; create N training example What M should we choose?Model Selection Given M, what w’s should we choose? Parameter Selection ...
... Experiment: Given a function; create N training example What M should we choose?Model Selection Given M, what w’s should we choose? Parameter Selection ...
Entropy (information theory)
In information theory, entropy (more specifically, Shannon entropy) is the expected value (average) of the information contained in each message received. 'Messages' don't have to be text; in this context a 'message' is simply any flow of information. The entropy of the message is its amount of uncertainty; it increases when the message is closer to random, and decreases when it is less random. The idea here is that the less likely an event is, the more information it provides when it occurs. This seems backwards at first: it seems like messages which have more structure would contain more information, but this is not true. For example, the message 'aaaaaaaaaa' (which appears to be very structured and not random at all [although in fact it could result from a random process]) contains much less information than the message 'alphabet' (which is somewhat structured, but more random) or even the message 'axraefy6h' (which is very random). In information theory, 'information' doesn't necessarily mean useful information; it simply describes the amount of randomness of the message, so in the example above the first message has the least information and the last message has the most information, even though in everyday terms we would say that the middle message, 'alphabet', contains more information than a stream of random letters. Therefore, we would say in information theory that the first message has low entropy, the second has higher entropy, and the third has the highest entropy.In a more technical sense, there are reasons (explained below) to define information as the negative of the logarithm of the probability distribution. The probability distribution of the events, coupled with the information amount of every event, forms a random variable whose average (also termed expected value) is the average amount of information, a.k.a. entropy, generated by this distribution. Units of entropy are the shannon, nat, or hartley, depending on the base of the logarithm used to define it, though the shannon is commonly referred to as a bit.The logarithm of the probability distribution is useful as a measure of entropy because it is additive for independent sources. For instance, the entropy of a coin toss is 1 shannon, whereas of m tosses it is m shannons. Generally, you need log2(n) bits to represent a variable that can take one of n values if n is a power of 2. If these values are equiprobable, the entropy (in shannons) is equal to the number of bits. Equality between number of bits and shannons holds only while all outcomes are equally probable. If one of the events is more probable than others, observation of that event is less informative. Conversely, observing rarer events compensate by providing more information when observed. Since observation of less probable events occurs more rarely, the net effect is that the entropy (thought of as the average information) received from non-uniformly distributed data is less than log2(n). Entropy is zero when one outcome is certain. Shannon entropy quantifies all these considerations exactly when a probability distribution of the source is known. The meaning of the events observed (a.k.a. the meaning of messages) do 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.Generally, entropy refers to disorder or uncertainty. Shannon entropy was introduced by Claude E. Shannon in his 1948 paper ""A Mathematical Theory of Communication"". Shannon entropy provides an absolute limit on the best possible average length of lossless encoding or compression of an information source. Rényi entropy generalizes Shannon entropy.