The randomized weighted majority algorithm is an algorithm in machine learning theory for aggregating expert predictions to a series of decision problems. It is a simple and effective method based on weighted voting which improves on the mistake bound of the deterministic weighted majority algorithm. In fact, in the limit, its prediction rate can be arbitrarily close to that of the best-predicting expert. == Example == Imagine that every morning before the stock market opens, we get a prediction from each of our "experts" about whether the stock market will go up or down. Our goal is to somehow combine this set of predictions into a single prediction that we then use to make a buy or sell decision for the day. The principal challenge is that we do not know which experts will give better or worse predictions. The RWMA gives us a way to do this combination such that our prediction record will be nearly as good as that of the single expert which, in hindsight, gave the most accurate predictions. == Motivation == In machine learning, the weighted majority algorithm (WMA) is a deterministic meta-learning algorithm for aggregating expert predictions. In pseudocode, the WMA is as follows: initialize all experts to weight 1 for each round: add each expert's weight to the option they predicted predict the option with the largest weighted sum multiply the weights of all experts who predicted wrongly by 1 2 {\displaystyle {\frac {1}{2}}} Suppose there are n {\displaystyle n} experts and the best expert makes m {\displaystyle m} mistakes. Then, the weighted majority algorithm (WMA) makes at most 2.4 ( log 2 n + m ) {\displaystyle 2.4(\log _{2}n+m)} mistakes. This bound is highly problematic in the case of highly error-prone experts. Suppose, for example, the best expert makes a mistake 20% of the time; that is, in N = 100 {\displaystyle N=100} rounds using n = 10 {\displaystyle n=10} experts, the best expert makes m = 20 {\displaystyle m=20} mistakes. Then, the weighted majority algorithm only guarantees an upper bound of 2.4 ( log 2 10 + 20 ) ≈ 56 {\displaystyle 2.4(\log _{2}10+20)\approx 56} mistakes. As this is a known limitation of the weighted majority algorithm, various strategies have been explored in order to improve the dependence on m {\displaystyle m} . In particular, we can do better by introducing randomization. Drawing inspiration from the Multiplicative Weights Update Method algorithm, we will probabilistically make predictions based on how the experts have performed in the past. Similarly to the WMA, every time an expert makes a wrong prediction, we will decrement their weight. Mirroring the MWUM, we will then use the weights to make a probability distribution over the actions and draw our action from this distribution (instead of deterministically picking the majority vote as the WMA does). == Randomized weighted majority algorithm (RWMA) == The randomized weighted majority algorithm is an attempt to improve the dependence of the mistake bound of the WMA on m {\displaystyle m} . Instead of predicting based on majority vote, the weights, are used as probabilities for choosing the experts in each round and are updated over time (hence the name randomized weighted majority). Precisely, if w i {\displaystyle w_{i}} is the weight of expert i {\displaystyle i} , let W = ∑ i w i {\displaystyle W=\sum _{i}w_{i}} . We will follow expert i {\displaystyle i} with probability w i W {\displaystyle {\frac {w_{i}}{W}}} . This results in the following algorithm: initialize all experts to weight 1. for each round: add all experts' weights together to obtain the total weight W {\displaystyle W} choose expert i {\displaystyle i} randomly with probability w i W {\displaystyle {\frac {w_{i}}{W}}} predict as the chosen expert predicts multiply the weights of all experts who predicted wrongly by β {\displaystyle \beta } The goal is to bound the worst-case expected number of mistakes, assuming that the adversary has to select one of the answers as correct before we make our coin toss. This is a reasonable assumption in, for instance, the stock market example provided above: the variance of a stock price should not depend on the opinions of experts that influence private buy or sell decisions, so we can treat the price change as if it was decided before the experts gave their recommendations for the day. The randomized algorithm is better in the worst case than the deterministic algorithm (weighted majority algorithm): in the latter, the worst case was when the weights were split 50/50. But in the randomized version, since the weights are used as probabilities, there would still be a 50/50 chance of getting it right. In addition, generalizing to multiplying the weights of the incorrect experts by β < 1 {\displaystyle \beta <1} instead of strictly 1 2 {\displaystyle {\frac {1}{2}}} allows us to trade off between dependence on m {\displaystyle m} and log 2 n {\displaystyle \log _{2}n} . This trade-off will be quantified in the analysis section. == Analysis == Let W t {\displaystyle W_{t}} denote the total weight of all experts at round t {\displaystyle t} . Also let F t {\displaystyle F_{t}} denote the fraction of weight placed on experts which predict the wrong answer at round t {\displaystyle t} . Finally, let N {\displaystyle N} be the total number of rounds in the process. By definition, F t {\displaystyle F_{t}} is the probability that the algorithm makes a mistake on round t {\displaystyle t} . It follows from the linearity of expectation that if M {\displaystyle M} denotes the total number of mistakes made during the entire process, E [ M ] = ∑ t = 1 N F t {\displaystyle E[M]=\sum _{t=1}^{N}F_{t}} . After round t {\displaystyle t} , the total weight is decreased by ( 1 − β ) F t W t {\displaystyle \ (1-\beta )F_{t}W_{t}} , since all weights corresponding to a wrong answer are multiplied by β < 1 {\displaystyle \ \beta <1} . It then follows that W t + 1 = W t ( 1 − ( 1 − β ) F t ) {\displaystyle W_{t+1}=W_{t}(1-(1-\beta )F_{t})} . By telescoping, since W 1 = n {\displaystyle W_{1}=n} , it follows that the total weight after the process concludes is On the other hand, suppose that m {\displaystyle \ m} is the number of mistakes made by the best-performing expert. At the end, this expert has weight β m {\displaystyle \ \beta ^{m}} . It follows, then, that the total weight is at least this much; in other words, W ≥ β m {\displaystyle \ W\geq \beta ^{m}} . This inequality and the above result imply Taking the natural logarithm of both sides yields Now, the Taylor series of the natural logarithm is In particular, it follows that ln ( 1 − ( 1 − β ) F t ) < − ( 1 − β ) F t {\displaystyle \ \ln(1-(1-\beta )F_{t})<-(1-\beta )F_{t}} . Thus, Recalling that E [ M ] = ∑ t = 1 N F t {\displaystyle E[M]=\sum _{t=1}^{N}F_{t}} and rearranging, it follows that Now, as β → 1 {\displaystyle \beta \to 1} from below, the first constant tends to 1 {\displaystyle 1} ; however, the second constant tends to + ∞ {\displaystyle +\infty } . To quantify this tradeoff, define ε = 1 − β {\displaystyle \varepsilon =1-\beta } to be the penalty associated with getting a prediction wrong. Then, again applying the Taylor series of the natural logarithm, It then follows that the mistake bound, for small ε {\displaystyle \varepsilon } , can be written in the form ( 1 + ϵ 2 + O ( ε 2 ) ) m + ϵ − 1 ln ( n ) {\displaystyle \ \left(1+{\frac {\epsilon }{2}}+O(\varepsilon ^{2})\right)m+\epsilon ^{-1}\ln(n)} . In English, the less that we penalize experts for their mistakes, the more that additional experts will lead to initial mistakes but the closer we get to capturing the predictive accuracy of the best expert as time goes on. In particular, given a sufficiently low value of ε {\displaystyle \varepsilon } and enough rounds, the randomized weighted majority algorithm can get arbitrarily close to the correct prediction rate of the best expert. In particular, as long as m {\displaystyle m} is sufficiently large compared to ln ( n ) {\displaystyle \ln(n)} (so that their ratio is sufficiently small), we can assign we can obtain an upper bound on the number of mistakes equal to This implies that the "regret bound" on the algorithm (that is, how much worse it performs than the best expert) is sublinear, at O ( m ln ( n ) ) {\displaystyle O({\sqrt {m\ln(n)}})} . == Revisiting the motivation == Recall that the motivation for the randomized weighted majority algorithm was given by an example where the best expert makes a mistake 20% of the time. Precisely, in N = 100 {\displaystyle N=100} rounds, with n = 10 {\displaystyle n=10} experts, where the best expert makes m = 20 {\displaystyle m=20} mistakes, the deterministic weighted majority algorithm only guarantees an upper bound of 2.4 ( log 2 10 + 20 ) ≈ 56 {\displaystyle 2.4(\log _{2}10+20)\approx 56} . By the analysis above, it follows that minimizing the number of worst-case expected mistakes is equivalent to minimizing the fun
Tensor operator
In pure and applied mathematics, quantum mechanics and computer graphics, a tensor operator generalizes the notion of operators which are scalars and vectors. A special class of these are spherical tensor operators which apply the notion of the spherical basis and spherical harmonics. The spherical basis closely relates to the description of angular momentum in quantum mechanics and spherical harmonic functions. The coordinate-free generalization of a tensor operator is known as a representation operator. == The general notion of scalar, vector, and tensor operators == In quantum mechanics, physical observables that are scalars, vectors, and tensors, must be represented by scalar, vector, and tensor operators, respectively. Whether something is a scalar, vector, or tensor depends on how it is viewed by two observers whose coordinate frames are related to each other by a rotation. Alternatively, one may ask how, for a single observer, a physical quantity transforms if the state of the system is rotated. Consider, for example, a system consisting of a molecule of mass M {\displaystyle M} , traveling with a definite center of mass momentum, p z ^ {\displaystyle p{\mathbf {\hat {z}} }} , in the z {\displaystyle z} direction. If we rotate the system by 90 ∘ {\displaystyle 90^{\circ }} about the y {\displaystyle y} axis, the momentum will change to p x ^ {\displaystyle p{\mathbf {\hat {x}} }} , which is in the x {\displaystyle x} direction. The center-of-mass kinetic energy of the molecule will, however, be unchanged at p 2 / 2 M {\displaystyle p^{2}/2M} . The kinetic energy is a scalar and the momentum is a vector, and these two quantities must be represented by a scalar and a vector operator, respectively. By the latter in particular, we mean an operator whose expected values in the initial and the rotated states are p z ^ {\displaystyle p{\mathbf {\hat {z}} }} and p x ^ {\displaystyle p{\mathbf {\hat {x}} }} . The kinetic energy on the other hand must be represented by a scalar operator, whose expected value must be the same in the initial and the rotated states. In the same way, tensor quantities must be represented by tensor operators. An example of a tensor quantity (of rank two) is the electrical quadrupole moment of the above molecule. Likewise, the octupole and hexadecapole moments would be tensors of rank three and four, respectively. Other examples of scalar operators are the total energy operator (more commonly called the Hamiltonian), the potential energy, and the dipole-dipole interaction energy of two atoms. Examples of vector operators are the momentum, the position, the orbital angular momentum, L {\displaystyle {\mathbf {L} }} , and the spin angular momentum, S {\displaystyle {\mathbf {S} }} . (Fine print: Angular momentum is a vector as far as rotations are concerned, but unlike position or momentum it does not change sign under space inversion, and when one wishes to provide this information, it is said to be a pseudovector.) Scalar, vector and tensor operators can also be formed by products of operators. For example, the scalar product L ⋅ S {\displaystyle {\mathbf {L} }\cdot {\mathbf {S} }} of the two vector operators, L {\displaystyle {\mathbf {L} }} and S {\displaystyle {\mathbf {S} }} , is a scalar operator, which figures prominently in discussions of the spin–orbit interaction. Similarly, the quadrupole moment tensor of our example molecule has the nine components Q i j = ∑ α q α ( 3 r α , i r α , j − r α 2 δ i j ) . {\displaystyle Q_{ij}=\sum _{\alpha }q_{\alpha }\left(3r_{\alpha ,i}r_{\alpha ,j}-r_{\alpha }^{2}\delta _{ij}\right).} Here, the indices i {\displaystyle i} and j {\displaystyle j} can independently take on the values 1, 2, and 3 (or x {\displaystyle x} , y {\displaystyle y} , and z {\displaystyle z} ) corresponding to the three Cartesian axes, the index α {\displaystyle \alpha } runs over all particles (electrons and nuclei) in the molecule, q α {\displaystyle q_{\alpha }} is the charge on particle α {\displaystyle \alpha } , and r α , i {\displaystyle r_{\alpha ,i}} is the i {\displaystyle i} -th component of the position of this particle. Each term in the sum is a tensor operator. In particular, the nine products r α , i r α , j {\displaystyle r_{\alpha ,i}r_{\alpha ,j}} together form a second rank tensor, formed by taking the outer product of the vector operator r α {\displaystyle {\mathbf {r} }_{\alpha }} with itself. == Rotations of quantum states == === Quantum rotation operator === The rotation operator about the unit vector n (defining the axis of rotation) through angle θ is U [ R ( θ , n ^ ) ] = exp ( − i θ ℏ n ^ ⋅ J ) {\displaystyle U[R(\theta ,{\hat {\mathbf {n} }})]=\exp \left(-{\frac {i\theta }{\hbar }}{\hat {\mathbf {n} }}\cdot \mathbf {J} \right)} where J = (Jx, Jy, Jz) are the rotation generators (also the angular momentum matrices): J x = ℏ 2 ( 0 1 0 1 0 1 0 1 0 ) J y = ℏ 2 ( 0 i 0 − i 0 i 0 − i 0 ) J z = ℏ ( − 1 0 0 0 0 0 0 0 1 ) {\displaystyle J_{x}={\frac {\hbar }{\sqrt {2}}}{\begin{pmatrix}0&1&0\\1&0&1\\0&1&0\end{pmatrix}}\,\quad J_{y}={\frac {\hbar }{\sqrt {2}}}{\begin{pmatrix}0&i&0\\-i&0&i\\0&-i&0\end{pmatrix}}\,\quad J_{z}=\hbar {\begin{pmatrix}-1&0&0\\0&0&0\\0&0&1\end{pmatrix}}} and let R ^ = R ^ ( θ , n ^ ) {\displaystyle {\widehat {R}}={\widehat {R}}(\theta ,{\hat {\mathbf {n} }})} be a rotation matrix. According to the Rodrigues' rotation formula, the rotation operator then amounts to U [ R ( θ , n ^ ) ] = 1 1 − i sin θ ℏ n ^ ⋅ J − 1 − cos θ ℏ 2 ( n ^ ⋅ J ) 2 . {\displaystyle U[R(\theta ,{\hat {\mathbf {n} }})]=1\!\!1-{\frac {i\sin \theta }{\hbar }}{\hat {\mathbf {n} }}\cdot \mathbf {J} -{\frac {1-\cos \theta }{\hbar ^{2}}}({\hat {\mathbf {n} }}\cdot \mathbf {J} )^{2}.} An operator Ω ^ {\displaystyle {\widehat {\Omega }}} is invariant under a unitary transformation U if Ω ^ = U † Ω ^ U ; {\displaystyle {\widehat {\Omega }}={U}^{\dagger }{\widehat {\Omega }}U;} in this case for the rotation U ^ ( R ) {\displaystyle {\widehat {U}}(R)} , Ω ^ = U ( R ) † Ω ^ U ( R ) = exp ( i θ ℏ n ^ ⋅ J ) Ω ^ exp ( − i θ ℏ n ^ ⋅ J ) . {\displaystyle {\widehat {\Omega }}={U(R)}^{\dagger }{\widehat {\Omega }}U(R)=\exp \left({\frac {i\theta }{\hbar }}{\hat {\mathbf {n} }}\cdot \mathbf {J} \right){\widehat {\Omega }}\exp \left(-{\frac {i\theta }{\hbar }}{\hat {\mathbf {n} }}\cdot \mathbf {J} \right).} === Angular momentum eigenkets === The orthonormal basis set for total angular momentum is | j , m ⟩ {\displaystyle |j,m\rangle } , where j is the total angular momentum quantum number and m is the magnetic angular momentum quantum number, which takes values −j, −j + 1, ..., j − 1, j. A general state within the j subspace | ψ ⟩ = ∑ m c j m | j , m ⟩ {\displaystyle |\psi \rangle =\sum _{m}c_{jm}|j,m\rangle } rotates to a new state by: | ψ ¯ ⟩ = U ( R ) | ψ ⟩ = ∑ m c j m U ( R ) | j , m ⟩ {\displaystyle |{\bar {\psi }}\rangle =U(R)|\psi \rangle =\sum _{m}c_{jm}U(R)|j,m\rangle } Using the completeness condition: I = ∑ m ′ | j , m ′ ⟩ ⟨ j , m ′ | {\displaystyle I=\sum _{m'}|j,m'\rangle \langle j,m'|} we have | ψ ¯ ⟩ = I U ( R ) | ψ ⟩ = ∑ m m ′ c j m | j , m ′ ⟩ ⟨ j , m ′ | U ( R ) | j , m ⟩ {\displaystyle |{\bar {\psi }}\rangle =IU(R)|\psi \rangle =\sum _{mm'}c_{jm}|j,m'\rangle \langle j,m'|U(R)|j,m\rangle } Introducing the Wigner D matrix elements: D ( R ) m ′ m ( j ) = ⟨ j , m ′ | U ( R ) | j , m ⟩ {\displaystyle {D(R)}_{m'm}^{(j)}=\langle j,m'|U(R)|j,m\rangle } gives the matrix multiplication: | ψ ¯ ⟩ = ∑ m m ′ c j m D m ′ m ( j ) | j , m ′ ⟩ ⇒ | ψ ¯ ⟩ = D ( j ) | ψ ⟩ {\displaystyle |{\bar {\psi }}\rangle =\sum _{mm'}c_{jm}D_{m'm}^{(j)}|j,m'\rangle \quad \Rightarrow \quad |{\bar {\psi }}\rangle =D^{(j)}|\psi \rangle } For one basis ket: | j , m ¯ ⟩ = ∑ m ′ D ( R ) m ′ m ( j ) | j , m ′ ⟩ {\displaystyle |{\overline {j,m}}\rangle =\sum _{m'}{D(R)}_{m'm}^{(j)}|j,m'\rangle } For the case of orbital angular momentum, the eigenstates | ℓ , m ⟩ {\displaystyle |\ell ,m\rangle } of the orbital angular momentum operator L and solutions of Laplace's equation on a 3d sphere are spherical harmonics: Y ℓ m ( θ , ϕ ) = ⟨ θ , ϕ | ℓ , m ⟩ = ( 2 ℓ + 1 ) 4 π ( ℓ − m ) ! ( ℓ + m ) ! P ℓ m ( cos θ ) e i m ϕ {\displaystyle Y_{\ell }^{m}(\theta ,\phi )=\langle \theta ,\phi |\ell ,m\rangle ={\sqrt {{(2\ell +1) \over 4\pi }{(\ell -m)! \over (\ell +m)!}}}\,P_{\ell }^{m}(\cos {\theta })\,e^{im\phi }} where Pℓm is an associated Legendre polynomial, ℓ is the orbital angular momentum quantum number, and m is the orbital magnetic quantum number which takes the values −ℓ, −ℓ + 1, ... ℓ − 1, ℓ The formalism of spherical harmonics have wide applications in applied mathematics, and are closely related to the formalism of spherical tensors, as shown below. Spherical harmonics are functions of the polar and azimuthal angles, ϕ and θ respectively, which can be conveniently collected into a unit vector n(θ, ϕ) pointing in the direction of those angles, in the Cartesian basis it is: n ^ ( θ , ϕ ) = cos ϕ sin θ e x + s
Psychology of reasoning
The psychology of reasoning (also known as the cognitive science of reasoning) is the study of how people reason, often broadly defined as the process of drawing conclusions to inform how people solve problems and make decisions. It overlaps with psychology, philosophy, linguistics, cognitive science, artificial intelligence, logic, and probability theory. Psychological experiments on how humans and other animals reason have been carried out for over 100 years. An enduring question is whether or not people have the capacity to be rational. Current research in this area addresses various questions about reasoning, rationality, judgments, intelligence, relationships between emotion and reasoning, and development. == Everyday reasoning == One of the most obvious areas in which people employ reasoning is with sentences in everyday language. Most experimentation on deduction has been carried out on hypothetical thought, in particular, examining how people reason about conditionals, e.g., If A then B. Participants in experiments make the modus ponens inference, given the indicative conditional If A then B, and given the premise A, they conclude B. However, given the indicative conditional and the minor premise for the modus tollens inference, not-B, about half of the participants in experiments conclude not-A and the remainder concludes that nothing follows. The ease with which people make conditional inferences is affected by context, as demonstrated in the well-known selection task developed by Peter Wason. Participants are better able to test a conditional in an ecologically relevant context, e.g., if the envelope is sealed then it must have a 50 cent stamp on it compared to one that contains symbolic content, e.g., if the letter is a vowel then the number is even. Background knowledge can also lead to the suppression of even the simple modus ponens inference Participants given the conditional if Lisa has an essay to write then she studies late in the library and the premise Lisa has an essay to write make the modus ponens inference 'she studies late in the library', but the inference is suppressed when they are also given a second conditional if the library stays open then she studies late in the library. Interpretations of the suppression effect are controversial Other investigations of propositional inference examine how people think about disjunctive alternatives, e.g., A or else B, and how they reason about negation, e.g., It is not the case that A and B. Many experiments have been carried out to examine how people make relational inferences, including comparisons, e.g., A is better than B. Such investigations also concern spatial inferences, e.g. A is in front of B and temporal inferences, e.g. A occurs before B. Other common tasks include categorical syllogisms, used to examine how people reason about quantifiers such as All or Some, e.g., Some of the A are not B. For example if all A are B and some B are C, what (if anything) follows? == Theories of reasoning == There are several alternative theories of the cognitive processes that human reasoning is based on. One view is that people rely on a mental logic consisting of formal (abstract or syntactic) inference rules similar to those developed by logicians in the propositional calculus. Another view is that people rely on domain-specific or content-sensitive rules of inference. A third view is that people rely on mental models, that is, mental representations that correspond to imagined possibilities. A fourth view is that people compute probabilities. One controversial theoretical issue is the identification of an appropriate competence model, or a standard against which to compare human reasoning. Initially classical logic was chosen as a competence model. Subsequently, some researchers opted for non-monotonic logic and Bayesian probability. Research on mental models and reasoning has led to the suggestion that people are rational in principle but err in practice. Connectionist approaches towards reasoning have also been proposed. Despite the ongoing debate about the cognitive processes involved in human reasoning, recent research has shown that multiple approaches can be useful in modeling human thinking. For instance, studies have found that people's reasoning is often influenced by their prior beliefs, which can be modeled using Bayesian probability theory. Additionally, research on mental models has shown that people tend to reason about problems by constructing multiple mental representations of the situation, which can help them to identify relevant features and make inferences based on their understanding of the problem. Moreover, connectionist approaches to reasoning have also gained attention, which focus on the neural network models that can learn from data and generalize to new situations. == Development of reasoning == It is an active question in psychology how, why, and when the ability to reason develops from infancy to adulthood. Jean Piaget's theory of cognitive development posited general mechanisms and stages in the development of reasoning from infancy to adulthood. According to the neo-Piagetian theories of cognitive development, changes in reasoning with development come from increasing working memory capacity, increasing speed of processing, and enhanced executive functions and control. Increasing self-awareness is also an important factor. In their book The Enigma of Reason, the cognitive scientists Hugo Mercier and Dan Sperber put forward an "argumentative" theory of reasoning, claiming that humans evolved to reason primarily to justify our beliefs and actions and to convince others in a social environment. Key evidence for their theory includes the errors in reasoning that solitary individuals are prone to when their arguments are not criticized, such as logical fallacies, and how groups become much better at performing cognitive reasoning tasks when they communicate with one another and can evaluate each other's arguments. Sperber and Mercier offer one attempt to resolve the apparent paradox that the confirmation bias is so strong despite the function of reasoning naively appearing to be to come to veridical conclusions about the world. The study of the development of reasoning abilities is an ongoing area of research in psychology, and multiple factors have been proposed to explain how, why, and when reasoning develops from infancy to adulthood. Recent research has suggested that early experiences and social interactions play a critical role in the development of reasoning abilities. For example, studies have shown that infants as young as six months old can engage in basic logical reasoning, such as reasoning about the relationship between objects and their properties. Furthermore, research has highlighted the importance of parental interaction and cognitive stimulation in the development of children's reasoning abilities. Additionally, studies have suggested that cultural factors, such as educational practices and the emphasis on critical thinking, can also influence the development of reasoning skills across different populations. == Different sorts of reasoning == Philip Johnson-Laird trying to taxonomize thought, distinguished between goal-directed thinking and thinking without goal, noting that association was involved in unrelated reading. He argues that goal directed reasoning can be classified based on the problem space involved in a solution, citing Allen Newell and Herbert A. Simon. Inductive reasoning makes broad generalizations from specific cases or observations. In this process of reasoning, general assertions are made based on past specific pieces of evidence. This kind of reasoning allows the conclusion to be false even if the original statement is true. For example, if one observes a college athlete, one makes predictions and assumptions about other college athletes based on that one observation. Scientists use inductive reasoning to create theories and hypotheses. Philip Johnson-Laird distinguished inductive from deductive reasoning, in that the former creates semantic information while the later does not . In opposition, deductive reasoning is a basic form of valid reasoning. In this reasoning process a person starts with a known claim or a general belief and from there asks what follows from these foundations or how will these premises influence other beliefs. In other words, deduction starts with a hypothesis and examines the possibilities to reach a conclusion. Deduction helps people understand why their predictions are wrong and indicates that their prior knowledge or beliefs are off track. An example of deduction can be seen in the scientific method when testing hypotheses and theories. Although the conclusion usually corresponds and therefore proves the hypothesis, there are some cases where the conclusion is logical, but the generalization is not. For example, the argument, "All young girls wear skirts; Julie is a young
.ai
.ai is the Internet country code top-level domain (ccTLD) for Anguilla, a British Overseas Territory in the Caribbean. It is administered by the government of Anguilla. It is a popular domain hack with companies and projects related to the artificial intelligence industry (AI). Google's ad targeting treats .ai as a generic top-level domain (gTLD) because "users and website owners frequently see [the domain] as being more generic than country-targeted." In 2021, Google Search analyst Gary Illyes announced that ".ai" had been added to Google’s list of generic country-code top-level domains, meaning that Google would no longer infer Anguilla-specific targeting from the ccTLD. Identity Digital began managing the domain as of January 2025. == Second and third level registrations == Registrations within off.ai, com.ai, net.ai, and org.ai are available worldwide without restriction. From 15 September 2009, second level registrations within .ai are available to everyone worldwide. == Registration == The minimum registration term allowed for .ai domains is 2 through 10 years for registration and renewal, and a 2-year renewal for domain transfer. Identity Digital is the authority in charge of managing this extension. Registrations began on 16 February 1995. The limits on the number of characters used for the domain name are, at a minimum, from 1 to 3, depending on the registrar, and always at most 63 characters. The character set supported for .ai domain names includes A–Z, a–z, 0–9, and hyphen. As of November 2022, .ai domains cannot accommodate IDN characters. There are no requirements for registering a domain, including local and foreign residents. A .ai domain can be suspended or revoked, if the domain is involved in illegal activity such as violating trademarks or copyrights. Usage must not violate the laws of Anguilla. Anguilla uses the UDRP. Filing a UDRP challenge requires using one of the ICANN Approved Dispute Resolution Service Providers. If the domain is with an ICANN accredited registrar, they should work with the arbitrator. Usually this means either doing nothing or transferring a domain. .ai domains are transferable to any desired registrars as the registration of domain is done maintaining EPP. There used to be a whois.ai-based platform of expired domains in which those could be procured and auctioned every ten days through a standard online process. The last auctions of such kind closed there in December 2024; the platform had been scheduled for shutdown on 30 June 2025, but remained online in the months following that date. == Valuation == Domains cost depends on the registrar, with yearly fees ranging from US$140 (the base fee, as established by Anguilla) to $200. As of July 2025, the highest-valued .ai domain is an undisclosed one sold on 8 November 2023, on Escrow.com, for US$1,500,000—months after an initial $300,000 sale to the same buyer. Among the publicly disclosed ones, the most valued, fin.ai, was sold for $1,000,000 in March 2025. On 16 December 2017, the .ai registry started supporting the Extensible Provisioning Protocol (EPP) and migrated all of its domains onto an EPP system. Consequently, many registrars are allowed to sell .ai domains. Since that date, the .ai ccTLD has also been popular with artificial intelligence companies and organisations. Though such trends are primarily seen among new AI based companies or startups, many established AI and Tech companies preferred not to opt for .ai domains. For example, DeepMind has its domain retained at .com; Meta has redirected its facebook.ai domain to ai.meta.com. == Impact on Anguilla's economy == The registration fees earned from the .ai domains go to the treasury of the Government of Anguilla. As per a 2018 New York Times report, the total revenue generated out of selling .ai domains was $2.9 million. In 2023, Anguilla's government made about US$32 million from fees collected for registering .ai domains; that amounted to over 10% of gross domestic product for the territory. "In the years before the real breakthrough of AI, revenue from .ai domains made up less than 1% of our state income, by 2025 it will be around 47%," explained Jose Vanterpool, Minister of Infrastructure and Communications (MICUHITES), in an interview with BBC. The high 90% renewal rate of .ai domains and the 2025 renewal wave of domains registered in 2023 are driving another surge in state revenues, according to Domaintechnik.
Active learning (machine learning)
Active learning is a special case of machine learning in which a learning algorithm can interactively query a human user (or some other information source) to label new data points with the desired outputs. The human user must possess expertise in the problem domain, including the ability to consult authoritative sources when necessary. In statistics literature, it is sometimes also called optimal experimental design. The information source is also called teacher or oracle. There are situations in which unlabeled data is abundant but manual labeling is expensive. In such a scenario, learning algorithms can actively query the teacher for labels. Since the learner chooses the examples, the number of examples to learn a concept can often be much lower than the number required in normal supervised learning. However, there is a risk that the algorithm is overwhelmed by uninformative examples. Recent developments are dedicated to multi-label active learning, hybrid active learning and active learning in a single-pass (on-line) context, combining concepts from the field of machine learning (e.g. conflict and ignorance) with adaptive, incremental learning policies in the field of online machine learning. Using active learning allows for faster development of a machine learning algorithm, when comparative updates would require a quantum or super computer. Large-scale active learning projects may benefit from crowdsourcing frameworks such as Amazon Mechanical Turk that include many humans in the active learning loop. == Definitions == Let T be the total set of all data under consideration. For example, in a protein engineering problem, T would include all proteins that are known to have a certain interesting activity and all additional proteins that one might want to test for that activity. During each iteration, i, T is broken up into three subsets T K , i {\displaystyle \mathbf {T} _{K,i}} : Data points where the label is known. T U , i {\displaystyle \mathbf {T} _{U,i}} : Data points where the label is unknown. T C , i {\displaystyle \mathbf {T} _{C,i}} : A subset of TU,i that is chosen to be labeled. Most of the current research in active learning involves the best method to choose the data points for TC,i. == Scenarios == Pool-based sampling: In this approach, which is the most well known scenario, the learning algorithm attempts to evaluate the entire dataset before selecting data points (instances) for labeling. It is often initially trained on a fully labeled subset of the data using a machine-learning method such as logistic regression or SVM that yields class-membership probabilities for individual data instances. The candidate instances are those for which the prediction is most ambiguous. Instances are drawn from the entire data pool and assigned a confidence score, a measurement of how well the learner "understands" the data. The system then selects the instances for which it is the least confident and queries the teacher for the labels. The theoretical drawback of pool-based sampling is that it is memory-intensive and is therefore limited in its capacity to handle enormous datasets, but in practice, the rate-limiting factor is that the teacher is typically a (fatiguable) human expert who must be paid for their effort, rather than computer memory. Stream-based selective sampling: Here, each consecutive unlabeled instance is examined one at a time with the machine evaluating the informativeness of each item against its query parameters. The learner decides for itself whether to assign a label or query the teacher for each datapoint. As contrasted with Pool-based sampling, the obvious drawback of stream-based methods is that the learning algorithm does not have sufficient information, early in the process, to make a sound assign-label-vs ask-teacher decision, and it does not capitalize as efficiently on the presence of already labeled data. Therefore, the teacher is likely to spend more effort in supplying labels than with the pool-based approach. Membership query synthesis: This is where the learner generates synthetic data from an underlying natural distribution. For example, if the dataset are pictures of humans and animals, the learner could send a clipped image of a leg to the teacher and query if this appendage belongs to an animal or human. This is particularly useful if the dataset is small. The challenge here, as with all synthetic-data-generation efforts, is in ensuring that the synthetic data is consistent in terms of meeting the constraints on real data. As the number of variables/features in the input data increase, and strong dependencies between variables exist, it becomes increasingly difficult to generate synthetic data with sufficient fidelity. For example, to create a synthetic data set for human laboratory-test values, the sum of the various white blood cell (WBC) components in a white blood cell differential must equal 100, since the component numbers are really percentages. Similarly, the enzymes alanine transaminase (ALT) and aspartate transaminase (AST) measure liver function (though AST is also produced by other tissues, e.g., lung, pancreas) A synthetic data point with AST at the lower limit of normal range (8–33 units/L) with an ALT several times above normal range (4–35 units/L) in a simulated chronically ill patient would be physiologically impossible. == Query strategies == Algorithms for determining which data points should be labeled can be organized into a number of different categories, based upon their purpose: Balance exploration and exploitation: the choice of examples to label is seen as a dilemma between the exploration and the exploitation over the data space representation. This strategy manages this compromise by modelling the active learning problem as a contextual bandit problem. For example, Bouneffouf et al. propose a sequential algorithm named Active Thompson Sampling (ATS), which, in each round, assigns a sampling distribution on the pool, samples one point from this distribution, and queries the oracle for this sample point label. Expected model change: label those points that would most change the current model. Expected error reduction: label those points that would most reduce the model's generalization error. Exponentiated Gradient Exploration for Active Learning: In this paper, the author proposes a sequential algorithm named exponentiated gradient (EG)-active that can improve any active learning algorithm by an optimal random exploration. Uncertainty sampling: label those points for which the current model is least certain as to what the correct output should be. Query by committee: a variety of models are trained on the current labeled data, and vote on the output for unlabeled data; label those points for which the "committee" disagrees the most Querying from diverse subspaces or partitions: When the underlying model is a forest of trees, the leaf nodes might represent (overlapping) partitions of the original feature space. This offers the possibility of selecting instances from non-overlapping or minimally overlapping partitions for labeling. Variance reduction: label those points that would minimize output variance, which is one of the components of error. Conformal prediction: predicts that a new data point will have a label similar to old data points in some specified way and degree of the similarity within the old examples is used to estimate the confidence in the prediction. Mismatch-first farthest-traversal: The primary selection criterion is the prediction mismatch between the current model and nearest-neighbour prediction. It targets on wrongly predicted data points. The second selection criterion is the distance to previously selected data, the farthest first. It aims at optimizing the diversity of selected data. User-centered labeling strategies: Learning is accomplished by applying dimensionality reduction to graphs and figures like scatter plots. Then the user is asked to label the compiled data (categorical, numerical, relevance scores, relation between two instances). A wide variety of algorithms have been studied that fall into these categories. While the traditional AL strategies can achieve remarkable performance, it is often challenging to predict in advance which strategy is the most suitable in a particular situation. In recent years, meta-learning algorithms have been gaining in popularity. Some of them have been proposed to tackle the problem of learning AL strategies instead of relying on manually designed strategies. A benchmark which compares 'meta-learning approaches to active learning' to 'traditional heuristic-based Active Learning' may give intuitions if 'Learning active learning' is at the crossroads == Minimum marginal hyperplane == Some active learning algorithms are built upon support-vector machines (SVMs) and exploit the structure of the SVM to determine which data points to label. Such methods usually calculate the margin, W, of each u
CamScanner
CamScanner is a Chinese mobile app first released in 2010 that allows iOS and Android devices to be used as image scanners. It allows users to 'scan' documents (by taking a photo with the device's camera) and share the photo as either a JPEG or PDF. This app is available free of charge on the Google Play Store and the Apple App Store. The app is based on freemium model, with ad-supported free version and a premium version with additional functions. == History == On August 27, 2019, Russian cyber security company Kaspersky Lab discovered that recent versions of the Android app distributed an advertising library containing a Trojan Dropper, which was also included in some apps preinstalled on several Chinese mobiles. The advertising library decrypts a Zip archive which subsequently downloads additional files from servers controlled by hackers, allowing the hackers to control the device, including by showing intrusive advertising or charging paid subscriptions. Google took the app down after Kaspersky reported its findings. An updated version of the app with the advertising library removed was made available on the Google Play Store as of September 5, 2019. Kaspersky later acknowledged "We appreciate the willingness to cooperate that we've seen from CamScanner representatives, as well as the responsible attitude to user safety they demonstrated while eliminating the threat…The malicious modules were removed from the app immediately upon Kaspersky's warning, and Google Play has restored the app." In June 2020, as tensions along the Line of Actual Control between China and India continued, the Government of India decided to ban 118 Chinese apps, including TikTok and CamScanner citing data and privacy issues. On January 5, 2021, US President Donald Trump signed Executive Order 13971 banning Alipay, Tencent's QQ, QQ Wallet, WeChat Pay, CamScanner, Shareit, VMate and WPS Office to conduct US transactions. The Trump administration explained this act by saying that this move helps prevent personal information such as text, phone calls and photos collected from rivals. However, the Biden administration did not meet the February 2021 deadline for implementing the executive order, allowing these apps to operate in the US and revoked the previous executive order Executive Order 14034 of June 9, 2021.
Data-driven model
Data-driven models are a class of computational models that primarily rely on historical data collected throughout a system's or process' lifetime to establish relationships between input, internal, and output variables. Commonly found in numerous articles and publications, data-driven models have evolved from earlier statistical models, overcoming limitations posed by strict assumptions about probability distributions. These models have gained prominence across various fields, particularly in the era of big data, artificial intelligence, and machine learning, where they offer valuable insights and predictions based on the available data. == Background == These models have evolved from earlier statistical models, which were based on certain assumptions about probability distributions that often proved to be overly restrictive. The emergence of data-driven models in the 1950s and 1960s coincided with the development of digital computers, advancements in artificial intelligence research, and the introduction of new approaches in non-behavioural modelling, such as pattern recognition and automatic classification. == Key Concepts == Data-driven models encompass a wide range of techniques and methodologies that aim to intelligently process and analyse large datasets. Examples include fuzzy logic, fuzzy and rough sets for handling uncertainty, neural networks for approximating functions, global optimization and evolutionary computing, statistical learning theory, and Bayesian methods. These models have found applications in various fields, including economics, customer relations management, financial services, medicine, and the military, among others. Machine learning, a subfield of artificial intelligence, is closely related to data-driven modelling as it also focuses on using historical data to create models that can make predictions and identify patterns. In fact, many data-driven models incorporate machine learning techniques, such as regression, classification, and clustering algorithms, to process and analyse data. In recent years, the concept of data-driven models has gained considerable attention in the field of water resources, with numerous applications, academic courses, and scientific publications using the term as a generalization for models that rely on data rather than physics. This classification has been featured in various publications and has even spurred the development of hybrid models in the past decade. Hybrid models attempt to quantify the degree of physically based information used in hydrological models and determine whether the process of building the model is primarily driven by physics or purely data-based. As a result, data-driven models have become an essential topic of discussion and exploration within water resources management and research. The term "data-driven modelling" (DDM) refers to the overarching paradigm of using historical data in conjunction with advanced computational techniques, including machine learning and artificial intelligence, to create models that can reveal underlying trends, patterns, and, in some cases, make predictions Data-driven models can be built with or without detailed knowledge of the underlying processes governing the system behavior, which makes them particularly useful when such knowledge is missing or fragmented.