Baum–Welch algorithm

In electrical engineering, statistical computing and bioinformatics, the Baum–Welch algorithm is a special case of the expectation–maximization algorithm used to find the unknown parameters of a hidden Markov model (HMM). It makes use of the forward-backward algorithm to compute the statistics for the expectation step. The Baum–Welch algorithm, the primary method for inference in hidden Markov models, is numerically unstable due to its recursive calculation of joint probabilities. As the number of variables grows, these joint probabilities become increasingly small, leading to the forward recursions rapidly approaching values below machine precision. == History == The Baum–Welch algorithm was named after its inventors Leonard E. Baum and Lloyd R. Welch. The algorithm and the Hidden Markov models were first described in a series of articles by Baum and his peers at the IDA Center for Communications Research, Princeton in the late 1960s and early 1970s. One of the first major applications of HMMs was to the field of speech processing. In the 1980s, HMMs were emerging as a useful tool in the analysis of biological systems and information, and in particular genetic information. They have since become an important tool in the probabilistic modeling of genomic sequences. == Description == A hidden Markov model describes the joint probability of a collection of "hidden" and observed discrete random variables. It relies on the assumption that the i-th hidden variable given the (i − 1)-th hidden variable is independent of previous hidden variables, and the current observation variables depend only on the current hidden state. The Baum–Welch algorithm uses the well known EM algorithm to find the maximum likelihood estimate of the parameters of a hidden Markov model given a set of observed feature vectors. Let X t {\displaystyle X_{t}} be a discrete hidden random variable with N {\displaystyle N} possible values (i.e. We assume there are N {\displaystyle N} states in total). We assume the P ( X t ∣ X t − 1 ) {\displaystyle P(X_{t}\mid X_{t-1})} is independent of time t {\displaystyle t} , which leads to the definition of the time-independent stochastic transition matrix A = { a i j } = P ( X t = j ∣ X t − 1 = i ) . {\displaystyle A=\{a_{ij}\}=P(X_{t}=j\mid X_{t-1}=i).} The initial state distribution (i.e. when t = 1 {\displaystyle t=1} ) is given by π i = P ( X 1 = i ) . {\displaystyle \pi _{i}=P(X_{1}=i).} The observation variables Y t {\displaystyle Y_{t}} can take one of K {\displaystyle K} possible values. We also assume the observation given the "hidden" state is time independent. The probability of a certain observation y i {\displaystyle y_{i}} at time t {\displaystyle t} for state X t = j {\displaystyle X_{t}=j} is given by b j ( y i ) = P ( Y t = y i ∣ X t = j ) . {\displaystyle b_{j}(y_{i})=P(Y_{t}=y_{i}\mid X_{t}=j).} Taking into account all the possible values of Y t {\displaystyle Y_{t}} and X t {\displaystyle X_{t}} , we obtain the N × K {\displaystyle N\times K} matrix B = { b j ( y i ) } {\displaystyle B=\{b_{j}(y_{i})\}} where b j {\displaystyle b_{j}} belongs to all the possible states and y i {\displaystyle y_{i}} belongs to all the observations. An observation sequence is given by Y = ( Y 1 = y 1 , Y 2 = y 2 , … , Y T = y T ) {\displaystyle Y=(Y_{1}=y_{1},Y_{2}=y_{2},\ldots ,Y_{T}=y_{T})} . Thus we can describe a hidden Markov chain by θ = ( A , B , π ) {\displaystyle \theta =(A,B,\pi )} . The Baum–Welch algorithm finds a local maximum for θ ∗ = a r g m a x θ ⁡ P ( Y ∣ θ ) {\displaystyle \theta ^{}=\operatorname {arg\,max} _{\theta }P(Y\mid \theta )} (i.e. the HMM parameters θ {\displaystyle \theta } that maximize the probability of the observation). === Algorithm === Set θ = ( A , B , π ) {\displaystyle \theta =(A,B,\pi )} with random initial conditions. They can also be set using prior information about the parameters if it is available; this can speed up the algorithm and also steer it toward the desired local maximum. ==== Forward procedure ==== Let α i ( t ) = P ( Y 1 = y 1 , … , Y t = y t , X t = i ∣ θ ) {\displaystyle \alpha _{i}(t)=P(Y_{1}=y_{1},\ldots ,Y_{t}=y_{t},X_{t}=i\mid \theta )} , the probability of seeing the observations y 1 , y 2 , … , y t {\displaystyle y_{1},y_{2},\ldots ,y_{t}} and being in state i {\displaystyle i} at time t {\displaystyle t} . This is found recursively: α i ( 1 ) = π i b i ( y 1 ) , {\displaystyle \alpha _{i}(1)=\pi _{i}b_{i}(y_{1}),} α i ( t + 1 ) = b i ( y t + 1 ) ∑ j = 1 N α j ( t ) a j i . {\displaystyle \alpha _{i}(t+1)=b_{i}(y_{t+1})\sum _{j=1}^{N}\alpha _{j}(t)a_{ji}.} Since this series converges exponentially to zero, the algorithm will numerically underflow for longer sequences. However, this can be avoided in a slightly modified algorithm by scaling α {\displaystyle \alpha } in the forward and β {\displaystyle \beta } in the backward procedure below. ==== Backward procedure ==== Let β i ( t ) = P ( Y t + 1 = y t + 1 , … , Y T = y T ∣ X t = i , θ ) {\displaystyle \beta _{i}(t)=P(Y_{t+1}=y_{t+1},\ldots ,Y_{T}=y_{T}\mid X_{t}=i,\theta )} that is the probability of the ending partial sequence y t + 1 , … , y T {\displaystyle y_{t+1},\ldots ,y_{T}} given starting state i {\displaystyle i} at time t {\displaystyle t} . We calculate β i ( t ) {\displaystyle \beta _{i}(t)} as, β i ( T ) = 1 , {\displaystyle \beta _{i}(T)=1,} β i ( t ) = ∑ j = 1 N β j ( t + 1 ) a i j b j ( y t + 1 ) . {\displaystyle \beta _{i}(t)=\sum _{j=1}^{N}\beta _{j}(t+1)a_{ij}b_{j}(y_{t+1}).} ==== Update ==== We can now calculate the temporary variables, according to Bayes' theorem: γ i ( t ) = P ( X t = i ∣ Y , θ ) = P ( X t = i , Y ∣ θ ) P ( Y ∣ θ ) = α i ( t ) β i ( t ) ∑ j = 1 N α j ( t ) β j ( t ) , {\displaystyle \gamma _{i}(t)=P(X_{t}=i\mid Y,\theta )={\frac {P(X_{t}=i,Y\mid \theta )}{P(Y\mid \theta )}}={\frac {\alpha _{i}(t)\beta _{i}(t)}{\sum _{j=1}^{N}\alpha _{j}(t)\beta _{j}(t)}},} which is the probability of being in state i {\displaystyle i} at time t {\displaystyle t} given the observed sequence Y {\displaystyle Y} and the parameters θ {\displaystyle \theta } ξ i j ( t ) = P ( X t = i , X t + 1 = j ∣ Y , θ ) = P ( X t = i , X t + 1 = j , Y ∣ θ ) P ( Y ∣ θ ) = α i ( t ) a i j β j ( t + 1 ) b j ( y t + 1 ) ∑ k = 1 N ∑ w = 1 N α k ( t ) a k w β w ( t + 1 ) b w ( y t + 1 ) , {\displaystyle \xi _{ij}(t)=P(X_{t}=i,X_{t+1}=j\mid Y,\theta )={\frac {P(X_{t}=i,X_{t+1}=j,Y\mid \theta )}{P(Y\mid \theta )}}={\frac {\alpha _{i}(t)a_{ij}\beta _{j}(t+1)b_{j}(y_{t+1})}{\sum _{k=1}^{N}\sum _{w=1}^{N}\alpha _{k}(t)a_{kw}\beta _{w}(t+1)b_{w}(y_{t+1})}},} which is the probability of being in state i {\displaystyle i} and j {\displaystyle j} at times t {\displaystyle t} and t + 1 {\displaystyle t+1} respectively given the observed sequence Y {\displaystyle Y} and parameters θ {\displaystyle \theta } . The denominators of γ i ( t ) {\displaystyle \gamma _{i}(t)} and ξ i j ( t ) {\displaystyle \xi _{ij}(t)} are the same ; they represent the probability of making the observation Y {\displaystyle Y} given the parameters θ {\displaystyle \theta } . The parameters of the hidden Markov model θ {\displaystyle \theta } can now be updated: π i ∗ = γ i ( 1 ) , {\displaystyle \pi _{i}^{}=\gamma _{i}(1),} which is the expected frequency spent in state i {\displaystyle i} at time 1 {\displaystyle 1} . a i j ∗ = ∑ t = 1 T − 1 ξ i j ( t ) ∑ t = 1 T − 1 γ i ( t ) , {\displaystyle a_{ij}^{}={\frac {\sum _{t=1}^{T-1}\xi _{ij}(t)}{\sum _{t=1}^{T-1}\gamma _{i}(t)}},} which is the expected number of transitions from state i to state j compared to the expected total number of transitions starting in state i, including from state i to itself. The number of transitions starting in state i is equivalent to the number of times state i is observed in the sequence from t = 1 to t = T − 1. b i ∗ ( v k ) = ∑ t = 1 T 1 y t = v k γ i ( t ) ∑ t = 1 T γ i ( t ) , {\displaystyle b_{i}^{}(v_{k})={\frac {\sum _{t=1}^{T}1_{y_{t}=v_{k}}\gamma _{i}(t)}{\sum _{t=1}^{T}\gamma _{i}(t)}},} where 1 y t = v k = { 1 if y t = v k , 0 otherwise {\displaystyle 1_{y_{t}=v_{k}}={\begin{cases}1&{\text{if }}y_{t}=v_{k},\\0&{\text{otherwise}}\end{cases}}} is an indicator function, and b i ∗ ( v k ) {\displaystyle b_{i}^{}(v_{k})} is the expected number of times the output observations have been equal to v k {\displaystyle v_{k}} while in state i {\displaystyle i} over the expected total number of times in state i {\displaystyle i} . These steps are now repeated iteratively until a desired level of convergence. Note: It is possible to over-fit a particular data set. That is, P ( Y ∣ θ final ) > P ( Y ∣ θ true ) {\displaystyle P(Y\mid \theta _{\text{final}})>P(Y\mid \theta _{\text{true}})} . The algorithm also does not guarantee a global maximum. ==== Multiple sequences ==== The algorithm described thus far assumes a single observed sequence Y = y 1 , … , y T {\displaystyle Y=y_{1},\ldots ,y_{T}} . However, in many situations, there are several sequences observed: Y 1 ,

Site Security Handbook

The Site Security Handbook, RFC 2196, is a guide on setting computer security policies and procedures for sites that have systems on the Internet (however, the information provided should also be useful to sites not yet connected to the Internet). The guide lists issues and factors that a site must consider when setting their own policies. It makes a number of recommendations and provides discussions of relevant areas. This guide is only a framework for setting security policies and procedures. In order to have an effective set of policies and procedures, a site will have to make many decisions, gain agreement, and then communicate and implement these policies. The guide is a product of the IETF SSH working group, and was published in 1997, obsoleting the earlier RFC 1244 from 1991.

Supreme Commander (video game)

Supreme Commander (sometimes SupCom) is a 2007 real-time strategy video game designed by Chris Taylor and developed by his company, Gas Powered Games. The game is considered to be a spiritual successor, not a direct sequel, to Taylor's 1997 game Total Annihilation. First announced in the August 2005 edition of PC Gamer magazine, the game was released in Europe on February 16, 2007, and in North America on February 20. The standalone expansion Supreme Commander: Forged Alliance was released on November 6 of the same year. The sequel, Supreme Commander 2, was released in 2010. Nowadays, the original Supreme Commander is played through the community client called Forged Alliance Forever; the game has been further developed and balanced, and offers a wide variety of community mods. The gameplay of Supreme Commander focuses on using a giant bipedal mech called an Armored Command Unit (ACU), the so-called "Supreme Commander", to build a base, upgrading units to reach higher technology tiers, and conquering opponents. The player can command one of three factions: the Aeon Illuminate, the Cybran Nation, or the United Earth Federation (UEF). The expansion game added the Seraphim faction. Supreme Commander was highly anticipated in pre-release previews, and was well received by critics, with a Metacritic average of 86 out of 100. == Gameplay == Supreme Commander, like its spiritual predecessors, Total Annihilation and Spring, begins with the player solely possessing a single, irreplaceable construction unit called the "Armored Command Unit," or ACU, the titular Supreme Commander. Normally the loss of this unit results in the loss of the game (Skirmish missions can be set for a variety of victory conditions). These mech suits are designed to be transported through quantum gateways across the galaxy and contain all the materials and blueprints necessary to create an army from a planet's native resources in hours. All standard units except Commanders and summoned Support Commanders (sACU) are self-sufficient robots. All units and structures belong to one of four technology tiers, or "Tech" levels, each tier being stronger and/or more efficient than the previous. Certain lower-tier structures can be upgraded into higher ones without having to rebuild them. The first tier is available at the start of the game and consists of small, relatively weak units and structures. The second tier expands the player's abilities greatly, especially in terms of stationary weapons and shielding, and introduces upgraded versions of tier one units. The third tier level has very powerful assault units designed to overcome the fortifications of the most entrenched player. The fourth tier is a limited range of "experimental" technology. These are usually massive units which take a lot of time and energy to produce, but provide a significant tactical advantage. Supreme Commander features a varied skirmish AI. The typical Easy' and Normal modes are present, but the Hard difficulty level has four possible variants. Horde AI will swarm the player with hordes of lower level units, Tech AI will upgrade its units as fast as possible and assault the player with advanced units, the Balanced AI attempts to find a balance between the two, and the Supreme AI decides which of the three hard strategies is best for the map. The single player campaign consists of eighteen missions, six for each faction. The player is an inexperienced Commander who plays a key role in their faction's campaign to bring the "Infinite War" to an end. Despite the low number of campaign missions, each mission can potentially last hours. At the start of a mission, objectives are assigned for the player to complete. Once the player accomplishes them, the map is expanded, sometimes doubling or tripling in size, and new objectives are assigned. As the mission is commonly divided into three segments, the player will often have to overcome several enemy positions to achieve victory. === Resource management === Because humans have developed replication technology, making advanced use of rapid prototyping and nanotechnology, only two types of resources are required to wage war: Energy and Mass. Energy is obtained by constructing power generators on any solid surface (except fuel generators, which can only be built on fuel deposits), while Mass is obtained either by placing mass extractors on limited mass deposit spots (the most efficient method, although it requires map control) or by building mass fabricators to convert energy into mass. Constructor units can gather energy by "reclaiming" it from organic debris such as trees and mass from rocks and wrecked units. Each player has a certain amount of resource storage, which can be expanded by the construction of storage structures. This gives the player reserves in times of shortage or allows them to stockpile resources. If the resource generation exceeds the player's capacity, the material is wasted. On the contrary, if the storages are depleted and the demand of one of the resources exceeds the production, then all the productions speed is reduced. In addition, if an energy deficit occurs, shields will stop working. An adjacency system allows certain structures to benefit from being built directly adjacent to others. Energy-consuming structures will use less energy when built adjacent to power generators and power generators will produce more energy when built adjacent to power storage structures. The same applies to their mass-producing equivalents. Likewise, factories will consume less energy and mass when built adjacent to power generators and mass fabricators/extractors, respectively. However, by placing structures in close proximity, they become more vulnerable to collateral damage if an adjacent structure is destroyed. Furthermore, most resource generation structures can cause chain reactions when destroyed (especially Tier III structures, which produce large amounts of resources but often have large detonations that can wipe out a nearby army). === Warfare === Supreme Commander uses a "strategic zoom" system that allows the player to seamlessly zoom from a detailed close up view of an individual unit all the way out to a view of the entire map, at which point it resembles a fullscreen version of the minimap denoting individual units with icons. The camera also has a free movement mode and can be slaved to track a selected unit and there is a split screen mode which also supports multiple monitors. This system allows Supreme Commander to use vast maps up to 80 km x 80 km, with players potentially controlling a thousand units each. Units in Supreme Commander are built to scale as they would be in the real world. For example, battleships dwarf submarines. Late into the game, the larger "experimental" units, such as the Cybran Monkeylord, an enormous spider-shaped assault unit, can actually crush smaller enemy units by stepping on them. Because of the wide range of planets colonized by humanity in the setting, the theatres of war range from desert to arctic, and all battlespaces are employed. Technologies emerging in modern warfare are frequently employed in Supreme Commander. For example, stealth technology and both tactical and strategic missile and missile defense systems can be used. Supreme Commander introduced several innovations designed to reduce the amount of micromanagement inherent in many RTS games. Engineers units have the command "assist", that will help follow other engineers and help them finish their orders or improve production rate of factories. In addition, engineers with the order "patrol" will repair units, buildings and recycle wrecks in their along their patrol route. Holding the shift key causes any orders given to a unit (or group of units) to be queued. In this manner a unit may be ordered to attack several targets in succession, or to make best speed to a given point on the map and then attack towards a specified location engaging any hostiles it encounters along the way. After orders have been issued, holding the shift key causes all issued orders to be displayed on the map where they can be subsequently modified to accommodate a change of plan. Further, when a unit is ordered to attack a target, the player can issue an order to perform a coordinated attack to another unit. This order coordinates the arrival time of the units at the target automatically by adjusting the speed of the units involved. As in other RTS games, air transports can be used to convey units to specified destinations, in Supreme Commander though by shift queuing orders a transport containing several units can be ordered to drop specific units at subsequent waypoints. An air transport can also be ordered to create a ferry route, an airbridge wherein any land units ordered to the start of the ferry route will be conveyed by the air transport to the specified destination. The output from a production factory can be routed to a ferry route causing all units co

With Folded Hands ...

"With Folded Hands ..." is a 1947 science fiction novelette by American writer Jack Williamson (1908–2006). In writing it, Williamson was influenced by the aftermath of World War II, the atomic bombings of Hiroshima and Nagasaki, and his concern that "some of the technological creations we had developed with the best intentions might have disastrous consequences in the long run." The novelette first appeared in the July 1947 issue of Astounding Science Fiction and was later included in The Science Fiction Hall of Fame, Volume Two (1973) after being voted one of the best novellas up to 1965. In 1950, it was the first of several Astounding stories adapted for NBC's radio series Dimension X. == Rewrite and sequel == The 1947 publication was followed by a novel-length rewrite, with a different setting and inventor. At the behest of Astounding editor-in-chief John W. Campbell, a new ending had the robots defeated by means of what Williamson and Campbell would later christen "psionics". This novel was serialized, also in Astounding (March, April, May 1948), as ... And Searching Mind, and finally published in hardback book form as The Humanoids (1949). Much later, in 1980, Williamson followed with another sequel, The Humanoid Touch. == Plot summary == Underhill, a seller of "Mechanicals" (unthinking robots that perform menial tasks) in the small town of Two Rivers, is startled to find a competitor's store on his way home. The competitors are not humans but are small black robots who appear more advanced than anything Underhill has encountered before. They describe themselves as "humanoids". Disturbed at his encounter, Underhill rushes home to discover that his wife has taken in a new lodger, a mysterious old man named Sledge. In the course of the next day, the new Mechanicals have appeared everywhere in town. They state that they only follow the Prime Directive: "to serve and obey and guard men from harm". Offering their services free of charge, they replace humans as police officers, bank tellers, and more, and eventually drive Underhill out of business. Despite the humanoids' benign appearance and mission, Underhill soon realizes that, in the name of their Prime Directive, the mechanicals have essentially taken over every aspect of human life. No humans may engage in any behavior that might endanger them, and every human action is carefully scrutinized. Suicide is prohibited. Humans who resist the Prime Directive are taken away and lobotomized, so that they may live happily under the direction of the humanoids. Underhill learns that his lodger Sledge is the creator of the humanoids and is on the run from them. Sledge explains that 60 years earlier he had discovered the force of "rhodomagnetics" on the planet Wing IV and that his discovery resulted in a war that destroyed his planet. In his grief, Sledge designed the humanoids to help humanity and be invulnerable to human exploitation. However, he eventually realized that they had instead taken control of humanity, in the name of their Prime Directive, to make humans happy. The humanoids are spreading out from Wing IV to every human-occupied planet to implement their Prime Directive. Sledge and Underhill attempt to stop the humanoids by aiming a rhodomagnetic beam at Wing IV, but fail. The humanoids take Sledge away for surgery. He returns with no memory of his prior life, stating that he is now happy under the humanoids' care. Underhill is driven home by the humanoids, sitting "with folded hands," as there is nothing left to do. == Origins == In a 1991 interview, Williamson revealed how the story construction reflected events of his childhood in addition to technological extrapolations: I wrote "With Folded Hands" immediately after World War II, when the shadow of the atomic bomb had just fallen over SF and was just beginning to haunt the imaginations of people in the US. The story grows out of that general feeling that some of the technological creations we had developed with the best intentions might have disastrous consequences in the long run (that idea, of course, still seems relevant today). The notion I was consciously working on specifically came out of a fragment of a story I had worked on for a while about an astronaut in space who is accompanied by a robot obviously superior to him physically—i.e., the robot wasn't hurt by gravity, extremes of temperature, radiation, or whatever. Just looking at the fragment gave me the sense of how inferior humanity is in many ways to mechanical creations. That basic recognition was the essence of the story, and as I wrote it up in my notes the theme was that the perfect machine would prove to be perfectly destructive... It was only when I looked back at the story much later on that I was able to realize that the emotional reach of the story undoubtedly derived from my own early childhood, when people were attempting to protect me from all those hazardous things a kid is going to encounter in the isolated frontier setting I grew up in. As a result, I felt frustrated and over protected by people whom I couldn't hate because I loved them. A sort of psychological trap. Specifically, the first three years of my life were spent on a ranch at the top of the Sierra Madre Mountains on the headwaters of the Yaqui River in Sonora, Mexico. ... [My mother] was terrified by this environment. My father built a crib that became a psychological prison for me, particularly because my mother apparently kept me in it too long, when I needed to get out and crawl on the floor. ... In retrospect, I'm certain I projected my fears and suspicions of this kind of conditioning, and these projections became the governing emotional principle of "With Folded Hands" and The Humanoids. == Reception == In 2024, Robert Silverberg wrote an essay in which he asserted that "With Folded Hands..." is "probably the best story ever written about robots" and suggested that Elon Musk's Optimus Generation 2 is the realization of the "humanoids" along with their worst drawbacks.

Argumentation theory

Argumentation theory is the interdisciplinary study of how conclusions can be supported or undermined by premises through logical reasoning. With historical origins in logic, dialectic, and rhetoric, argumentation theory includes the arts and sciences of civil debate, dialogue, conversation, and persuasion. It studies rules of inference, logic, and procedural rules in both artificial and real-world settings. Argumentation includes various forms of dialogue such as deliberation and negotiation which are concerned with collaborative decision-making procedures. It also encompasses eristic dialogue, the branch of social debate in which victory over an opponent is the primary goal, and didactic dialogue used for teaching. This discipline also studies the means by which people can express and rationally resolve or at least manage their disagreements. Argumentation is a daily occurrence, such as in public debate, science, and law. For example in law, in courts by the judge, the parties and the prosecutor, in presenting and testing the validity of evidences. Also, argumentation scholars study the post hoc rationalizations by which organizational actors try to justify decisions they have made irrationally. Argumentation is one of four rhetorical modes (also known as modes of discourse), along with exposition, description, and narration. == Key components of argumentation == Some key components of argumentation are: Understanding and identifying arguments, either explicit or implied, and the goals of the participants in the different types of dialogue. Identifying the premises from which conclusions are derived. Establishing the "burden of proof" – determining who made the initial claim and is thus responsible for providing evidence why their position merits acceptance. For the one carrying the "burden of proof", the advocate, to marshal evidence for their position in order to convince or force the opponent's acceptance. The method by which this is accomplished is producing valid, sound, and cogent arguments, devoid of weaknesses, and not easily attacked. In a debate, fulfillment of the burden of proof creates a burden of rejoinder. One must try to identify faulty reasoning in the opponent's argument, to attack the reasons/premises of the argument, to provide counterexamples if possible, to identify any fallacies, and to show why a valid conclusion cannot be derived from the reasons provided for their argument. For example, consider the following exchange, illustrating the No true Scotsman fallacy: Argument: "No Scotsman puts sugar on his porridge." Reply: "But my friend Angus, who is a Scotsman, likes sugar with his porridge." Rebuttal: "Well perhaps, but no true Scotsman puts sugar on his porridge." In this dialogue, the proposer first offers a premise, the premise is challenged by the interlocutor, and so the proposer offers a modification of the premise, which is designed only to evade the challenge provided. == Internal structure of arguments == Typically an argument has an internal structure, comprising the following: a set of assumptions or premises, a method of reasoning or deduction, and a conclusion or point. An argument has one or more premises and one conclusion. Often classical logic is used as the method of reasoning so that the conclusion follows logically from the assumptions or support. One challenge is that if the set of assumptions is inconsistent then anything can follow logically from inconsistency. Therefore, it is common to insist that the set of assumptions be consistent. It is also good practice to require the set of assumptions to be the minimal set, with respect to set inclusion, necessary to infer the consequent. Such arguments are called MINCON arguments, short for minimal consistent. Such argumentation has been applied to the fields of law and medicine. A non-classical approach to argumentation investigates abstract arguments, where 'argument' is considered a primitive term, so no internal structure of arguments is taken into account. == Types of dialogue == In its most common form, argumentation involves an individual and an interlocutor or opponent engaged in dialogue, each contending differing positions and trying to persuade each other, but there are various types of dialogue: Persuasion dialogue aims to resolve conflicting points of view of different positions. Negotiation aims to resolve conflicts of interests by cooperation and dealmaking. Inquiry aims to resolve general ignorance by the growth of knowledge. Deliberation aims to resolve a need to take action by reaching a decision. Information seeking aims to reduce one party's ignorance by requesting information from another party that is in a position to know something. Eristic aims to resolve a situation of antagonism through verbal fighting. == Argumentation and the grounds of knowledge == Argumentation theory had its origins in foundationalism, a theory of knowledge (epistemology) in the field of philosophy. It sought to find the grounds for claims in the forms (logic) and materials (factual laws) of a universal system of knowledge. The dialectical method was made famous by Plato and his use of Socrates critically questioning various characters and historical figures. But argument scholars gradually rejected Aristotle's systematic philosophy and the idealism in Plato and Kant. They questioned and ultimately discarded the idea that argument premises take their soundness from formal philosophical systems. The field thus broadened. One of the original contributors to this trend was the philosopher Chaïm Perelman, who together with Lucie Olbrechts-Tyteca introduced the French term la nouvelle rhetorique in 1958 to describe an approach to argument which is not reduced to application of formal rules of inference. Perelman's view of argumentation is much closer to a juridical one, in which rules for presenting evidence and rebuttals play an important role. Karl R. Wallace's seminal essay, "The Substance of Rhetoric: Good Reasons" in the Quarterly Journal of Speech (1963) 44, led many scholars to study "marketplace argumentation" – the ordinary arguments of ordinary people. The seminal essay on marketplace argumentation is Ray Lynn Anderson's and C. David Mortensen's "Logic and Marketplace Argumentation" Quarterly Journal of Speech 53 (1967): 143–150. This line of thinking led to a natural alliance with late developments in the sociology of knowledge. Some scholars drew connections with recent developments in philosophy, namely the pragmatism of John Dewey and Richard Rorty. Rorty has called this shift in emphasis "the linguistic turn". In this new hybrid approach argumentation is used with or without empirical evidence to establish convincing conclusions about issues which are moral, scientific, epistemic, or of a nature in which science alone cannot answer. Out of pragmatism and many intellectual developments in the humanities and social sciences, "non-philosophical" argumentation theories grew which located the formal and material grounds of arguments in particular intellectual fields. These theories include informal logic, social epistemology, ethnomethodology, speech acts, the sociology of knowledge, the sociology of science, and social psychology. These new theories are not non-logical or anti-logical. They find logical coherence in most communities of discourse. These theories are thus often labeled "sociological" in that they focus on the social grounds of knowledge. == Kinds of argumentation == === Conversational argumentation === The study of naturally occurring conversation arose from the field of sociolinguistics. It is usually called conversation analysis (CA). Inspired by ethnomethodology, it was developed in the late 1960s and early 1970s principally by the sociologist Harvey Sacks and, among others, his close associates Emanuel Schegloff and Gail Jefferson. Sacks died early in his career, but his work was championed by others in his field, and CA has now become an established force in sociology, anthropology, linguistics, speech-communication and psychology. It is particularly influential in interactional sociolinguistics, discourse analysis and discursive psychology, as well as being a coherent discipline in its own right. Recently CA techniques of sequential analysis have been employed by phoneticians to explore the fine phonetic details of speech. Empirical studies and theoretical formulations by Sally Jackson and Scott Jacobs, and several generations of their students, have described argumentation as a form of managing conversational disagreement within communication contexts and systems that naturally prefer agreement. === Mathematical argumentation === The basis of mathematical truth has been the subject of long debate. Frege in particular sought to demonstrate (see Gottlob Frege, The Foundations of Arithmetic, 1884, and Begriffsschrift, 1879) that arithmetical truths can be derived from purely logical axioms and therefore are, in th

Pixel shift

Pixel shift is a method in digital cameras for producing a super-resolution image. The method works by taking several images, after each such capture moving ("shifting") the sensor to a new position. In digital colour cameras that employ pixel shift, this avoids a major limitation inherent in using Bayer pattern for obtaining colour, and instead produces an image with increased colour resolution and, assuming a static subject or additional computational steps, an image free of colour moiré. Taking this idea further, sub-pixel shifting may increase the resolution of the final image beyond that suggested by the specified resolution of the image sensor. Additionally, assuming that the various individual captures are taken at the same sensitivity, the final combined image will have less image noise than a single capture. This can be thought of as an averaging effect (for instance, in a pixel shift image composed of four individual frames with a classic Bayer pattern, every pixel in the final colour image is based on two measurements of the green channel). == List of cameras implementing pixel shift == All of the following cameras are fabricated with one imaging sensor, thus any kind of pixel shift requires a movement of the whole sensor. === Canon === Canon R5: Contains a 45 Mpixel sensor. The High-Resolution Mode shifts the sensor by one pixel to obtain a sequence of nine images that are merged into a 400 Mpixel image. === Fujifilm === Fujifilm GFX50S II: contains a 51 Mpixel sensor. The Pixel Shift Multi-Shot mode shifts the imaging sensor by 0.5-pixel movements to obtain a sequence of 16 images that are subsequently merged into a 200 Mpixel image. Fujifilm GFX100, Fujifilm GFX100 II: contains a 102 Mpixel sensor. A sequence of 16 pixel shifted images are merged into a 400 Mpixel image. Fujifilm GFX100S, Fujifilm GFX100S II: contains a 102 Mpixel sensor. A sequence of 16 pixel shifted images are merged into a 400 Mpixel image Fujifilm GFX100IR: contains a 102 Mpixel sensor. A sequence of 16 pixel shifted images are merged into a 400 Mpixel image Fujifilm X-H2: contains a 40 Mpixel sensor. A sequence of 20 shifted images are merged into a 160 Mpixel image. Fujifilm X-T5: contains a 40 Mpixel sensor. A sequence of 20 shifted images are merged into a 160 Mpixel image. === Nikon === Nikon Z8: contains a 47.5 Mpixel sensor. The High Res shot mode shifts the imaging sensor by 0.5-pixel movements to obtain a sequence of up to 32 images that can be merged in Nikon's NX studio software. Nikon Zf: contains a 24 Mpixel sensor. The High Res shot mode shifts the imaging sensor by 0.5-pixel movements to obtain a sequence of up to 32 images that can be merged in Nikon's NX studio software. === Olympus === Olympus OM-D E-M1 Mark II: contains a 20.4 Mpixel sensor. The High Res shot mode produces a 50 Mpixel image. Olympus OM-D E-M5 Mark II: contains a 16 Mpixel sensor. The High Res shot mode shifts the imaging sensor by 0.5-pixel movements to obtain a sequence of 8 images that are subsequently merged into a 40 Mpixel image. Olympus OM-D E-M5 Mark III: contains a 20.4 Mpixel sensor. The High Res shot mode shifts the imaging sensor by 0.5-pixel movements to obtain a sequence of 8 images that are subsequently merged into a 50 Mpixel image. Olympus OM-D E-M1X: contains a 20.4 Mpixel sensor. The camera sports two pixel shift mode: (a) the 80Mp Tripod mode produces an 80 Mpixel image, (b) the Handheld High Res shot mode produces a 50 Mpixel image. Olympus PEN-F: contains a 20.4 Mpixel sensor. The High Res Shot mode takes multiple images, continually shifting the position of the sensor in sub-pixel increments. Combining these images results in either a 50MP JPEG or an 80MP Raw file. ==== OM System ==== OM System OM-1: contains a 20MPix sensor. The High Res Shot mode takes multiple images, and it can be used handheld or on a tripod. Handheld it will internally produce 50 Mpix files and 80 Mpix when mounted on a tripod. OM System OM-5: contains a 20MPix sensor. The High Res Shot mode takes multiple images, and it can be used handheld or on a tripod. Handheld it will internally produce 50 Mpix files and 80 Mpix when mounted on a tripod. === Panasonic === Panasonic Lumix DC-G9: contains a 20.3 Mpixel sensor. The High Resolution Mode takes a sequence of 8 shots in quick succession between which the sensor is shifted by 0.5 pixel for each image. These are subsequently merged into an 80 Mpixel image. Panasonic Lumix DC-S1: contains a 24.2 Mpixel sensor. The High Resolution Mode takes a sequence of shots in quick succession between which the sensor is shifted by a small amount. These are subsequently merged into a 96 Mpixel image. Panasonic Lumix DC-S1R: contains a 47.3 Mpixel sensor. The High Resolution Mode shifts the imaging sensor by a small increments to obtain a sequence of 8 images that are subsequently merged into a 187 Mpixel image. Panasonic Lumix DC-S1H Panasonic Lumix DC-S5 === Pentax === Pentax K-70: contains a 24.3 Mpixel sensor. The pixel shift mode takes a sequence of 4 shots between which the sensor is shifted by 1 pixel. These are subsequently merged into an image sporting 'all color data in each pixel to deliver super-high-resolution images'. Pentax KP: contains a 24.3 Mpixel sensor. The pixel shift mode takes a sequence of 4 shots between which the sensor is shifted by 1 pixel. These are subsequently merged into an image sporting 'high-resolution images with more accurate colours and much finer details'. Pentax K-3 II: contains a 24.3 Mpixel sensor. The pixel shift mode takes a sequence of 4 shots between which the sensor is shifted by 1 pixel. These are subsequently merged into an image sporting 'super-high-resolution images with far more truthful color reproduction and much finer details'. Pentax K-3 III: contains a 25.7 Mpixel sensor. The pixel shift mode takes a sequence of 4 shots between which the sensor is shifted by 1 pixel. These are subsequently merged into an image sporting 'a cancelling out of the Bayer pattern and removal of the need for sharpness-sapping demosaicing'. Pentax K-1: contains a 36.4 Mpixel sensor. The pixel shift mode takes a sequence of 4 shots between which the sensor is shifted by 1 pixel. These are subsequently merged into an image sporting 'improved detail and colour resolution'. Pentax K-1 II: contains a 36.4 Mpixel sensor. The camera sports two pixel shift mode: (a) a series of 4 tripod-stabilised images shifted by 1 pixel each are subsequently combined into a 47.3 Mpixel image, (b) a series of images taken in handheld mode are combined into a 47.3 Mpixel image that is, within limits, able to cope even with moving subjects. === Sony === Sony a6600: contains a 24.3 Mpixel sensor. The pixel shift mode takes a sequence of 4 shots between which the sensor is shifted by 1 pixel. These are subsequently merged into an image sporting 'all color data in each pixel to deliver super-high-resolution images'. Sony α7R III: contains a 42.4 Mpixel sensor. The pixel shift mode takes a sequence of 4 shots between which the sensor is shifted by 1 pixel. These are subsequently merged into a 42.4 Mpixel image with improved tonal resolution. Sony α7R IV: contains a 61 Mpixel sensor. The camera has two pixel shift modes, (a) the first takes a sequence of 4 shots between which the sensor is shifted by 1 pixel. These are subsequently merged into a 61 Mpixel image with improved tonal resolution, (b) the other takes a sequence of 16 shots between which the sensor is shifted by 0.5 pixel. These are subsequently merged into a 240 Mpixel image with both enhanced detail and improved tonal resolution. Sony α1: contains a 50 Mpixel sensor. The camera has two pixel shift modes, (a) the first takes a sequence of 4 shots between which the sensor is shifted by 1 pixel. These are subsequently merged into a 50 Mpixel image with improved tonal resolution, (b) the other takes a sequence of 16 shots between which the sensor is shifted by 0.5 pixel. These are subsequently merged into a 200 Mpixel image with both enhanced detail and improved tonal resolution. === Hasselblad === Hasselblad H3DII: the model H3DII-39 sports a 39 Mpixel sensor, the model H3DII-50 a 50 Mpixel sensor. Both enable a pixel shift mode which takes a sequence of 4 shots between which the sensor is shifted by 1 pixel. These are subsequently merged into a single image. Hasselblad H4D series: the model H4D-200MS contains a 50 Mpixel sensor. The sensor sports 3 different pixel shift modes which take (a) a sequence of 6 shots taken at slight offsets, (b) a sequence of 4 shots between which the sensor is shifted by 1 pixel, (c) a sequence of 4 shots between which the sensor is shifted by 0.5 pixels. Images obtained by all three modes are subsequently merged into 200 Mpixel images. Hasselblad H5D series: both models H5D-50c MS and H5D-200c MS contain a 50 Mpixel sensor. This sensor sports 2 different pixel shift modes which take (a) a sequence of 6 shots with full and half pixel moveme

Estimation of distribution algorithm

Estimation of distribution algorithms (EDAs), sometimes called probabilistic model-building genetic algorithms (PMBGAs), are stochastic optimization methods that guide the search for the optimum by building and sampling explicit probabilistic models of promising candidate solutions. Optimization is viewed as a series of incremental updates of a probabilistic model, starting with the model encoding an uninformative prior over admissible solutions and ending with the model that generates only the global optima. EDAs belong to the class of evolutionary algorithms. The main difference between EDAs and most conventional evolutionary algorithms is that evolutionary algorithms generate new candidate solutions using an implicit distribution defined by one or more variation operators, whereas EDAs use an explicit probability distribution encoded by a Bayesian network, a multivariate normal distribution, or another model class. Similarly as other evolutionary algorithms, EDAs can be used to solve optimization problems defined over a number of representations from vectors to LISP style S expressions, and the quality of candidate solutions is often evaluated using one or more objective functions. The general procedure of an EDA is outlined in the following: t := 0 initialize model M(0) to represent uniform distribution over admissible solutions while (termination criteria not met) do P := generate N>0 candidate solutions by sampling M(t) F := evaluate all candidate solutions in P M(t + 1) := adjust_model(P, F, M(t)) t := t + 1 Using explicit probabilistic models in optimization allowed EDAs to feasibly solve optimization problems that were notoriously difficult for most conventional evolutionary algorithms and traditional optimization techniques, such as problems with high levels of epistasis. Nonetheless, the advantage of EDAs is also that these algorithms provide an optimization practitioner with a series of probabilistic models that reveal a lot of information about the problem being solved. This information can in turn be used to design problem-specific neighborhood operators for local search, to bias future runs of EDAs on a similar problem, or to create an efficient computational model of the problem. For example, if the population is represented by bit strings of length 4, the EDA can represent the population of promising solution using a single vector of four probabilities (p1, p2, p3, p4) where each component of p defines the probability of that position being a 1. Using this probability vector it is possible to create an arbitrary number of candidate solutions. == Estimation of distribution algorithms (EDAs) == This section describes the models built by some well known EDAs of different levels of complexity. It is always assumed a population P ( t ) {\displaystyle P(t)} at the generation t {\displaystyle t} , a selection operator S {\displaystyle S} , a model-building operator α {\displaystyle \alpha } and a sampling operator β {\displaystyle \beta } . == Univariate factorizations == The most simple EDAs assume that decision variables are independent, i.e. p ( X 1 , X 2 ) = p ( X 1 ) ⋅ p ( X 2 ) {\displaystyle p(X_{1},X_{2})=p(X_{1})\cdot p(X_{2})} . Therefore, univariate EDAs rely only on univariate statistics and multivariate distributions must be factorized as the product of N {\displaystyle N} univariate probability distributions, D Univariate := p ( X 1 , … , X N ) = ∏ i = 1 N p ( X i ) . {\displaystyle D_{\text{Univariate}}:=p(X_{1},\dots ,X_{N})=\prod _{i=1}^{N}p(X_{i}).} Such factorizations are used in many different EDAs, next we describe some of them. === Univariate marginal distribution algorithm (UMDA) === The UMDA is a simple EDA that uses an operator α U M D A {\displaystyle \alpha _{UMDA}} to estimate marginal probabilities from a selected population S ( P ( t ) ) {\displaystyle S(P(t))} . By assuming S ( P ( t ) ) {\displaystyle S(P(t))} contain λ {\displaystyle \lambda } elements, α U M D A {\displaystyle \alpha _{UMDA}} produces probabilities: p t + 1 ( X i ) = 1 λ ∑ x ∈ S ( P ( t ) ) x i , ∀ i ∈ 1 , 2 , … , N . {\displaystyle p_{t+1}(X_{i})={\dfrac {1}{\lambda }}\sum _{x\in S(P(t))}x_{i},~\forall i\in 1,2,\dots ,N.} Every UMDA step can be described as follows D ( t + 1 ) = α UMDA ∘ S ∘ β λ ( D ( t ) ) . {\displaystyle D(t+1)=\alpha _{\text{UMDA}}\circ S\circ \beta _{\lambda }(D(t)).} === Population-based incremental learning (PBIL) === The PBIL, represents the population implicitly by its model, from which it samples new solutions and updates the model. At each generation, μ {\displaystyle \mu } individuals are sampled and λ ≤ μ {\displaystyle \lambda \leq \mu } are selected. Such individuals are then used to update the model as follows p t + 1 ( X i ) = ( 1 − γ ) p t ( X i ) + ( γ / λ ) ∑ x ∈ S ( P ( t ) ) x i , ∀ i ∈ 1 , 2 , … , N , {\displaystyle p_{t+1}(X_{i})=(1-\gamma )p_{t}(X_{i})+(\gamma /\lambda )\sum _{x\in S(P(t))}x_{i},~\forall i\in 1,2,\dots ,N,} where γ ∈ ( 0 , 1 ] {\displaystyle \gamma \in (0,1]} is a parameter defining the learning rate, a small value determines that the previous model p t ( X i ) {\displaystyle p_{t}(X_{i})} should be only slightly modified by the new solutions sampled. PBIL can be described as D ( t + 1 ) = α PIBIL ∘ S ∘ β μ ( D ( t ) ) {\displaystyle D(t+1)=\alpha _{\text{PIBIL}}\circ S\circ \beta _{\mu }(D(t))} === Compact genetic algorithm (cGA) === The CGA, also relies on the implicit populations defined by univariate distributions. At each generation t {\displaystyle t} , two individuals x , y {\displaystyle x,y} are sampled, P ( t ) = β 2 ( D ( t ) ) {\displaystyle P(t)=\beta _{2}(D(t))} . The population P ( t ) {\displaystyle P(t)} is then sorted in decreasing order of fitness, S Sort ( f ) ( P ( t ) ) {\displaystyle S_{{\text{Sort}}(f)}(P(t))} , with u {\displaystyle u} being the best and v {\displaystyle v} being the worst solution. The CGA estimates univariate probabilities as follows p t + 1 ( X i ) = p t ( X i ) + γ ( u i − v i ) , ∀ i ∈ 1 , 2 , … , N , {\displaystyle p_{t+1}(X_{i})=p_{t}(X_{i})+\gamma (u_{i}-v_{i}),\quad \forall i\in 1,2,\dots ,N,} where, γ ∈ ( 0 , 1 ] {\displaystyle \gamma \in (0,1]} is a constant defining the learning rate, usually set to γ = 1 / N {\displaystyle \gamma =1/N} . The CGA can be defined as D ( t + 1 ) = α CGA ∘ S Sort ( f ) ∘ β 2 ( D ( t ) ) {\displaystyle D(t+1)=\alpha _{\text{CGA}}\circ S_{{\text{Sort}}(f)}\circ \beta _{2}(D(t))} == Bivariate factorizations == Although univariate models can be computed efficiently, in many cases they are not representative enough to provide better performance than GAs. In order to overcome such a drawback, the use of bivariate factorizations was proposed in the EDA community, in which dependencies between pairs of variables could be modeled. A bivariate factorization can be defined as follows, where π i {\displaystyle \pi _{i}} contains a possible variable dependent to X i {\displaystyle X_{i}} , i.e. | π i | = 1 {\displaystyle |\pi _{i}|=1} . D Bivariate := p ( X 1 , … , X N ) = ∏ i = 1 N p ( X i | π i ) . {\displaystyle D_{\text{Bivariate}}:=p(X_{1},\dots ,X_{N})=\prod _{i=1}^{N}p(X_{i}|\pi _{i}).} Bivariate and multivariate distributions are usually represented as probabilistic graphical models (graphs), in which edges denote statistical dependencies (or conditional probabilities) and vertices denote variables. To learn the structure of a PGM from data linkage-learning is employed. === Mutual information maximizing input clustering (MIMIC) === The MIMIC factorizes the joint probability distribution in a chain-like model representing successive dependencies between variables. It finds a permutation of the decision variables, r : i ↦ j {\displaystyle r:i\mapsto j} , such that x r ( 1 ) x r ( 2 ) , … , x r ( N ) {\displaystyle x_{r(1)}x_{r(2)},\dots ,x_{r(N)}} minimizes the Kullback–Leibler divergence in relation to the true probability distribution, i.e. π r ( i + 1 ) = { X r ( i ) } {\displaystyle \pi _{r(i+1)}=\{X_{r(i)}\}} . MIMIC models a distribution p t + 1 ( X 1 , … , X N ) = p t ( X r ( N ) ) ∏ i = 1 N − 1 p t ( X r ( i ) | X r ( i + 1 ) ) . {\displaystyle p_{t+1}(X_{1},\dots ,X_{N})=p_{t}(X_{r(N)})\prod _{i=1}^{N-1}p_{t}(X_{r(i)}|X_{r(i+1)}).} New solutions are sampled from the leftmost to the rightmost variable, the first is generated independently and the others according to conditional probabilities. Since the estimated distribution must be recomputed each generation, MIMIC uses concrete populations in the following way P ( t + 1 ) = β μ ∘ α MIMIC ∘ S ( P ( t ) ) . {\displaystyle P(t+1)=\beta _{\mu }\circ \alpha _{\text{MIMIC}}\circ S(P(t)).} === Bivariate marginal distribution algorithm (BMDA) === The BMDA factorizes the joint probability distribution in bivariate distributions. First, a randomly chosen variable is added as a node in a graph, the most dependent variable to one of those in the graph is chosen among those not yet in the graph, this procedure is repeated until no remain