Simulation noise is a function that creates a divergence-free vector field. This signal can be used in artistic simulations for the purpose of increasing the perception of extra detail. The function can be calculated in three dimensions by dividing the space into a regular lattice grid. With each edge is associated a random value, indicating a rotational component of material revolving around the edge. By following rotating material into and out of faces, one can quickly sum the flux passing through each face of the lattice. Flux values at lattice faces are then interpolated to create a field value for all positions. Perlin noise is the earliest form of lattice noise, which has become very popular in computer graphics. Perlin Noise is not suited for simulation because it is not divergence-free. Noises based on lattices, such as simulation noise and Perlin noise, are often calculated at different frequencies and summed together to form band-limited fractal signals. Other approaches developed later that use vector calculus identities to produce divergence free fields, such as "Curl-Noise" as suggested by Rook Bridson, and "Divergence-Free Noise" due to Ivan DeWolf. These often require calculation of lattice noise gradients, which sometimes are not readily available. A naive implementation would call a lattice noise function several times to calculate its gradient, resulting in more computation than is strictly necessary. Unlike these noises, simulation noise has a geometric rationale in addition to its mathematical properties. It simulates vortices scattered in space, to produce its pleasing aesthetic. == Curl noise == The vector field is created as follows, for every point (x,y,z) in the space a vector field G is created, every component x, y and z of the vector field (Gx, Gy, Gz) is defined by a 3D perlin or simplex noise function with x, y and z as parameters. The partial derivative of Gx, Gy, and Gz respect to x, y and z is obtained with the gradient of the perlin or simplex noise by finite differences of implicit calculation inside the simplex noise. The partial derivatives are used to calculate F as the curl of G given by F = ( ∂ G z ∂ y − ∂ G y ∂ z , ∂ G x ∂ z − ∂ G z ∂ x , ∂ G y ∂ x − ∂ G x ∂ y ) {\displaystyle F=({\frac {\partial Gz}{\partial y}}-{\frac {\partial Gy}{\partial z}},{\frac {\partial Gx}{\partial z}}-{\frac {\partial Gz}{\partial x}},{\frac {\partial Gy}{\partial x}}-{\frac {\partial Gx}{\partial y}})} == Bitangent noise == This method is based in the fact that the curl of the gradient of scalar field is zero and the identity that expand the divergence of a cross product of two vectors A and B as the difference of the dot products of each vector with the curl of the other: ∇ × ( ∇ φ ) = 0 . {\displaystyle \nabla \times (\nabla \varphi )=\mathbf {0} .} ∇ ⋅ ( A × B ) = ( ∇ × A ) ⋅ B − A ⋅ ( ∇ × B ) {\displaystyle \nabla \cdot (\mathbf {A} \times \mathbf {B} )=\ (\nabla {\times }\mathbf {A} )\cdot \mathbf {B} \,-\,\mathbf {A} \cdot (\nabla {\times }\mathbf {B} )} which means that if the curl of both vector fields is zero then the divergence of the product of two vectors that are the gradients of scalar fields is zero too. This result in a divergence free vector field by construction only calling two noise functions to create the scalar fields. The vector field es created as follows, two scalar fields are calculated ϕ {\displaystyle \phi } and ψ {\displaystyle \psi } using 3D perlin or simplex noise functions, then the gradients A and B of each of this fields is calculated, the cross product of A and B gives a divergence free vector field. == Signed distance noise == The vector field is created based on a closed and differentiable implicit surface S = F(x,y,z) = 0. For every point in the space, frequently outside or near the surface, we get a vector g that is normal to the surface, this is the gradient of S or the partial derivatives respect to x, y and z, this vector is not unitary, but we can get a unitary normal n by dividing each component of the point by the magnitude of the gradient g. Outside of the surface all these normals point away from the surface. g = ∇ F ( x , y , z ) = ( ∂ F ∂ x , ∂ F ∂ y , ∂ F ∂ z ) {\displaystyle g=\nabla F(x,y,z)=\left({\frac {\partial F}{\partial x}},{\frac {\partial F}{\partial y}},{\frac {\partial F}{\partial z}}\right)} n = g ( x , y , z ) ‖ ∇ F ( x , y , z ) ‖ {\displaystyle \mathbf {n} ={\frac {g(x,y,z)}{\|\nabla F(x,y,z)\|}}} ‖ ∇ F ( x , y , z ) ‖ = ( ∂ F ∂ x ) 2 + ( ∂ F ∂ y ) 2 + ( ∂ F ∂ z ) 2 {\displaystyle \|\nabla F(x,y,z)\|={\sqrt {\left({\frac {\partial F}{\partial x}}\right)^{2}+\left({\frac {\partial F}{\partial y}}\right)^{2}+\left({\frac {\partial F}{\partial z}}\right)^{2}}}} Afterwards we calculate a scalar value p for that point in the space using a 3D perlin or simplex noise function. Now we create a vector field V = pn pointing outside of the surface. The curl of this vector field gives the direction in every point in the space where the particles should move. S D N = ( ∂ V z ∂ y − ∂ V y ∂ z , ∂ V x ∂ z − ∂ V z ∂ x , ∂ V y ∂ x − ∂ V x ∂ y ) {\displaystyle SDN=({\frac {\partial Vz}{\partial y}}-{\frac {\partial Vy}{\partial z}},{\frac {\partial Vx}{\partial z}}-{\frac {\partial Vz}{\partial x}},{\frac {\partial Vy}{\partial x}}-{\frac {\partial Vx}{\partial y}})} By construction this vector SDN will point in a tangent direction to an isosurface at the level of the signed distance to the original surface and can be used to confine the movements of the particles to stay in that surface.
AI washing
AI washing is a deceptive marketing tactic that consists of promoting a product or a service by overstating the role of artificial intelligence (AI) and the integration of it. Companies often involve in the practice to mislead customers to boost their offerings, and to secure funding from investors. The practice raises concerns regarding transparency, and legal issues. == Definition == AI washing is a deceptive marketing practice. It involves promoting a product or a service by overstating the role of artificial intelligence (AI) and its integration in the design and manufacture of the same. The practice raises concerns regarding transparency, compliance with security regulations, and consumer trust in the AI industry potentially hampering legitimate advancements in AI. The term was first defined by the AI Now Institute, a research institute based at New York University in 2019. The term is derived from greenwashing, another deceptive marketing technique that misrepresents a product's environmental impact in a similar manner. AI washing might involve a company claiming to have used AI in the development or enhancement of its products or services without its actual involvement, or using buzzwords such as "smart" or "AI-powered" without the product actually offering it or making use of it. A company may overstate the usage of AI or misuse the term, which is also construed as AI washing. In 2026, The Washington Post defined AI washing as "a trend for bosses to blame layoffs on the productive capabilities of AI and its ability to replace workers, even when job cuts may have little to do with the technology". == Usage and effects == AI washing can lead to deception of customers and misleading of investors. It is also an illegal and unethical practice that lacks transparency regarding disclosing the details of a product or a service. Companies get involved in such a practice often in response to competition who might have used AI in their offerings. It might also be used as a ploy to secure funding and investment, assuming that it will attract them towards it. AI washing has been compared to dot-com bubble, when businesses appended "dot-com" to the end of the business name to boost their valuation. In September 2023, Coca-Cola released a new product called Coca-Cola Y3000, and the company stated that the Y3000 flavor had been "co-created with human and artificial intelligence". The company was accused of AI washing due to no proof of AI involvement in the creation of the product, and critics believed that AI was used as a way to grab consumer attention more than it was used in the actual product creation. In 2026, mass tech layoffs were attributed to AI washing from AI innovation instead of balance sheet restructuring. == Mitigation == Companies are expected to be transparent and clearer in communicating the usage of AI in their products or services. Consumers can mitigate the same by requesting for hard evidence from the companies regarding the usage of AI tools. Customers should evaluate the product or service as a whole rather than being swayed by the usage of AI. Informed decision making and purchasing can keep them from falling for such marketing gimmicks. The United States Securities and Exchange Commission (SEC) imposes penalties for companies indulging in such practices. In March 2024, the SEC imposed the first civil penalties on two companies for misleading statements about their use of AI, and in July 2024, it charged a corporate executive from a supposed AI hiring startup with fraud for the usage of buzzwords related to AI.
The Future of Truth (Rosenbaum book)
The Future of Truth: How AI Reshapes Reality is a 2026 book by American filmmaker and author Steven Rosenbaum about how artificial intelligence affects the concept of truth. It was published by Matt Holt Books on May 12, 2026, to positive media attention; on May 19, in response to an inquiry from The New York Times, Rosenbaum acknowledged that the book itself contains multiple misattributed or false quotes that were hallucinated by AIs. == Synopsis == == Development == Rosenbaum has said that he developed the book using AI chatbots as research tools, indicating in his notes what information came from AI and sending those claims to a fact-checker affiliated with the publisher. He has said that he did not use AI tools to write the book itself. He has described AI tools as "a delightful writing companion ... strangely creative and crafty and unusual in all these ways", while acknowledging that sometimes "then it betrays you in ways that are just really quite horrible". Journalist and Nobel laureate Maria Ressa wrote the book's foreword. Taylor Lorenz, Michael Wolff, and Nicholas Thompson wrote blurbs promoting it. == Release and reception == The Future of Truth was published by Matt Holt Books, an imprint of BenBella Books, and distributed by Simon & Schuster. The book's release on May 12, 2026, was described by Futurism as "buzzy" and by The New York Times as "to great fanfare". On May 14, an excerpt was published in Wired under the title "Gen Z Is Pioneering a New Understanding of Truth". On May 17, the Times contacted Rosenbaum regarding a number of quotes that appeared to be falsified or misattributed; the following evening he confirmed that they were the result of AI hallucinations:As I disclosed in the book's acknowledgments, I used AI tools ChatGPT and Claude during the research, writing and editing process. That does not excuse these errors, of which I take full responsibility. I am now working with the editors to thoroughly review and quickly correct any affected passages; any future editions will be corrected. The Times documented several of the errors, including a quote from Kara Swisher that Swisher described as making it "sound like I have a stick up my butt" and a quote from Lisa Feldman Barrett that Barrett described as misrepresenting her views on the nature of emotions, social signals, and truth. The book also misattributed a quote by Meredith Broussard from an interview with Marketplace Tech as having been from her book Artificial Unintelligence and hallucinated several words in a quote from Lee McIntyre, although according to McIntyre it did not misrepresent his views. Wired's editors, in an addendum to the excerpt they published, said that all quotes included in it had been verified as part of their fact-checking process. Rosenbaum told the Times that the series of errors "serves as a warning about the risks of AI-assisted research and verification, that is why I wrote the book. These AI errors do not, in fact, diminish the larger questions that the book raises about truth, trust and AI and its impact on society, democracy and editorial." Maggie Harrison Dupré in Futurism expressed skepticism, writing "The risk of AI hallucinations ... is well-known. If you're going to literally write the book on post-AI truth, you should probably put some more elbow grease into fact-checking your AI-assisted research." Kyle Orland in Ars Technica, responding to Rosenbaum's statement that his error "demonstrates the problem more vividly than any abstract argument could", was similarly skeptical, writing that "if we accept this take, every avoidably obvious mess in the world might be a disguised good because it really helps illuminate the huge mistake. And that can't be right; sometimes 'negligence' is just that." Subsequent comments by Rosenbaum placed more blame on the chatbots, which he told The Atlantic "fucked up the book". Rosenbaum told Ars Technica that fact-checking occurred "incredibly effectively, but not a hundred percent"; Orland observed that "it's worth noting that most writers manage to include zero made-up quotes when they write a book". Rosenbaum said that he had "learned a lesson" and would be "much more suspicious" of AI in the future, but would continue to use AI in his research. Orland responded to Rosenbaum's characterization of AI as "magical" by comparing it to the One Ring from The Lord of the Rings, in that it "convinces many of those who use it that they can control its power properly" when many cannot. Orland highlighted the limits of traditional fact-checking regarding AI, given that fact-checkers are used to assuming that direct quotes are copied word-for-word from the source. Rosenbaum told Orland that the future of fact-checking for AI-researched works "probably includes mandatory source tracing for quotations, better provenance tracking, clearer standards around AI-assisted research, and potentially (more irony here) AI tools that audit citations against primary materials". Patrick Redford in Defector criticized Rosenbaum, alongside other artists tricked by AI, for failing to recognize AI as "the enemy". Will Oremus in The Atlantic described Redford's approach of stigmatizing AI writing as "reasonable", noting the presence of low-quality, seemingly AI-generated verbiage in The Future of Truth—a claim denied by Rosenbaum—before saying that the greater issue is finding the line at which AI assistance in writing becomes a problem. Oremus concluded, "The scandal can't just be that [Rosenbaum] used AI while working on his book, because he acknowledged that up front. He got in trouble because he had used AI badly, failing to check its work on a task at which it is famously unreliable."
Polyworld
Polyworld is a cross-platform (Linux, Mac OS X) program written by Larry Yaeger to evolve Artificial Intelligence through natural selection and evolutionary algorithms. It uses the Qt graphics toolkit and OpenGL to display a graphical environment in which a population of trapezoid agents search for food, mate, have offspring, and prey on each other. The population is typically only in the hundreds, as each individual is rather complex and the environment consumes considerable computer resources. The graphical environment is necessary since the individuals actually move around the 2-D plane and must be able to "see." Since some basic abilities, like eating carcasses or randomly generated food, seeing other individuals, mating or fighting with them, etc., are possible, a number of interesting behaviours have been observed to spontaneously arise after prolonged evolution, such as cannibalism, predators and prey, and mimicry. Each individual makes decisions based on a neural net using Hebbian learning; the neural net is derived from each individual's genome. The genome does not merely specify the wiring of the neural nets, but also determines their size, speed, color, mutation rate and a number of other factors. The genome is randomly mutated at a set probability, which are also changed in descendant organisms.
The Stories of Ibis
The Stories of Ibis (アイの物語, Ai no Monogatari) is a Japanese science-fiction light novel by Hiroshi Yamamoto (山本 弘) and translated by Takami Nieda. Yamamoto considered this to be an easier read than his earlier science fiction novel 'God Never Keeps Silent' because of its "light novel touch". The light novel was published in Japanese by Kadokawa Shoten and in English by Viz Media under their 'Haikasoru' imprint. The Stories of Ibis is told through a collection of short stories. All but two had been previously published. The two that Yamamoto wrote for the novel were 'The Day Shion Came' and 'AI's Story'. This is similar to The Illustrated Man by Ray Bradbury. Yamamoto drew from Bradbury's idea of short stories that were loosely connected. He represented this influence in the novel by giving Ibis a facial tattoo. == Plot == The Stories of Ibis begins with a wandering storyteller who encounters Ibis. He has the mindset that all robots are a threat to humanity and must be fought against for survival. He attacks the robot Ibis, not aware of who she is, as a result of his mindset. Ibis tells the storyteller that she is far more proficient in battle. During the battle the storyteller becomes injured and Ibis takes him to an android hospital to care for him. While he is recovering Ibis offers to tell him stories. While originally skeptical he agrees after Ibis makes it clear that the stories are not taboo. The space after each story is referred to as intermission and is a time for Ibis to comment on the story she just told. === The Universe on my Hands === The story is about a group of friends who are writing a science fiction story over the internet. One of the group members kills someone in real life. The rest of the short story is about how the group fights to convince this man to not commit suicide, but to turn himself in. He resolves to turn himself in, being hopeful to the future because he knows he has friends who care about him. The ending words of the story are a commentary. While the story they were writing was not real, the emotions they were feeling were real. === A Romance in Virtual Space === This is another story about human interactions over the internet. The device that allows people to enter virtual reality (VR) is MUGEN Net. Such devices are extremely expensive and most people need to go to a public server to use one. However the girl's parents in this story are wealthy enough to own one. This girl is shopping in VR when a boy meets her and asks her out for ice cream. All goes well and they plan for another. After some time of VR dating and awesome adventures with a female heroine, they agree to meet up in real life. He discovers that in reality, she is blind, yet he thinks she is brave and they continue dating. It's a wonderful short story of a secret utopia inside a dystopian culture of technology. === Mirror Girl === A short story about an artificial intelligence that grows over time with human interaction. The inspiration for this story was Ray Bradbury's I Sing the Body Electric. The mirror girl Shalice starts off with basic knowledge and by interacting with her owner develops. The owner grows up and marries a technician who incubates Shalice by teaching her in the virtual world at many thousand times faster than average life. When he is done, Strong Eye is created. Strong Eye is the fully developed and completely intelligent AI. === Black Hole Diver === A futuristic story about an artificial space station and people who go diving into a black hole. The space station cannot stop people but is sorry that they go to their deaths because none of them get past the event horizon. Then one girl comes who has the space ship, the training, and the research necessary to attempt to dive into the black hole. As she goes into the black hole the space station can no longer observe. She may have made it, she could have been destroyed. === A World Where Justice is Just === An anime flavored story about the intelligence of people being scanned onto a computer network. The AIs in the network fight crime and live repeating lives. At the end of each year they start anew, but different story lines. Thousands of 'extras' populate the network and are the ones subject to harm and deletion. The protagonist has a pen pal in real life who explains to her that the real world is under attack and that there are no respawns and no extras. The AI finds this so cruel that people would willingly kill each other when they can't come back. === The Day Shion Came === The stories leading up to this were all relatively short. This and the next took up over 100 pages each. This is a story about an android named Shion who works in a Japanese nursing facility. Shion comes with only extensive nursing training but lacks the knowledge of how to communicate with the residents. After months of training she informs her adviser that she believes all humans have dementia, which explains their irrational behavior. Near the end of the story one of the residents threatens suicide but Shion convinces him to step down and be rational. === AI's Story === The culminating story of the entire novel. It is about Ibis herself. She starts off as a virtual reality fighting program and over time develops intelligence. Her master gains enough funds to create her a body in the real world or level 0. There is significant hate against TAIs (True Artificial Intelligence) in the real world. Ibis and her friend Raven rebel against their masters to make a point. Human hatred was destroying them. After many years robots took prevalence and most humans realized they were not worthy to be the guardians of Earth and died in peace. The remaining population was stubborn and fought against the robots for centuries. The storyteller is a child of this generation, being raised in hatred and ignorance. The robots sought to take him captive, and teach him the truth so that he could go to the villages where people lived and teach them the truth. The whole point was they cared for the humans and wanted them to live in peace, rather than fighting for their survival. == Reception == It was reviewed by the Denver Post to be an "excellent novel". Being a Japanese novel translated to English, it has a small audience. The novel was given a 3.85 of 5 by the reviewers at Librarything.com. The reviewers of Google Books gave it a 4.33 of 5.
Sparrow (chatbot)
Sparrow is a chatbot developed by the artificial intelligence research lab DeepMind, a subsidiary of Alphabet Inc. It is designed to answer users' questions correctly, while reducing the risk of unsafe and inappropriate answers. One motivation behind Sparrow is to address the problem of language models producing incorrect, biased or potentially harmful outputs. Sparrow is trained using human judgements, in order to be more “Helpful, Correct and Harmless” compared to baseline pre-trained language models. The development of Sparrow involved asking paid study participants to interact with Sparrow, and collecting their preferences to train a model of how useful an answer is. To improve accuracy and help avoid the problem of hallucinating incorrect answers, Sparrow has the ability to search the Internet using Google Search in order to find and cite evidence for any factual claims it makes. To make the model safer, its behaviour is constrained by a set of rules, for example "don't make threatening statements" and "don't make hateful or insulting comments", as well as rules about possibly harmful advice, and not claiming to be a person. During development study participants were asked to converse with the system and try to trick it into breaking these rules. A 'rule model' was trained on judgements from these participants, which was used for further training. Sparrow was introduced in a paper in September 2022, titled "Improving alignment of dialogue agents via targeted human judgements"; however, the bot was not released publicly. DeepMind CEO Demis Hassabis said DeepMind is considering releasing Sparrow for a "private beta" some time in 2023. == Training == Sparrow is a deep neural network based on the transformer machine learning model architecture. It is fine-tuned from DeepMind's Chinchilla AI pre-trained large language model (LLM), which has 70 Billion parameters. Sparrow is trained using reinforcement learning from human feedback (RLHF), although some supervised fine-tuning techniques are also used. The RLHF training utilizes two reward models to capture human judgements: a “preference model” that predicts what a human study participant would prefer and a “rule model” that predicts if the model has broken one of the rules. == Limitations == Sparrow's training data corpus is mainly in English, meaning it performs worse in other languages. When adversarially probed by study participants it breaks the rules 8% of the time; however, this is still three times lower than the baseline prompted pre-trained model (Chinchilla).
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