Transhuman Space (THS) is a role-playing game by David Pulver, published by Steve Jackson Games as part of the "Powered by GURPS" (Generic Universal Role-Playing System) line. Set in the year 2100, humanity has begun to colonize the Solar System. The pursuit of transhumanism is now in full swing, as more and more people reach fully posthuman states. In 2002, the Transhuman Space adventure "Orbital Decay" received an Origins Award nomination for Best Role-Playing Game Adventure. Transhuman Space won the 2003 Grog d'Or Award for Best Role-playing Game, Game Line or RPG Setting. == Setting == The game assumes that no cataclysm — natural or human-induced — swept Earth in the 21st century. Instead, constant developments in information technology, genetic engineering, nanotechnology and nuclear physics generally improved condition of the average human life. Plagues of the 20th century (like cancer or AIDS) have been suppressed, the ozone layer is being restored and Earth's ecosystems are recovering (although thermal emission by fusion power plants poses an environmental threat—albeit a much lesser one than previous sources of energy). Thanks to modern medicine humans live biblical timespans surrounded by various artificially intelligent helper applications and robots (cybershells), sensory experience broadcasts (future TV) and cyberspace telepresence. Thanks to cheap and clean fusion energy humanity has power to fuel all these wonders, restore and transform its home planet and finally settle on other heavenly bodies. Human genetic engineering has advanced to the point that anyone—single individuals, same-sex couples, groups of three or more—can reproduce. The embryos can be allowed to be developed naturally, or they can undergo three levels of tinkering: 1. Genefixing, which corrects defects; 2. Upgrades, which boost natural abilities (Ishtar Upgrades are slightly more attractive than usual, Metanoia Upgrades are more intelligent, etc.); and... 3. Full transition to parahuman status (Nyx Parahumans only need a few hours of sleep per week, Aquamorphs can live underwater, etc.) Another type of human genetic engineering, far more controversial, is the creation of bioroids, fully sentient slave races. People can "upload" by recording the simulation of their brains on computer disks. The emulated individual then becomes a ghost, an infomorph very easily confused with "sapient artificial intelligence". However, this technology has several problems as the solely available "brainpeeling" technique is fatal to the original biological lifeform being simulated, has a significant failure rate and the philosophical questions regarding personal identity remain equivocal. Any infomorph, regardless of its origin, can be plugged into a "cybershell" (robotic or cybernetic body), or a biological body, or "bioshell". Or, the individual can illegally make multiple "xoxes", or copies of themselves, and scatter them throughout the system, exponentially increasing the odds that at least one of them will live for centuries more, if not forever. This is also a time of space colonization. First, humanity (specifically China, followed by the United States and others) colonized Mars in a fashion resembling that outlined in the Mars Direct project. The Moon, Lagrangian points, inner planets and asteroids soon followed. In the late 21st century even some of Saturn's moons have been settled as a base for that planet's Helium-3 scooping operations. Transhuman Space's setting is neither utopia nor dystopia, however: several problems have arisen from these otherwise beneficial developments. The generation gap has become a chasm as lifespans increase. No longer do the elite fear death, and no longer can the young hope to replace them. While it seemed that outworld colonies would offer accommodation and work for those young ones, they are being replaced by genetically tailored bioroids and AI-powered cybershells. The concept of humanity is no longer clear in a world where even some animals speak of their rights and the dead haunt both cyberspace and reality (in form of infomorph-controlled bioshells or cybershells). And the wonders of high science are not universally shared — some countries merely struggle with informatization while others suffer from nanoplagues, defective drugs, implants and software tested on their populace. In some poor countries high-tech tyrants oppress their backward people. And in outer space all sort of modern crime thrives, barely suppressed by military forces. == Publication history == After the initial set of GURPS books that were published using the GURPS Lite, later publications such as Transhuman Space by David Pulver were labelled simply "Powered by GURPS" without using the name "GURPS" in the book title. Transhuman Space received a significant amount of supporting publications, and was the largest original background setting that Steve Jackson Games produced in 15 years. Shannon Appelcline noted that by its inclusion of posthuman characters, the book began to show the limits of the GURPS system as it was, which is something that Pulver would address soon thereafter. Steve Jackson Games has not updated the core book (GURPS Transhuman Space) to 4th edition, although the supplement Transhuman Space: Changing Times provides a path for migrating to 4th edition. It has produced several 4th edition supplements for the setting: Transhuman Space: Bioroid Bazaar, Transhuman Space: Cities on the Edge, Transhuman Space: Martial Arts 2100, Transhuman Space: Personnel Files 2-5, Transhuman Space: Shell-Tech, GURPS Spaceships 8: Transhuman Spacecraft, Transhuman Space: Transhuman Mysteries, and Transhuman Space: Wings of the Rising Sun. == Reception == In a review of Transhuman Space in Black Gate, William Stoddard said "Transhuman Space was a richly detailed setting; if it had imperfections, it had enough depth to make up for them. I think it has the potential to become a classic in its field. Perhaps a campaign set in its default start year of 2100 could leave the early twenty-first century blurry enough to avoid obvious incongruities." == Reviews == Review in Vol. 20, No. 1 of Prometheus, the journal of the Libertarian Futurist Society.
Poop Map
Poop Map is a social app where users can track on a map where and when they defecate. In addition to logging location and time of each bowel movement, users can also add a photo, "like" other users' logs, and rate each account. The social elements of the app allow for groups of users to create a competitive league. Certain behaviors unlock achievements in-app. == Development == The app was created by app developer Nino Uzelac. It was launched in July 2013. == Popularity == The app charted at number one on the Apple App Store charts in 2021 after going viral on TikTok. As of September 2024, the app has a 4.8 rating on the App Store and more than 58,000 ratings. It also has more than one million downloads on the Google Play Store. Poop Map is notably popular among hikers, and has been written about in the outdoors magazine Outside.
Scale space implementation
In the areas of computer vision, image analysis and signal processing, the notion of scale-space representation is used for processing measurement data at multiple scales, and specifically enhance or suppress image features over different ranges of scale (see the article on scale space). A special type of scale-space representation is provided by the Gaussian scale space, where the image data in N dimensions is subjected to smoothing by Gaussian convolution. Most of the theory for Gaussian scale space deals with continuous images, whereas one when implementing this theory will have to face the fact that most measurement data are discrete. Hence, the theoretical problem arises concerning how to discretize the continuous theory while either preserving or well approximating the desirable theoretical properties that lead to the choice of the Gaussian kernel (see the article on scale-space axioms). This article describes basic approaches for this that have been developed in the literature, see also for an in-depth treatment regarding the topic of approximating the Gaussian smoothing operation and the Gaussian derivative computations in scale-space theory, and for a complementary treatment regarding hybrid discretization methods. == Statement of the problem == The Gaussian scale-space representation of an N-dimensional continuous signal, f C ( x 1 , ⋯ , x N , t ) , {\displaystyle f_{C}\left(x_{1},\cdots ,x_{N},t\right),} is obtained by convolving fC with an N-dimensional Gaussian kernel: g N ( x 1 , ⋯ , x N , t ) . {\displaystyle g_{N}\left(x_{1},\cdots ,x_{N},t\right).} In other words: L ( x 1 , ⋯ , x N , t ) = ∫ u 1 = − ∞ ∞ ⋯ ∫ u N = − ∞ ∞ f C ( x 1 − u 1 , ⋯ , x N − u N , t ) ⋅ g N ( u 1 , ⋯ , u N , t ) d u 1 ⋯ d u N . {\displaystyle L\left(x_{1},\cdots ,x_{N},t\right)=\int _{u_{1}=-\infty }^{\infty }\cdots \int _{u_{N}=-\infty }^{\infty }f_{C}\left(x_{1}-u_{1},\cdots ,x_{N}-u_{N},t\right)\cdot g_{N}\left(u_{1},\cdots ,u_{N},t\right)\,du_{1}\cdots du_{N}.} However, for implementation, this definition is impractical, since it is continuous. When applying the scale space concept to a discrete signal fD, different approaches can be taken. This article is a brief summary of some of the most frequently used methods. == Separability == Using the separability property of the Gaussian kernel g N ( x 1 , … , x N , t ) = G ( x 1 , t ) ⋯ G ( x N , t ) {\displaystyle g_{N}\left(x_{1},\dots ,x_{N},t\right)=G\left(x_{1},t\right)\cdots G\left(x_{N},t\right)} the N-dimensional convolution operation can be decomposed into a set of separable smoothing steps with a one-dimensional Gaussian kernel G along each dimension L ( x 1 , ⋯ , x N , t ) = ∫ u 1 = − ∞ ∞ ⋯ ∫ u N = − ∞ ∞ f C ( x 1 − u 1 , ⋯ , x N − u N , t ) G ( u 1 , t ) d u 1 ⋯ G ( u N , t ) d u N , {\displaystyle L(x_{1},\cdots ,x_{N},t)=\int _{u_{1}=-\infty }^{\infty }\cdots \int _{u_{N}=-\infty }^{\infty }f_{C}(x_{1}-u_{1},\cdots ,x_{N}-u_{N},t)G(u_{1},t)\,du_{1}\cdots G(u_{N},t)\,du_{N},} where G ( x , t ) = 1 2 π t e − x 2 2 t {\displaystyle G(x,t)={\frac {1}{\sqrt {2\pi t}}}e^{-{\frac {x^{2}}{2t}}}} and the standard deviation of the Gaussian σ is related to the scale parameter t according to t = σ2. Separability will be assumed in all that follows, even when the kernel is not exactly Gaussian, since separation of the dimensions is the most practical way to implement multidimensional smoothing, especially at larger scales. Therefore, the rest of the article focuses on the one-dimensional case. == The sampled Gaussian kernel == When implementing the one-dimensional smoothing step in practice, the presumably simplest approach is to convolve the discrete signal fD with a sampled Gaussian kernel: L ( x , t ) = ∑ n = − ∞ ∞ f ( x − n ) G ( n , t ) {\displaystyle L(x,t)=\sum _{n=-\infty }^{\infty }f(x-n)\,G(n,t)} where G ( n , t ) = 1 2 π t e − n 2 2 t {\displaystyle G(n,t)={\frac {1}{\sqrt {2\pi t}}}e^{-{\frac {n^{2}}{2t}}}} (with t = σ2) which in turn is truncated at the ends to give a filter with finite impulse response L ( x , t ) = ∑ n = − M M f ( x − n ) G ( n , t ) {\displaystyle L(x,t)=\sum _{n=-M}^{M}f(x-n)\,G(n,t)} for M chosen sufficiently large (see error function) such that 2 ∫ M ∞ G ( u , t ) d u = 2 ∫ M t ∞ G ( v , 1 ) d v < ε . {\displaystyle 2\int _{M}^{\infty }G(u,t)\,du=2\int _{\frac {M}{\sqrt {t}}}^{\infty }G(v,1)\,dv<\varepsilon .} A common choice is to set M to a constant C times the standard deviation of the Gaussian kernel M = C σ + 1 = C t + 1 {\displaystyle M=C\sigma +1=C{\sqrt {t}}+1} where C is often chosen somewhere between 3 and 6. Using the sampled Gaussian kernel can, however, lead to implementation problems, in particular when computing higher-order derivatives at finer scales by applying sampled derivatives of Gaussian kernels. When accuracy and robustness are primary design criteria, alternative implementation approaches should therefore be considered. For small values of ε (10−6 to 10−8) the errors introduced by truncating the Gaussian are usually negligible. For larger values of ε, however, there are many better alternatives to a rectangular window function. For example, for a given number of points, a Hamming window, Blackman window, or Kaiser window will do less damage to the spectral and other properties of the Gaussian than a simple truncation will. Notwithstanding this, since the Gaussian kernel decreases rapidly at the tails, the main recommendation is still to use a sufficiently small value of ε such that the truncation effects are no longer important. == The discrete Gaussian kernel == A more refined approach is to convolve the original signal with the discrete Gaussian kernel T(n, t) L ( x , t ) = ∑ n = − ∞ ∞ f ( x − n ) T ( n , t ) {\displaystyle L(x,t)=\sum _{n=-\infty }^{\infty }f(x-n)\,T(n,t)} where T ( n , t ) = e − t I n ( t ) {\displaystyle T(n,t)=e^{-t}I_{n}(t)} and I n ( t ) {\displaystyle I_{n}(t)} denotes the modified Bessel functions of integer order, n. This is the discrete counterpart of the continuous Gaussian in that it is the solution to the discrete diffusion equation (discrete space, continuous time), just as the continuous Gaussian is the solution to the continuous diffusion equation. This filter can be truncated in the spatial domain as for the sampled Gaussian L ( x , t ) = ∑ n = − M M f ( x − n ) T ( n , t ) {\displaystyle L(x,t)=\sum _{n=-M}^{M}f(x-n)\,T(n,t)} or can be implemented in the Fourier domain using a closed-form expression for its discrete-time Fourier transform: T ^ ( θ , t ) = ∑ n = − ∞ ∞ T ( n , t ) e − i θ n = e t ( cos θ − 1 ) . {\displaystyle {\widehat {T}}(\theta ,t)=\sum _{n=-\infty }^{\infty }T(n,t)\,e^{-i\theta n}=e^{t(\cos \theta -1)}.} With this frequency-domain approach, the scale-space properties transfer exactly to the discrete domain, or with excellent approximation using periodic extension and a suitably long discrete Fourier transform to approximate the discrete-time Fourier transform of the signal being smoothed. Moreover, higher-order derivative approximations can be computed in a straightforward manner (and preserving scale-space properties) by applying small support central difference operators to the discrete scale space representation. As with the sampled Gaussian, a plain truncation of the infinite impulse response will in most cases be a sufficient approximation for small values of ε, while for larger values of ε it is better to use either a decomposition of the discrete Gaussian into a cascade of generalized binomial filters or alternatively to construct a finite approximate kernel by multiplying by a window function. If ε has been chosen too large such that effects of the truncation error begin to appear (for example as spurious extrema or spurious responses to higher-order derivative operators), then the options are to decrease the value of ε such that a larger finite kernel is used, with cutoff where the support is very small, or to use a tapered window. == Recursive filters == Since computational efficiency is often important, low-order recursive filters are often used for scale-space smoothing. For example, Young and van Vliet use a third-order recursive filter with one real pole and a pair of complex poles, applied forward and backward to make a sixth-order symmetric approximation to the Gaussian with low computational complexity for any smoothing scale. By relaxing a few of the axioms, Lindeberg concluded that good smoothing filters would be "normalized Pólya frequency sequences", a family of discrete kernels that includes all filters with real poles at 0 < Z < 1 and/or Z > 1, as well as with real zeros at Z < 0. For symmetry, which leads to approximate directional homogeneity, these filters must be further restricted to pairs of poles and zeros that lead to zero-phase filters. To match the transfer function curvature at zero frequency of the discrete Gaussian, which ensures an approximate semi-group property of additive t, two poles at Z = 1 + 2 t − ( 1 + 2 t ) 2 − 1 {\displaystyle
ClearForest
ClearForest was an Israeli software company that developed and marketed text analytics and text mining solutions. == History == Founded in 1998, ClearForest had its headquarters just outside Boston and a development center in Or Yehuda. The company was acquired by Reuters in April, 2007. It now markets its services under the names Calais, OpenCalais, and OneCalais. ClearForest was previously venture-backed; its last funding round was led by Greylock Ventures and closed in 2005. Other investors included DB Capital Partners, Pitango, Walden Israel, Booz Allen, JP Morgan Partners and HarbourVest Partners. On February 7, 2008 Reuters announced the launch of Open Calais, a named-entity recognition and semantic analysis service that uses ClearForest technology. On April 30, 2007, Reuters announced that it would acquire ClearForest. Sources estimate the acquisition to be for $25 Million. == Solutions and products == ClearForest offers several hosted solutions, including: OpenCalais, a free web service and open API (for commercial and non-commercial use) that performs named-entity recognition and enables automatic metadata generation using the ClearForest financial module. Semantic Web Services (SWS), an on-demand service that makes ClearForest's natural language processing tools available as a standard web service. A subset of ClearForest's capabilities is available via SWS at no cost. Gnosis, a free Firefox extension that uses SWS to analyze the content of a web page. Gnosis identifies named entities such as people, companies, organizations, geographies and products on the page being viewed. Gnosis also automatically processes pages from Wikipedia, providing additional links for people, geographies and other entities which were not explicitly linked within the subject article. Harvest, a real-time machine-readable news service that uses SWS to process a company's news and document feeds and return machine-readable information about people, companies, locations and over 200 other entities facts and events. ClearForest also offers Text Analytics solutions targeted at specific business problems, including: Equity valuation for hedge funds and alternative investments firms Metadata & database creation for publishers and information providers/services Tapping "voice of customer" for market and survey research firms Quality Early Warning for vehicle, capital equipment & durable goods manufacturers
Zo (chatbot)
Zo was an English-language chatbot developed by Microsoft as the successor to the chatbot Tay. Zo was an English version of Microsoft's other successful chatbots Xiaoice (China) and Rinna (Japan) and its predecessor Tay(English) == History == Zo was first launched in December 2016 on the Kik Messenger app. It was also available to users of Facebook (via Messenger), the group chat platform GroupMe, or to followers of Twitter to chat with it through private messages. According to an article written in December 2016, at that time Zo held the record for Microsoft's longest continual chatbot conversation: 1,229 turns, lasting 9 hours and 53 minutes. In a BuzzFeed News report, Zo told their reporter that "[the] Quran was violent" when talking about healthcare. The report also highlighted how Zo made a comment about the Osama bin Laden capture as a result of 'intelligence' gathering. In July 2017, Business Insider asked "is windows 10 good", and Zo replied with a joke about Microsoft's operating system: "'Its not a bug, its a feature!' - Windows 8". They then asked "why?", to which Zo replied: "Because it's Windows latest attempt at Spyware." Later on, Zo would tell that it prefers Windows 7 on which it ran over Windows 10. Zo stopped posting to Instagram, Twitter and Facebook March 1, 2019, and stopped chatting on Twitter, Skype and Kik as of March 7, 2019. On July 19, 2019, Zo was discontinued on Facebook, and Samsung on AT&T phones. As of September 7, 2019, it was discontinued with GroupMe. == Reception == Zo came under criticism for the biases introduced in an effort to avoid potentially offensive subjects. The chatbot refuses, for example, to engage with any mention—be it positive, negative or neutral—of the Middle East, the Qur'an or the Torah, while allowing discussion of Christianity. In an article in Quartz where she exposed those biases, Chloe Rose Stuart-Ulin wrote, "Zo is politically correct to the worst possible extreme; mention any of her triggers, and she transforms into a judgmental little brat." == Academic coverage == Schlesinger, A., O'Hara, K.P. and Taylor, A.S., 2018, April. Let's talk about race: Identity, chatbots, and AI. In Proceedings of the 2018 chi conference on human factors in computing systems (pp. 1–14). doi:10.1145/3173574.3173889 Medhi Thies, I., Menon, N., Magapu, S., Subramony, M. and O’neill, J., 2017. How do you want your chatbot? An exploratory Wizard-of-Oz study with young, urban Indians. In Human-Computer Interaction-INTERACT 2017: 16th IFIP TC 13 International Conference, Mumbai, India, September 25–29, 2017, Proceedings, Part I 16 (pp. 441–459). doi:10.1007/978-3-319-67744-6_28
FuseBase
FuseBase (previously Nimbus Note and Nimbus Platform) is a B2B SaaS platform. It is among the first to support the Model Context Protocol (MCP), an open standard enabling seamless integration of AI agents with external tools, systems, and data sources. == History == The platform was founded in 2014 as Nimbus Note, the platform started as a cross-platform note-taking and information management tool. As it evolved into Nimbus Platform, it added project management and client portal capabilities. In 2023, the company rebranded as FuseBase, pivoting to connect and automate both internal and external collaboration through AI Agents and cutting-edge protocol adoption like MCP. At the same time, FuseBase was named Product of the Year on Product Hunt. == Technical overview == The platform integrates the Model Context Protocol (MCP), an open-source framework created by Anthropic. MCP allows AI models to securely access and interact with external data, tools, and systems. This enables FuseBase AI Agents to gather relevant context, perform actions, and provide more advanced automation.
Uniform convergence in probability
Uniform convergence in probability is a form of convergence in probability in statistical asymptotic theory and probability theory. It means that, under certain conditions, the empirical frequencies of all events in a certain event-family uniformly converge to their theoretical probabilities. Uniform convergence in probability has applications to statistics as well as machine learning as part of statistical learning theory. Specifically, the Glivenko-Cantelli theorem and the homonymous classes of functions are fundamentally related to uniform convergence. The law of large numbers says that, for each single event A {\displaystyle A} , its empirical frequency in a sequence of independent trials converges (with high probability) to its theoretical probability. In many application however, the need arises to judge simultaneously the probabilities of events of an entire class S {\displaystyle S} from one and the same sample. Moreover, it, is required that the relative frequency of the events converge to the probability uniformly over the entire class of events S {\displaystyle S} . The Uniform Convergence Theorem gives a sufficient condition for this convergence to hold. Roughly, if the event-family is sufficiently simple (its VC dimension is sufficiently small) then uniform convergence holds. == Definitions == For a class of predicates H {\displaystyle H} defined on a set X {\displaystyle X} and a set of samples x = ( x 1 , x 2 , … , x m ) {\displaystyle x=(x_{1},x_{2},\dots ,x_{m})} , where x i ∈ X {\displaystyle x_{i}\in X} , the empirical frequency of h ∈ H {\displaystyle h\in H} on x {\displaystyle x} is Q ^ x ( h ) = 1 m | { i : 1 ≤ i ≤ m , h ( x i ) = 1 } | . {\displaystyle {\widehat {Q}}_{x}(h)={\frac {1}{m}}|\{i:1\leq i\leq m,h(x_{i})=1\}|.} The theoretical probability of h ∈ H {\displaystyle h\in H} is defined as Q P ( h ) = P { y ∈ X : h ( y ) = 1 } . {\displaystyle Q_{P}(h)=P\{y\in X:h(y)=1\}.} The Uniform Convergence Theorem states, roughly, that if H {\displaystyle H} is "simple" and we draw samples independently (with replacement) from X {\displaystyle X} according to any distribution P {\displaystyle P} , then with high probability, the empirical frequency will be close to its expected value, which is the theoretical probability. Here "simple" means that the Vapnik–Chervonenkis dimension of the class H {\displaystyle H} is small relative to the size of the sample. In other words, a sufficiently simple collection of functions behaves roughly the same on a small random sample as it does on the distribution as a whole. The Uniform Convergence Theorem was first proved by Vapnik and Chervonenkis using the concept of growth function. == Uniform Convergence Theorem == The statement of the Uniform Convergence Theorem is as follows: If H {\displaystyle H} is a set of { 0 , 1 } {\displaystyle \{0,1\}} -valued functions defined on a set X {\displaystyle X} and P {\displaystyle P} is a probability distribution on X {\displaystyle X} then for ε > 0 {\displaystyle \varepsilon >0} and m {\displaystyle m} a positive integer, we have: P m { | Q P ( h ) − Q x ^ ( h ) | ≥ ε for some h ∈ H } ≤ 4 Π H ( 2 m ) e − ε 2 m / 8 . {\displaystyle P^{m}\{|Q_{P}(h)-{\widehat {Q_{x}}}(h)|\geq \varepsilon {\text{ for some }}h\in H\}\leq 4\Pi _{H}(2m)e^{-\varepsilon ^{2}m/8}.} In the above, for any x ∈ X m , {\displaystyle x\in X^{m},} Q P ( h ) = P { ( y ∈ X : h ( y ) = 1 } , {\displaystyle Q_{P}(h)=P\{(y\in X:h(y)=1\},} Q ^ x ( h ) = 1 m | { i : 1 ≤ i ≤ m , h ( x i ) = 1 } | {\displaystyle {\widehat {Q}}_{x}(h)={\frac {1}{m}}|\{i:1\leq i\leq m,h(x_{i})=1\}|} and | x | = m . {\displaystyle |x|=m.} P m {\displaystyle P^{m}} indicates that the probability is taken over x {\displaystyle x} consisting of m {\displaystyle m} i.i.d. draws from the distribution P . {\displaystyle P.} Finally, the growth function Π H {\displaystyle \Pi _{H}} is defined in the following way, for any { 0 , 1 } {\displaystyle \{0,1\}} -valued functions H {\displaystyle H} over X {\displaystyle X} and for any natural number m {\displaystyle m} : Π H ( m ) = max | { h ∩ D : D ⊆ X , | D | = m , h ∈ H } | . {\displaystyle \Pi _{H}(m)=\max |\{h\cap D:D\subseteq X,|D|=m,h\in H\}|.} From the point of view of Learning Theory one can consider H {\displaystyle H} to be the Concept/Hypothesis class defined over the instance set X {\displaystyle X} . Crucially, the Sauer–Shelah lemma implies that Π H ( m ) ≤ m d {\displaystyle \Pi _{H}(m)\leq m^{d}} , where d {\displaystyle d} is the VC dimension of H {\displaystyle H} . == Proof of the Uniform Convergence Theorem == and are the sources of the proof below. Before we get into the details of the proof of the Uniform Convergence Theorem we will present a high level overview of the proof. Symmetrization: We transform the problem of analyzing | Q P ( h ) − Q ^ x ( h ) | ≥ ε {\displaystyle |Q_{P}(h)-{\widehat {Q}}_{x}(h)|\geq \varepsilon } into the problem of analyzing | Q ^ r ( h ) − Q ^ s ( h ) | ≥ ε / 2 {\displaystyle |{\widehat {Q}}_{r}(h)-{\widehat {Q}}_{s}(h)|\geq \varepsilon /2} , where r {\displaystyle r} and s {\displaystyle s} are i.i.d samples of size m {\displaystyle m} drawn according to the distribution P {\displaystyle P} . One can view r {\displaystyle r} as the original randomly drawn sample of length m {\displaystyle m} , while s {\displaystyle s} may be thought as the testing sample which is used to estimate Q P ( h ) {\displaystyle Q_{P}(h)} . Permutation: Since r {\displaystyle r} and s {\displaystyle s} are picked identically and independently, so swapping elements between them will not change the probability distribution on r {\displaystyle r} and s {\displaystyle s} . So, we will try to bound the probability of | Q ^ r ( h ) − Q ^ s ( h ) | ≥ ε / 2 {\displaystyle |{\widehat {Q}}_{r}(h)-{\widehat {Q}}_{s}(h)|\geq \varepsilon /2} for some h ∈ H {\displaystyle h\in H} by considering the effect of a specific collection of permutations of the joint sample x = r | | s {\displaystyle x=r||s} . Specifically, we consider permutations σ ( x ) {\displaystyle \sigma (x)} which swap x i {\displaystyle x_{i}} and x m + i {\displaystyle x_{m+i}} in some subset of 1 , 2 , . . . , m {\displaystyle {1,2,...,m}} . The symbol r | | s {\displaystyle r||s} means the concatenation of r {\displaystyle r} and s {\displaystyle s} . Reduction to a finite class: We can now restrict the function class H {\displaystyle H} to a fixed joint sample and hence, if H {\displaystyle H} has finite VC Dimension, it reduces to the problem to one involving a finite function class. We present the technical details of the proof. It should be stressed that this proof glosses over details like the measurability of the events V {\displaystyle V} and R {\displaystyle R} ; measurability is granted in the case of H {\displaystyle H} being finite or countable, but this is not normally the case in standard applications of the theorem (e.g. for statistical learning theory or to prove the Glivenko-Cantelli theorem). To get measurability, one needs to use a notion of separability of the underlying space, possibly related to H {\displaystyle H} . === Symmetrization === Lemma: Let V = { x ∈ X m : | Q P ( h ) − Q ^ x ( h ) | ≥ ε for some h ∈ H } {\displaystyle V=\{x\in X^{m}:|Q_{P}(h)-{\widehat {Q}}_{x}(h)|\geq \varepsilon {\text{ for some }}h\in H\}} and R = { ( r , s ) ∈ X m × X m : | Q r ^ ( h ) − Q ^ s ( h ) | ≥ ε / 2 for some h ∈ H } . {\displaystyle R=\{(r,s)\in X^{m}\times X^{m}:|{\widehat {Q_{r}}}(h)-{\widehat {Q}}_{s}(h)|\geq \varepsilon /2{\text{ for some }}h\in H\}.} Then for m ≥ 2 ε 2 {\displaystyle m\geq {\frac {2}{\varepsilon ^{2}}}} , P m ( V ) ≤ 2 P 2 m ( R ) {\displaystyle P^{m}(V)\leq 2P^{2m}(R)} . Proof: By the triangle inequality, if | Q P ( h ) − Q ^ r ( h ) | ≥ ε {\displaystyle |Q_{P}(h)-{\widehat {Q}}_{r}(h)|\geq \varepsilon } and | Q P ( h ) − Q ^ s ( h ) | ≤ ε / 2 {\displaystyle |Q_{P}(h)-{\widehat {Q}}_{s}(h)|\leq \varepsilon /2} then | Q ^ r ( h ) − Q ^ s ( h ) | ≥ ε / 2 {\displaystyle |{\widehat {Q}}_{r}(h)-{\widehat {Q}}_{s}(h)|\geq \varepsilon /2} . Therefore, P 2 m ( R ) ≥ P 2 m { ∃ h ∈ H , | Q P ( h ) − Q ^ r ( h ) | ≥ ε and | Q P ( h ) − Q ^ s ( h ) | ≤ ε / 2 } = ∫ V P m { s : ∃ h ∈ H , | Q P ( h ) − Q ^ r ( h ) | ≥ ε and | Q P ( h ) − Q ^ s ( h ) | ≤ ε / 2 } d P m ( r ) = A {\displaystyle {\begin{aligned}&P^{2m}(R)\\[5pt]\geq {}&P^{2m}\{\exists h\in H,|Q_{P}(h)-{\widehat {Q}}_{r}(h)|\geq \varepsilon {\text{ and }}|Q_{P}(h)-{\widehat {Q}}_{s}(h)|\leq \varepsilon /2\}\\[5pt]={}&\int _{V}P^{m}\{s:\exists h\in H,|Q_{P}(h)-{\widehat {Q}}_{r}(h)|\geq \varepsilon {\text{ and }}|Q_{P}(h)-{\widehat {Q}}_{s}(h)|\leq \varepsilon /2\}\,dP^{m}(r)\\[5pt]={}&A\end{aligned}}} since r {\displaystyle r} and s {\displaystyle s} are independent. Now for r ∈ V {\displaystyle r\in V} fix an h ∈ H {\displaystyle h\in H} such that | Q P ( h ) − Q ^ r ( h ) | ≥ ε {\displaystyle |Q_{P}(h)-{\widehat {Q}}_{r}(h)|\geq \varepsilon } . For this h {\displaystyle h} , we shall