In database systems, atomicity (; from Ancient Greek: ἄτομος, romanized: átomos, lit. 'undividable') is the property of a database transaction consisting of an indivisible and irreducible series of database operations such that either all occur, or none occur. It is one of the ACID transaction properties: Atomicity, Consistency, Isolation, Durability. A guarantee of atomicity prevents partial database updates from occurring, because they can cause greater problems than rejecting the whole series outright. As a consequence, an atomic transaction cannot be observed to be in progress by another database client: at one moment in time, it has not yet happened, and at the next it has already occurred in whole (or nothing happened if the transaction was cancelled in progress). An example of transaction atomicity could be a digital monetary transfer from bank account A to account B. It consists of two operations, debiting the money from account A and crediting it to account B. Performing both of these operations inside of an atomic transaction ensures that the database remains in a consistent state, if either operation fails there will not be any unaccountable credits or debits affecting either account. The same term is also used in the definition of First normal form in database systems, where it instead refers to the concept that the values for fields may not consist of multiple smaller values to be decomposed, such as a string into which multiple names, numbers, dates, or other types may be packed. == Orthogonality == Atomicity does not behave completely orthogonally with regard to the other ACID properties of transactions. For example, isolation relies on atomicity to roll back the enclosing transaction in the event of an isolation violation such as a deadlock; consistency also relies on atomicity to roll back the enclosing transaction in the event of a consistency violation by an illegal transaction. As a result of this, a failure to detect a violation and roll back the enclosing transaction may cause an isolation or consistency failure. == Implementation == Typically, systems implement Atomicity by providing some mechanism to indicate which transactions have started and which finished; or by keeping a copy of the data before any changes occurred (Read-copy-update). Several filesystems have developed methods for avoiding the need to keep multiple copies of data, using journaling (see journaling file system). Databases usually implement this using some form of logging/journaling to track changes. The system synchronizes the logs (often the metadata) as necessary after changes have successfully taken place. Afterwards, crash recovery ignores incomplete entries. Although implementations vary depending on factors such as concurrency issues, the principle of atomicity – i.e. complete success or complete failure – remain. Ultimately, any application-level implementation relies on operating-system functionality. At the file-system level, POSIX-compliant systems provide system calls such as open(2) and flock(2) that allow applications to atomically open or lock a file. At the process level, POSIX Threads provide adequate synchronization primitives. The hardware level requires atomic operations such as Test-and-set, Fetch-and-add, Compare-and-swap, or Load-Link/Store-Conditional, together with memory barriers. Portable operating systems cannot simply block interrupts to implement synchronization, since hardware that lacks concurrent execution such as hyper-threading or multi-processing is now extremely rare. In distributed and sharded databases, atomicity is complicated by network latency and the potential for partial failures. While traditional distributed systems often employ locking protocols (like 2PC) to ensure cross-shard atomicity, these can introduce performance bottlenecks. Recent research into distributed ledger consensus suggests alternative models, such as "braided synchronization". This technique, utilized in protocols like Cerberus, intertwines the consensus phases of multiple shards to enforce atomic guarantees without a global ordering of all transactions.
Data item
A data item describes an atomic state of a particular object concerning a specific property at a certain time point. A collection of data items for the same object at the same time forms an object instance (or table row). Any type of complex information can be broken down to elementary data items (atomic state). Data items are identified by object (o), property (p) and time (t), while the value (v) is a function of o, p and t: v = F(o,p,t). Values typically are represented by symbols like numbers, texts, images, sounds or videos. Values are not necessarily atomic. A value's complexity depends on the complexity of the property and time component. When looking at databases or XML files, the object is usually identified by an object name or other type of object identifier, which is part of the "data". Properties are defined as columns (table row), properties (object instance) or tags (XML). Often, time is not explicitly expressed and is an attribute applying to the complete data set. Other data collections provide time on the instance level (time series), column level, or even attribute/property level.
Mountain car problem
Mountain Car, a standard testing domain in Reinforcement learning, is a problem in which an under-powered car must drive up a steep hill. Since gravity is stronger than the car's engine, even at full throttle, the car cannot simply accelerate up the steep slope. The car is situated in a valley and must learn to leverage potential energy by driving up the opposite hill before the car is able to make it to the goal at the top of the rightmost hill. The domain has been used as a test bed in various reinforcement learning papers. == Introduction == The mountain car problem, although fairly simple, is commonly applied because it requires a reinforcement learning agent to learn on two continuous variables: position and velocity. For any given state (position and velocity) of the car, the agent is given the possibility of driving left, driving right, or not using the engine at all. In the standard version of the problem, the agent receives a negative reward at every time step when the goal is not reached; the agent has no information about the goal until an initial success. == History == The mountain car problem appeared first in Andrew Moore's PhD thesis (1990). It was later more strictly defined in Singh and Sutton's reinforcement learning paper with eligibility traces. The problem became more widely studied when Sutton and Barto added it to their book Reinforcement Learning: An Introduction (1998). Throughout the years many versions of the problem have been used, such as those which modify the reward function, termination condition, and the start state. == Techniques used to solve mountain car == Q-learning and similar techniques for mapping discrete states to discrete actions need to be extended to be able to deal with the continuous state space of the problem. Approaches often fall into one of two categories, state space discretization or function approximation. === Discretization === In this approach, two continuous state variables are pushed into discrete states by bucketing each continuous variable into multiple discrete states. This approach works with properly tuned parameters but a disadvantage is information gathered from one state is not used to evaluate another state. Tile coding can be used to improve discretization and involves continuous variables mapping into sets of buckets offset from one another. Each step of training has a wider impact on the value function approximation because when the offset grids are summed, the information is diffused. === Function approximation === Function approximation is another way to solve the mountain car. By choosing a set of basis functions beforehand, or by generating them as the car drives, the agent can approximate the value function at each state. Unlike the step-wise version of the value function created with discretization, function approximation can more cleanly estimate the true smooth function of the mountain car domain. === Eligibility traces === One aspect of the problem involves the delay of actual reward. The agent is not able to learn about the goal until a successful completion. Given a naive approach for each trial the car can only backup the reward of the goal slightly. This is a problem for naive discretization because each discrete state will only be backed up once, taking a larger number of episodes to learn the problem. This problem can be alleviated via the mechanism of eligibility traces, which will automatically backup the reward given to states before, dramatically increasing the speed of learning. Eligibility traces can be viewed as a bridge from temporal difference learning methods to Monte Carlo methods. == Technical details == The mountain car problem has undergone many iterations. This section focuses on the standard well-defined version from Sutton (2008). === State variables === Two-dimensional continuous state space. V e l o c i t y = ( − 0.07 , 0.07 ) {\displaystyle Velocity=(-0.07,0.07)} P o s i t i o n = ( − 1.2 , 0.6 ) {\displaystyle Position=(-1.2,0.6)} === Actions === One-dimensional discrete action space. m o t o r = ( l e f t , n e u t r a l , r i g h t ) {\displaystyle motor=(left,neutral,right)} === Reward === For every time step: r e w a r d = − 1 {\displaystyle reward=-1} === Update function === For every time step: A c t i o n = [ − 1 , 0 , 1 ] {\displaystyle Action=[-1,0,1]} V e l o c i t y = V e l o c i t y + ( A c t i o n ) ∗ 0.001 + cos ( 3 ∗ P o s i t i o n ) ∗ ( − 0.0025 ) {\displaystyle Velocity=Velocity+(Action)0.001+\cos(3Position)(-0.0025)} P o s i t i o n = P o s i t i o n + V e l o c i t y {\displaystyle Position=Position+Velocity} === Starting condition === Optionally, many implementations include randomness in both parameters to show better generalized learning. P o s i t i o n = − 0.5 {\displaystyle Position=-0.5} V e l o c i t y = 0.0 {\displaystyle Velocity=0.0} === Termination condition === End the simulation when: P o s i t i o n ≥ 0.6 {\displaystyle Position\geq 0.6} == Variations == There are many versions of the mountain car which deviate in different ways from the standard model. Variables that vary include but are not limited to changing the constants (gravity and steepness) of the problem so specific tuning for specific policies become irrelevant and altering the reward function to affect the agent's ability to learn in a different manner. An example is changing the reward to be equal to the distance from the goal, or changing the reward to zero everywhere and one at the goal. Additionally, a 3D mountain car can be used, with a 4D continuous state space.
Psychology of reasoning
The psychology of reasoning (also known as the cognitive science of reasoning) is the study of how people reason, often broadly defined as the process of drawing conclusions to inform how people solve problems and make decisions. It overlaps with psychology, philosophy, linguistics, cognitive science, artificial intelligence, logic, and probability theory. Psychological experiments on how humans and other animals reason have been carried out for over 100 years. An enduring question is whether or not people have the capacity to be rational. Current research in this area addresses various questions about reasoning, rationality, judgments, intelligence, relationships between emotion and reasoning, and development. == Everyday reasoning == One of the most obvious areas in which people employ reasoning is with sentences in everyday language. Most experimentation on deduction has been carried out on hypothetical thought, in particular, examining how people reason about conditionals, e.g., If A then B. Participants in experiments make the modus ponens inference, given the indicative conditional If A then B, and given the premise A, they conclude B. However, given the indicative conditional and the minor premise for the modus tollens inference, not-B, about half of the participants in experiments conclude not-A and the remainder concludes that nothing follows. The ease with which people make conditional inferences is affected by context, as demonstrated in the well-known selection task developed by Peter Wason. Participants are better able to test a conditional in an ecologically relevant context, e.g., if the envelope is sealed then it must have a 50 cent stamp on it compared to one that contains symbolic content, e.g., if the letter is a vowel then the number is even. Background knowledge can also lead to the suppression of even the simple modus ponens inference Participants given the conditional if Lisa has an essay to write then she studies late in the library and the premise Lisa has an essay to write make the modus ponens inference 'she studies late in the library', but the inference is suppressed when they are also given a second conditional if the library stays open then she studies late in the library. Interpretations of the suppression effect are controversial Other investigations of propositional inference examine how people think about disjunctive alternatives, e.g., A or else B, and how they reason about negation, e.g., It is not the case that A and B. Many experiments have been carried out to examine how people make relational inferences, including comparisons, e.g., A is better than B. Such investigations also concern spatial inferences, e.g. A is in front of B and temporal inferences, e.g. A occurs before B. Other common tasks include categorical syllogisms, used to examine how people reason about quantifiers such as All or Some, e.g., Some of the A are not B. For example if all A are B and some B are C, what (if anything) follows? == Theories of reasoning == There are several alternative theories of the cognitive processes that human reasoning is based on. One view is that people rely on a mental logic consisting of formal (abstract or syntactic) inference rules similar to those developed by logicians in the propositional calculus. Another view is that people rely on domain-specific or content-sensitive rules of inference. A third view is that people rely on mental models, that is, mental representations that correspond to imagined possibilities. A fourth view is that people compute probabilities. One controversial theoretical issue is the identification of an appropriate competence model, or a standard against which to compare human reasoning. Initially classical logic was chosen as a competence model. Subsequently, some researchers opted for non-monotonic logic and Bayesian probability. Research on mental models and reasoning has led to the suggestion that people are rational in principle but err in practice. Connectionist approaches towards reasoning have also been proposed. Despite the ongoing debate about the cognitive processes involved in human reasoning, recent research has shown that multiple approaches can be useful in modeling human thinking. For instance, studies have found that people's reasoning is often influenced by their prior beliefs, which can be modeled using Bayesian probability theory. Additionally, research on mental models has shown that people tend to reason about problems by constructing multiple mental representations of the situation, which can help them to identify relevant features and make inferences based on their understanding of the problem. Moreover, connectionist approaches to reasoning have also gained attention, which focus on the neural network models that can learn from data and generalize to new situations. == Development of reasoning == It is an active question in psychology how, why, and when the ability to reason develops from infancy to adulthood. Jean Piaget's theory of cognitive development posited general mechanisms and stages in the development of reasoning from infancy to adulthood. According to the neo-Piagetian theories of cognitive development, changes in reasoning with development come from increasing working memory capacity, increasing speed of processing, and enhanced executive functions and control. Increasing self-awareness is also an important factor. In their book The Enigma of Reason, the cognitive scientists Hugo Mercier and Dan Sperber put forward an "argumentative" theory of reasoning, claiming that humans evolved to reason primarily to justify our beliefs and actions and to convince others in a social environment. Key evidence for their theory includes the errors in reasoning that solitary individuals are prone to when their arguments are not criticized, such as logical fallacies, and how groups become much better at performing cognitive reasoning tasks when they communicate with one another and can evaluate each other's arguments. Sperber and Mercier offer one attempt to resolve the apparent paradox that the confirmation bias is so strong despite the function of reasoning naively appearing to be to come to veridical conclusions about the world. The study of the development of reasoning abilities is an ongoing area of research in psychology, and multiple factors have been proposed to explain how, why, and when reasoning develops from infancy to adulthood. Recent research has suggested that early experiences and social interactions play a critical role in the development of reasoning abilities. For example, studies have shown that infants as young as six months old can engage in basic logical reasoning, such as reasoning about the relationship between objects and their properties. Furthermore, research has highlighted the importance of parental interaction and cognitive stimulation in the development of children's reasoning abilities. Additionally, studies have suggested that cultural factors, such as educational practices and the emphasis on critical thinking, can also influence the development of reasoning skills across different populations. == Different sorts of reasoning == Philip Johnson-Laird trying to taxonomize thought, distinguished between goal-directed thinking and thinking without goal, noting that association was involved in unrelated reading. He argues that goal directed reasoning can be classified based on the problem space involved in a solution, citing Allen Newell and Herbert A. Simon. Inductive reasoning makes broad generalizations from specific cases or observations. In this process of reasoning, general assertions are made based on past specific pieces of evidence. This kind of reasoning allows the conclusion to be false even if the original statement is true. For example, if one observes a college athlete, one makes predictions and assumptions about other college athletes based on that one observation. Scientists use inductive reasoning to create theories and hypotheses. Philip Johnson-Laird distinguished inductive from deductive reasoning, in that the former creates semantic information while the later does not . In opposition, deductive reasoning is a basic form of valid reasoning. In this reasoning process a person starts with a known claim or a general belief and from there asks what follows from these foundations or how will these premises influence other beliefs. In other words, deduction starts with a hypothesis and examines the possibilities to reach a conclusion. Deduction helps people understand why their predictions are wrong and indicates that their prior knowledge or beliefs are off track. An example of deduction can be seen in the scientific method when testing hypotheses and theories. Although the conclusion usually corresponds and therefore proves the hypothesis, there are some cases where the conclusion is logical, but the generalization is not. For example, the argument, "All young girls wear skirts; Julie is a young
80 Million Tiny Images
80 Million Tiny Images is a dataset intended for training machine-learning systems constructed by Antonio Torralba, Rob Fergus, and William T. Freeman in a collaboration between MIT and New York University. It was published in 2008. The dataset has size 760 GB. It contains 79,302,017 32×32-pixel color images, scaled down from images scraped from the World Wide Web over 8 months. The images are classified into 75,062 classes. Each class is a non-abstract noun in WordNet. Images may appear in more than one class. The dataset was motivated by non-parametric models of neural activations in the visual cortex upon seeing images. The CIFAR-10 dataset uses a subset of the images in this dataset, but with independently generated labels, as the original labels were not reliable. The CIFAR-10 set has 6000 examples of each of 10 classes, and the CIFAR-100 set has 600 examples of each of 100 non-overlapping classes. == Construction == It was first reported in a technical report in April 2007, during the middle of the construction process, when there were only 73 million images. The full dataset was published in 2008. They began with all 75,846 non-abstract nouns in WordNet, and then for each of these nouns, they scraped 7 image search engines: Altavista, Ask.com, Flickr, Cydral, Google, Picsearch, and Webshots. After 8 months of scraping, they obtained 97,245,098 images. Since they did not have enough storage, they downsized the images to 32×32 as they were scraped. After gathering, they removed images with zero variance and intra-word duplicate images, resulting in the final dataset. Out of the 75,846 nouns, only 75,062 classes had any results, so the other nouns did not appear in the final dataset. The number of images per noun follows a Zipf-like distribution, with 1056 images per noun on average. To prevent a few nouns taking up too many images, they put an upper bound of at most 3000 images per noun. == Retirement == The 80 Million Tiny Images dataset was retired from use by its creators in 2020, after a paper by researchers Abeba Birhane and Vinay Prabhu found that some of the labeling of several publicly available image datasets, including 80 Million Tiny Images, contained racist and misogynistic slurs which were causing models trained on them to exhibit racial and sexual bias. The dataset also contained offensive images. Following the release of the paper, the dataset's creators removed the dataset from distribution, and requested that other researchers not use it for further research and to delete their copies of the dataset.
Computer vision dazzle
Computer vision dazzle, also known as CV dazzle, dazzle makeup, or anti-surveillance makeup, is a type of camouflage used to hamper facial recognition software, inspired by dazzle camouflage used by vehicles such as ships and planes. == Methods == CV dazzle combines stylized makeup, asymmetric hair, and sometimes infrared lights built in to glasses or clothing to break up detectable facial patterns recognized by computer vision algorithms in much the same way that warships contrasted color and used sloping lines and curves to distort the structure of a vessel. It has been shown to be somewhat successful at defeating face detection software in common use, including that employed by Facebook. CV dazzle attempts to block detection by facial recognition technologies such as DeepFace "by creating an 'anti-face'". It uses occlusion, covering certain facial features; transformation, altering the shape or colour of parts of the face; and a combination of the two. Prominent artists employing this technique include Adam Harvey and Jillian Mayer. == Use in protests == Computer vision dazzle makeup has been used by protestors in several different protest movements. Its use as a protesting aid has often been found ineffective. It may be effective to thwart computer technology, but draws human attention, is easy for human monitors to spot on security cameras, and makes it hard for protestors to blend in within a crowd. Advances in facial recognition technology make dazzle makeup increasingly ineffective.
Algorithmic inference
Algorithmic inference gathers new developments in the statistical inference methods made feasible by the powerful computing devices widely available to any data analyst. Cornerstones in this field are computational learning theory, granular computing, bioinformatics, and, long ago, structural probability (Fraser 1966). The main focus is on the algorithms which compute statistics rooting the study of a random phenomenon, along with the amount of data they must feed on to produce reliable results. This shifts the interest of mathematicians from the study of the distribution laws to the functional properties of the statistics, and the interest of computer scientists from the algorithms for processing data to the information they process. == The Fisher parametric inference problem == Concerning the identification of the parameters of a distribution law, the mature reader may recall lengthy disputes in the mid 20th century about the interpretation of their variability in terms of fiducial distribution (Fisher 1956), structural probabilities (Fraser 1966), priors/posteriors (Ramsey 1925), and so on. From an epistemology viewpoint, this entailed a companion dispute as to the nature of probability: is it a physical feature of phenomena to be described through random variables or a way of synthesizing data about a phenomenon? Opting for the latter, Fisher defines a fiducial distribution law of parameters of a given random variable that he deduces from a sample of its specifications. With this law he computes, for instance "the probability that μ (mean of a Gaussian variable – omeur note) is less than any assigned value, or the probability that it lies between any assigned values, or, in short, its probability distribution, in the light of the sample observed". == The classic solution == Fisher fought hard to defend the difference and superiority of his notion of parameter distribution in comparison to analogous notions, such as Bayes' posterior distribution, Fraser's constructive probability and Neyman's confidence intervals. For half a century, Neyman's confidence intervals won out for all practical purposes, crediting the phenomenological nature of probability. With this perspective, when you deal with a Gaussian variable, its mean μ is fixed by the physical features of the phenomenon you are observing, where the observations are random operators, hence the observed values are specifications of a random sample. Because of their randomness, you may compute from the sample specific intervals containing the fixed μ with a given probability that you denote confidence. === Example === Let X be a Gaussian variable with parameters μ {\displaystyle \mu } and σ 2 {\displaystyle \sigma ^{2}} and { X 1 , … , X m } {\displaystyle \{X_{1},\ldots ,X_{m}\}} a sample drawn from it. Working with statistics S μ = ∑ i = 1 m X i {\displaystyle S_{\mu }=\sum _{i=1}^{m}X_{i}} and S σ 2 = ∑ i = 1 m ( X i − X ¯ ) 2 , where X ¯ = S μ m {\displaystyle S_{\sigma ^{2}}=\sum _{i=1}^{m}(X_{i}-{\overline {X}})^{2},{\text{ where }}{\overline {X}}={\frac {S_{\mu }}{m}}} is the sample mean, we recognize that T = S μ − m μ S σ 2 m − 1 m = X ¯ − μ S σ 2 / ( m ( m − 1 ) ) {\displaystyle T={\frac {S_{\mu }-m\mu }{\sqrt {S_{\sigma ^{2}}}}}{\sqrt {\frac {m-1}{m}}}={\frac {{\overline {X}}-\mu }{\sqrt {S_{\sigma ^{2}}/(m(m-1))}}}} follows a Student's t distribution (Wilks 1962) with parameter (degrees of freedom) m − 1, so that f T ( t ) = Γ ( m / 2 ) Γ ( ( m − 1 ) / 2 ) 1 π ( m − 1 ) ( 1 + t 2 m − 1 ) m / 2 . {\displaystyle f_{T}(t)={\frac {\Gamma (m/2)}{\Gamma ((m-1)/2)}}{\frac {1}{\sqrt {\pi (m-1)}}}\left(1+{\frac {t^{2}}{m-1}}\right)^{m/2}.} Gauging T between two quantiles and inverting its expression as a function of μ {\displaystyle \mu } you obtain confidence intervals for μ {\displaystyle \mu } . With the sample specification: x = { 7.14 , 6.3 , 3.9 , 6.46 , 0.2 , 2.94 , 4.14 , 4.69 , 6.02 , 1.58 } {\displaystyle \mathbf {x} =\{7.14,6.3,3.9,6.46,0.2,2.94,4.14,4.69,6.02,1.58\}} having size m = 10, you compute the statistics s μ = 43.37 {\displaystyle s_{\mu }=43.37} and s σ 2 = 46.07 {\displaystyle s_{\sigma ^{2}}=46.07} , and obtain a 0.90 confidence interval for μ {\displaystyle \mu } with extremes (3.03, 5.65). == Inferring functions with the help of a computer == From a modeling perspective the entire dispute looks like a chicken-egg dilemma: either fixed data by first and probability distribution of their properties as a consequence, or fixed properties by first and probability distribution of the observed data as a corollary. The classic solution has one benefit and one drawback. The former was appreciated particularly back when people still did computations with sheet and pencil. Per se, the task of computing a Neyman confidence interval for the fixed parameter θ is hard: you do not know θ, but you look for disposing around it an interval with a possibly very low probability of failing. The analytical solution is allowed for a very limited number of theoretical cases. Vice versa a large variety of instances may be quickly solved in an approximate way via the central limit theorem in terms of confidence interval around a Gaussian distribution – that's the benefit. The drawback is that the central limit theorem is applicable when the sample size is sufficiently large. Therefore, it is less and less applicable with the sample involved in modern inference instances. The fault is not in the sample size on its own part. Rather, this size is not sufficiently large because of the complexity of the inference problem. With the availability of large computing facilities, scientists refocused from isolated parameters inference to complex functions inference, i.e. re sets of highly nested parameters identifying functions. In these cases we speak about learning of functions (in terms for instance of regression, neuro-fuzzy system or computational learning) on the basis of highly informative samples. A first effect of having a complex structure linking data is the reduction of the number of sample degrees of freedom, i.e. the burning of a part of sample points, so that the effective sample size to be considered in the central limit theorem is too small. Focusing on the sample size ensuring a limited learning error with a given confidence level, the consequence is that the lower bound on this size grows with complexity indices such as VC dimension or detail of a class to which the function we want to learn belongs. === Example === A sample of 1,000 independent bits is enough to ensure an absolute error of at most 0.081 on the estimation of the parameter p of the underlying Bernoulli variable with a confidence of at least 0.99. The same size cannot guarantee a threshold less than 0.088 with the same confidence 0.99 when the error is identified with the probability that a 20-year-old man living in New York does not fit the ranges of height, weight and waistline observed on 1,000 Big Apple inhabitants. The accuracy shortage occurs because both the VC dimension and the detail of the class of parallelepipeds, among which the one observed from the 1,000 inhabitants' ranges falls, are equal to 6. == The general inversion problem solving the Fisher question == With insufficiently large samples, the approach: fixed sample – random properties suggests inference procedures in three steps: === Definition === For a random variable and a sample drawn from it a compatible distribution is a distribution having the same sampling mechanism M X = ( Z , g θ ) {\displaystyle {\mathcal {M}}_{X}=(Z,g_{\boldsymbol {\theta }})} of X with a value θ {\displaystyle {\boldsymbol {\theta }}} of the random parameter Θ {\displaystyle \mathbf {\Theta } } derived from a master equation rooted on a well-behaved statistic s. === Example === You may find the distribution law of the Pareto parameters A and K as an implementation example of the population bootstrap method as in the figure on the left. Implementing the twisting argument method, you get the distribution law F M ( μ ) {\displaystyle F_{M}(\mu )} of the mean M of a Gaussian variable X on the basis of the statistic s M = ∑ i = 1 m x i {\textstyle s_{M}=\sum _{i=1}^{m}x_{i}} when Σ 2 {\displaystyle \Sigma ^{2}} is known to be equal to σ 2 {\displaystyle \sigma ^{2}} (Apolloni, Malchiodi & Gaito 2006). Its expression is: F M ( μ ) = Φ ( m μ − s M σ m ) , {\displaystyle F_{M}(\mu )=\Phi {\left({\frac {m\mu -s_{M}}{\sigma {\sqrt {m}}}}\right)},} shown in the figure on the right, where Φ {\displaystyle \Phi } is the cumulative distribution function of a standard normal distribution. Computing a confidence interval for M given its distribution function is straightforward: we need only find two quantiles (for instance δ / 2 {\displaystyle \delta /2} and 1 − δ / 2 {\displaystyle 1-\delta /2} quantiles in case we are interested in a confidence interval of level δ symmetric in the tail's probabilities) as indicated on the left in the diagram showing the behavior of