Shopping for the best AI copywriting tool? An AI copywriting tool is software that uses machine learning to help you get more done — it keeps getting smarter as the underlying models improve. Pricing, accuracy, and the size of the model behind the tool are the three factors that most affect daily usefulness. Whether you are a beginner or a pro, the right AI copywriting tool slots into your workflow and pays for itself fast. Below we compare features, pricing, and real output so you can choose with confidence.
Toggl Track
Toggl Track (formerly Toggl) is a time tracking software developed by Toggl OÜ which is headquartered in Tallinn, Estonia. The company offers online time tracking and reporting services through their website along with mobile and desktop applications. Time can be tracked through a start/stop button, manual entry, or dragging and resizing time blocks in a calendar view. == History == According to Alari Aho, Toggl's CEO and founder, the application has been fully self-funded from the start. The name was created using a random name generator.
Ballie
Ballie is an AI robot created by Samsung to be released in 2026. It is an autonomous robot which has the ability to control smart home devices. Ballie can text, send pictures and follow commands through SmartThings. It can also show workout information shared from a Galaxy Watch. Ballie can make video calls and welcome you home. == History == It was first unveiled at Samsung's CES event in CES 2020, and later updated the design in CES 2024, and will be later released in 2026. == Design ==
Ishikawa diagram
Ishikawa diagrams (also called fishbone diagrams, herringbone diagrams, cause-and-effect diagrams) are causal diagrams created by Kaoru Ishikawa that show the potential causes of a specific event. Common uses of the Ishikawa diagram are product design and quality defect prevention to identify potential factors causing an overall effect. Each cause or reason for imperfection is a source of variation. Causes are usually grouped into major categories to identify and classify these sources of variation. == Overview == The defect, or the problem to be solved, is shown as the fish's head, facing to the right, with the causes extending to the left as fishbones; the ribs branch off the backbone for major causes, with sub-branches for root-causes, to as many levels as required. Ishikawa diagrams were popularized in the 1960s by Kaoru Ishikawa, who pioneered quality management processes in the Kawasaki shipyards, and in the process became one of the founding fathers of modern management. The basic concept was first used in the 1920s, and is considered one of the seven basic tools of quality control. It is known as a fishbone diagram because of its shape, similar to the side view of a fish skeleton. Mazda Motors famously used an Ishikawa diagram in the development of the Miata (MX5) sports car. == Root causes == Root-cause analysis is intended to reveal key relationships among various variables, and the possible causes provide additional insight into process behavior. It shows high-level causes that lead to the problem encountered by providing a snapshot of the current situation. There can be confusion about the relationships between problems, causes, symptoms and effects. Smith highlights this and the common question “Is that a problem or a symptom?” which mistakenly presumes that problems and symptoms are mutually exclusive categories. A problem is a situation that bears improvement; a symptom is the effect of a cause: a situation can be both a problem and a symptom. At a practical level, a cause is whatever is responsible for, or explains, an effect - a factor "whose presence makes a critical difference to the occurrence of an outcome". The causes emerge by analysis, often through brainstorming sessions, and are grouped into categories on the main branches off the fishbone. To help structure the approach, the categories are often selected from one of the common models shown below, but may emerge as something unique to the application in a specific case. Each potential cause is traced back to find the root cause, often using the 5 Whys technique. Typical categories include: === The 5 Ms (used in manufacturing) === Originating with lean manufacturing and the Toyota Production System, the 5 Ms is one of the most common frameworks for root-cause analysis: Manpower / Mindpower (physical or knowledge work, includes: kaizens, suggestions) Machine (equipment, technology) Material (includes raw material, consumables, and information) Method (process) Measurement / medium (inspection, environment) These have been expanded by some to include an additional three, and are referred to as the 8 Ms: Mission / mother nature (purpose, environment) Management / money power (leadership) Maintenance === The 8 Ps (used in product marketing) === This common model for identifying crucial attributes for planning in product marketing is often also used in root-cause analysis as categories for the Ishikawa diagram: Product (or service) Price Place Promotion People (personnel) Process Physical evidence (proof) Performance === The 4 or 5 Ss (used in service industries) === An alternative used for service industries, uses four categories of possible cause: Surroundings: Refers to the environment in which the process occurs. Suppliers: Refers to external parties that provide inputs—raw materials, components, or services. Systems: Refers to the procedures, processes, and technologies used to perform the work. Skill: Refers to the human factor, particularly the knowledge and abilities of employees. Safety: Refers to physical and psychological well-being in the workplace. == Use in specific industries == The Ishikawa diagram has been widely adopted across various industries as an effective tool for root cause analysis in quality, efficiency, and safety-related issues. Its versatility allows it to be applied in both manufacturing and service contexts. In the manufacturing industry, particularly in the automotive and electronics sectors, the diagram is frequently used in continuous improvement initiatives such as Six Sigma and Lean Manufacturing. Quality teams use it to identify causes related to materials, methods, machinery, manpower, environment, and measurement, facilitating informed decision-making to reduce defects and optimize processes. In the food industry, the Ishikawa diagram is applied to analyze issues related to food safety, temperature control, cross-contamination, and regulatory compliance. Its use enables companies to identify improvement opportunities in production, packaging, and distribution stages. In the pharmaceutical sector, it is a key tool in process validation, quality control, and compliance with Good Manufacturing Practices (GMP). It helps visualize factors affecting product quality from formulation to storage. It has also been successfully implemented in sectors such as aerospace, pulp and paper, construction, education, and healthcare, where it supports structured problem-solving and promotes continuous improvement and a culture of quality.
User modeling
User modeling is the subdivision of human–computer interaction which describes the process of building up and modifying a conceptual understanding of the user. The main goal of user modeling is customization and adaptation of systems to the user's specific needs. The system needs to "say the 'right' thing at the 'right' time in the 'right' way". To do so it needs an internal representation of the user. Another common purpose is modeling specific kinds of users, including modeling of their skills and declarative knowledge, for use in automatic software-tests. User-models can thus serve as a cheaper alternative to user testing but should not replace user testing. == Background == A user model is the collection and categorization of personal data associated with a specific user. A user model is a (data) structure that is used to capture certain characteristics about an individual user, and a user profile is the actual representation in a given user model. The process of obtaining the user profile is called user modeling. Therefore, it is the basis for any adaptive changes to the system's behavior. Which data is included in the model depends on the purpose of the application. It can include personal information such as users' names and ages, their interests, their skills and knowledge, their goals and plans, their preferences and their dislikes or data about their behavior and their interactions with the system. There are different design patterns for user models, though often a mixture of them is used. Static user models Static user models are the most basic kinds of user models. Once the main data is gathered they are normally not changed again, they are static. Shifts in users' preferences are not registered and no learning algorithms are used to alter the model. Dynamic user models Dynamic user models allow a more up to date representation of users. Changes in their interests, their learning progress or interactions with the system are noticed and influence the user models. The models can thus be updated and take the current needs and goals of the users into account. Stereotype based user models Stereotype based user models are based on demographic statistics. Based on the gathered information users are classified into common stereotypes. The system then adapts to this stereotype. The application therefore can make assumptions about a user even though there might be no data about that specific area, because demographic studies have shown that other users in this stereotype have the same characteristics. Thus, stereotype based user models mainly rely on statistics and do not take into account that personal attributes might not match the stereotype. However, they allow predictions about a user even if there is rather little information about him or her. Highly adaptive user models Highly adaptive user models try to represent one particular user and therefore allow a very high adaptivity of the system. In contrast to stereotype based user models they do not rely on demographic statistics but aim to find a specific solution for each user. Although users can take great benefit from this high adaptivity, this kind of model needs to gather a lot of information first. == Data gathering == Information about users can be gathered in several ways. There are three main methods: Asking for specific facts while (first) interacting with the system Mostly this kind of data gathering is linked with the registration process. While registering users are asked for specific facts, their likes and dislikes and their needs. Often the given answers can be altered afterwards. Learning users' preferences by observing and interpreting their interactions with the system In this case users are not asked directly for their personal data and preferences, but this information is derived from their behavior while interacting with the system. The ways they choose to accomplish a tasks, the combination of things they takes interest in, these observations allow inferences about a specific user. The application dynamically learns from observing these interactions. Different machine learning algorithms may be used to accomplish this task. A hybrid approach which asks for explicit feedback and alters the user model by adaptive learning This approach is a mixture of the ones above. Users have to answer specific questions and give explicit feedback. Furthermore, their interactions with the system are observed and the derived information are used to automatically adjust the user models. Though the first method is a good way to quickly collect main data it lacks the ability to automatically adapt to shifts in users' interests. It depends on the users' readiness to give information and it is unlikely that they are going to edit their answers once the registration process is finished. Therefore, there is a high likelihood that the user models are not up to date. However, this first method allows the users to have full control over the collected data about them. It is their decision which information they are willing to provide. This possibility is missing in the second method. Adaptive changes in a system that learns users' preferences and needs only by interpreting their behavior might appear a bit opaque to the users, because they cannot fully understand and reconstruct why the system behaves the way it does. Moreover, the system is forced to collect a certain amount of data before it is able to predict the users' needs with the required accuracy. Therefore, it takes a certain learning time before a user can benefit from adaptive changes. However, afterwards these automatically adjusted user models allow a quite accurate adaptivity of the system. The hybrid approach tries to combine the advantages of both methods. Through collecting data by directly asking its users it gathers a first stock of information which can be used for adaptive changes. By learning from the users' interactions it can adjust the user models and reach more accuracy. Yet, the designer of the system has to decide, which of these information should have which amount of influence and what to do with learned data that contradicts some of the information given by a user. == System adaptation == Once a system has gathered information about a user it can evaluate that data by preset analytical algorithm and then start to adapt to the user's needs. These adaptations may concern every aspect of the system's behavior and depend on the system's purpose. Information and functions can be presented according to the user's interests, knowledge or goals by displaying only relevant features, hiding information the user does not need, making proposals what to do next and so on. One has to distinguish between adaptive and adaptable systems. In an adaptable system the user can manually change the system's appearance, behavior or functionality by actively selecting the corresponding options. Afterwards the system will stick to these choices. In an adaptive system a dynamic adaption to the user is automatically performed by the system itself, based on the built user model. Thus, an adaptive system needs ways to interpret information about the user in order to make these adaptations. One way to accomplish this task is implementing rule-based filtering. In this case a set of IF... THEN... rules is established that covers the knowledge base of the system. The IF-conditions can check for specific user-information and if they match the THEN-branch is performed which is responsible for the adaptive changes. Another approach is based on collaborative filtering. In this case information about a user is compared to that of other users of the same systems. Thus, if characteristics of the current user match those of another, the system can make assumptions about the current user by presuming that he or she is likely to have similar characteristics in areas where the model of the current user is lacking data. Based on these assumption the system then can perform adaptive changes. == Usages == Adaptive hypermedia: In an adaptive hypermedia system the displayed content and the offered hyperlinks are chosen on basis of users' specific characteristics, taking their goals, interests, knowledge and abilities into account. Thus, an adaptive hypermedia system aims to reduce the "lost in hyperspace" syndrome by presenting only relevant information. Adaptive educational hypermedia: Being a subdivision of adaptive hypermedia the main focus of adaptive educational hypermedia lies on education, displaying content and hyperlinks corresponding to the user's knowledge on the field of study. Intelligent tutoring system: Unlike adaptive educational hypermedia systems intelligent tutoring systems are stand-alone systems. Their aim is to help students in a specific field of study. To do so, they build up a user model where they store information about abilities, knowledge and needs of the user. The system can now adapt to this user by presenting approp
Competition in artificial intelligence
Competition in artificial intelligence refers to the rivalry among companies, research institutions, and governments to develop and deploy the most capable artificial intelligence (AI) systems. The competition spans multiple domains, including large language models (LLMs), autonomous vehicles, robotics, computer vision systems, natural language processing (NLP), and AI-optimized hardware. == Background == Competition in AI is driven by potential economic, strategic, and scientific advantages. Breakthroughs in AI can enhance productivity, enable new products and services, and provide geopolitical leverage. The field has experienced rapid progress since the mid-2010s, particularly in machine learning and artificial neural networks, leading to intense rivalry among leading actors. == Corporate competition == Major technology companies are among the most visible competitors in AI. In the United States, firms such as OpenAI, Google DeepMind, Meta Platforms, Microsoft, Anthropic, and Nvidia compete in building advanced LLMs, generative AI platforms, and AI-optimized graphics processing units (GPUs). In China, companies such as Baidu, Alibaba Group, Tencent, and startups such DeepSeek have become leaders in AI deployment, often with state backing. The "[war for talent]" in AI research has become a defining feature of corporate competition. Leading firms often recruit top AI researchers from rivals, sometimes offering multi-million-dollar compensation packages. == National competition == Governments see leadership in AI as a strategic priority. The United States has funded AI research for military, economic, and societal applications, while China has set a target to lead the world in AI by 2030 through its "New Generation Artificial Intelligence Development Plan". Other nations, including the UK, India, Israel, Russia, South Korea, and members of the European Union, have launched national AI strategies. In February 2026 Anthropic said Chinese companies - DeepSeek, Moonshot AI, and MiniMax - were conducting "distillation attacks" in an attempt to copy their model's capabilities, and warned that business wars were closely tied to geopolitical ones: "foreign labs that illicitly distill American models can remove safeguards, feeding model capabilities into their own military, intelligence, and surveillance systems." == Sectors of competition == === Large language models and chatbots competition === Competition to produce the most capable generative text models, with benchmarks such as MMLU and ARC used to evaluate performance has been on scale since the emergence of AI. These systems leverage deep learning, especially transformer architectures, to understand and generate human-like language. Companies and research groups globally compete to develop chatbots that are more capable, reliable, and context-aware. Among the most well-known chatbots is ChatGPT, developed by OpenAI. Since its public release in 2022, ChatGPT has rapidly gained widespread attention for its ability to engage in coherent and versatile conversations, assist with creative writing, and solve complex problems. In response, technology firms introduced competing chatbots aiming to challenge or surpass ChatGPT's capabilities. Notably, DeepSeek, a Chinese AI company, launched an advanced chatbot integrated with their R1 language model, emphasizing strong natural language understanding and multilingual support. Similarly, Grok, developed by xAI (company), integrates conversational AI into vehicles and digital assistants, combining natural language processing with real-time data for personalized user interaction. These chatbots not only compete in language tasks but also demonstrate strategic reasoning capabilities by playing complex games such as chess and Go. This form of competition is reminiscent of historic AI milestones set by programs such as Deep Blue and AlphaGo. The OpenAI’s ChatGPT has been tested in playing chess at various levels, while DeepSeek’s chatbot showcased its prowess in online chess tournaments in early 2024, winning several matches against human and AI opponents. Grok, leveraging Tesla's vast data infrastructure, has demonstrated real-time strategic decision-making in simulation environments that include chess-like games. The competition pushes rapid innovation, with firms racing to improve chatbot conversational depth, reduce biases, increase factual accuracy, and integrate multimodal inputs like images and videos. At the same time, the competition raises questions about AI safety, ethical use, and the societal impacts of increasingly human-like chatbots. === Autonomous vehicles === Companies such as Waymo, Tesla, and Baidu are racing to deploy safe and reliable self-driving car technology. === AI chips === Rivalry between Nvidia, AMD, Intel, and Huawei in designing processors optimized for AI workloads. === Military applications === Development of AI-enabled drones, surveillance systems, and decision-support tools, with associated ethical debates. == Events == In 2023, OpenAI released GPT-4, prompting competitors such as Google DeepMind to accelerate the release of their own models, including Gemini. In 2024, Chinese AI company DeepSeek launched the R1 model, leading OpenAI to release an open-source system, GPT-OSS, as a strategic countermeasure. In 2022, Tesla and Waymo both expanded autonomous taxi services in U.S. cities, competing for regulatory approval and public trust. The U.S. Department of Defense's Project Maven and China's AI-enabled surveillance programs have been cited as examples of military AI rivalry. In 2025, Microsoft hired several senior engineers from Google DeepMind, highlighting the ongoing "talent poaching" competition in the AI sector. == Risks and concerns == Critics warn that unrestrained competition in AI can undermine safety, ethics, and governance. Concerns include the proliferation of biased or unsafe models, escalation in autonomous weapons, and reduced cooperation on safety standards.
Linear belief function
Linear belief functions are an extension of the Dempster–Shafer theory of belief functions to the case when variables of interest are continuous. Examples of such variables include financial asset prices, portfolio performance, and other antecedent and consequent variables. The theory was originally proposed by Arthur P. Dempster in the context of Kalman Filters and later was elaborated, refined, and applied to knowledge representation in artificial intelligence and decision making in finance and accounting by Liping Liu. == Concept == A linear belief function intends to represent our belief regarding the location of the true value as follows: We are certain that the truth is on a so-called certainty hyperplane but we do not know its exact location; along some dimensions of the certainty hyperplane, we believe the true value could be anywhere from –∞ to +∞ and the probability of being at a particular location is described by a normal distribution; along other dimensions, our knowledge is vacuous, i.e., the true value is somewhere from –∞ to +∞ but the associated probability is unknown. A belief function in general is defined by a mass function over a class of focal elements, which may have nonempty intersections. A linear belief function is a special type of belief function in the sense that its focal elements are exclusive, parallel sub-hyperplanes over the certainty hyperplane and its mass function is a normal distribution across the sub-hyperplanes. Based on the above geometrical description, Shafer and Liu propose two mathematical representations of a LBF: a wide-sense inner product and a linear functional in the variable space, and as their duals over a hyperplane in the sample space. Monney proposes still another structure called Gaussian hints. Although these representations are mathematically neat, they tend to be unsuitable for knowledge representation in expert systems. == Knowledge representation == A linear belief function can represent both logical and probabilistic knowledge for three types of variables: deterministic such as an observable or controllable, random whose distribution is normal, and vacuous on which no knowledge bears. Logical knowledge is represented by linear equations, or geometrically, a certainty hyperplane. Probabilistic knowledge is represented by a normal distribution across all parallel focal elements. In general, assume X is a vector of multiple normal variables with mean μ and covariance Σ. Then, the multivariate normal distribution can be equivalently represented as a moment matrix: M ( X ) = ( μ Σ ) . {\displaystyle M(X)=\left({\begin{array}{{20}c}\mu \\\Sigma \end{array}}\right).} If the distribution is non-degenerate, i.e., Σ has a full rank and its inverse exists, the moment matrix can be fully swept: M ( X → ) = ( μ Σ − 1 − Σ − 1 ) {\displaystyle M({\vec {X}})=\left({\begin{array}{{20}c}\mu \Sigma ^{-1}\\-\Sigma ^{-1}\end{array}}\right)} Except for normalization constant, the above equation completely determines the normal density function for X. Therefore, M ( X → ) {\displaystyle M({\vec {X}})} represents the probability distribution of X in the potential form. These two simple matrices allow us to represent three special cases of linear belief functions. First, for an ordinary normal probability distribution M(X) represents it. Second, suppose one makes a direct observation on X and obtains a value μ. In this case, since there is no uncertainty, both variance and covariance vanish, i.e., Σ = 0. Thus, a direct observation can be represented as: M ( X ) = ( μ 0 ) {\displaystyle M(X)=\left({\begin{array}{{20}c}\mu \\0\end{array}}\right)} Third, suppose one is completely ignorant about X. This is a very thorny case in Bayesian statistics since the density function does not exist. By using the fully swept moment matrix, we represent the vacuous linear belief functions as a zero matrix in the swept form follows: M ( X → ) = [ 0 0 ] {\displaystyle M({\vec {X}})=\left[{\begin{array}{{20}c}0\\0\end{array}}\right]} One way to understand the representation is to imagine complete ignorance as the limiting case when the variance of X approaches to ∞, where one can show that Σ−1 = 0 and hence M ( X → ) {\displaystyle M({\vec {X}})} vanishes. However, the above equation is not the same as an improper prior or normal distribution with infinite variance. In fact, it does not correspond to any unique probability distribution. For this reason, a better way is to understand the vacuous linear belief functions as the neutral element for combination (see later). To represent the remaining three special cases, we need the concept of partial sweeping. Unlike a full sweeping, a partial sweeping is a transformation on a subset of variables. Suppose X and Y are two vectors of normal variables with the joint moment matrix: M ( X , Y ) = [ μ 1 Σ 11 Σ 21 μ 2 Σ 12 Σ 22 ] {\displaystyle M(X,Y)=\left[{\begin{array}{{20}c}{\begin{array}{{20}c}\mu _{1}\\\Sigma _{11}\\\Sigma _{21}\end{array}}&{\begin{array}{{20}c}\mu _{2}\\\Sigma _{12}\\\Sigma _{22}\end{array}}\end{array}}\right]} Then M(X, Y) may be partially swept. For example, we can define the partial sweeping on X as follows: M ( X → , Y ) = [ μ 1 ( Σ 11 ) − 1 − ( Σ 11 ) − 1 Σ 21 ( Σ 11 ) − 1 μ 2 − μ 1 ( Σ 11 ) − 1 Σ 12 ( Σ 11 ) − 1 Σ 12 Σ 22 − Σ 21 ( Σ 11 ) − 1 Σ 12 ] {\displaystyle M({\vec {X}},Y)=\left[{\begin{array}{{20}c}{\begin{array}{{20}c}\mu _{1}(\Sigma _{11})^{-1}\\-(\Sigma _{11})^{-1}\\\Sigma _{21}(\Sigma _{11})^{-1}\end{array}}&{\begin{array}{{20}c}\mu _{2}-\mu _{1}(\Sigma _{11})^{-1}\Sigma _{12}\\(\Sigma _{11})^{-1}\Sigma _{12}\\\Sigma _{22}-\Sigma _{21}(\Sigma _{11})^{-1}\Sigma _{12}\end{array}}\end{array}}\right]} If X is one-dimensional, a partial sweeping replaces the variance of X by its negative inverse and multiplies the inverse with other elements. If X is multidimensional, the operation involves the inverse of the covariance matrix of X and other multiplications. A swept matrix obtained from a partial sweeping on a subset of variables can be equivalently obtained by a sequence of partial sweepings on each individual variable in the subset and the order of the sequence does not matter. Similarly, a fully swept matrix is the result of partial sweepings on all variables. We can make two observations. First, after the partial sweeping on X, the mean vector and covariance matrix of X are respectively μ 1 ( Σ 11 ) − 1 {\displaystyle \mu _{1}(\Sigma _{11})^{-1}} and − ( Σ 11 ) − 1 {\displaystyle -(\Sigma _{11})^{-1}} , which are the same as that of a full sweeping of the marginal moment matrix of X. Thus, the elements corresponding to X in the above partial sweeping equation represent the marginal distribution of X in potential form. Second, according to statistics, μ 2 − μ 1 ( Σ 11 ) − 1 Σ 12 {\displaystyle \mu _{2}-\mu _{1}(\Sigma _{11})^{-1}\Sigma _{12}} is the conditional mean of Y given X = 0; Σ 22 − Σ 21 ( Σ 11 ) − 1 Σ 12 {\displaystyle \Sigma _{22}-\Sigma _{21}(\Sigma _{11})^{-1}\Sigma _{12}} is the conditional covariance matrix of Y given X = 0; and ( Σ 11 ) − 1 Σ 12 {\displaystyle (\Sigma _{11})^{-1}\Sigma _{12}} is the slope of the regression model of Y on X. Therefore, the elements corresponding to Y indices and the intersection of X and Y in M ( X → , Y ) {\displaystyle M({\vec {X}},Y)} represents the conditional distribution of Y given X = 0. These semantics render the partial sweeping operation a useful method for manipulating multivariate normal distributions. They also form the basis of the moment matrix representations for the three remaining important cases of linear belief functions, including proper belief functions, linear equations, and linear regression models. === Proper linear belief functions === For variables X and Y, assume there exists a piece of evidence justifying a normal distribution for variables Y while bearing no opinions for variables X. Also, assume that X and Y are not perfectly linearly related, i.e., their correlation is less than 1. This case involves a mix of an ordinary normal distribution for Y and a vacuous belief function for X. Thus, we represent it using a partially swept matrix as follows: M ( X → , Y ) = [ 0 0 0 μ 2 0 Σ 22 ] {\displaystyle M({\vec {X}},Y)=\left[{\begin{array}{{20}c}{\begin{array}{{20}c}0\\0\\0\end{array}}&{\begin{array}{{20}c}\mu _{2}\\0\\\Sigma _{22}\\\end{array}}\end{array}}\right]} This is how we could understand the representation. Since we are ignorant on X, we use its swept form and set μ 1 ( Σ 11 ) − 1 = 0 {\displaystyle \mu _{1}(\Sigma _{11})^{-1}=0} and − ( Σ 11 ) − 1 = 0 {\displaystyle -(\Sigma _{11})^{-1}=0} . Since the correlation between X and Y is less than 1, the regression coefficient of X on Y approaches to 0 when the variance of X approaches to ∞. Therefore, ( Σ 11 ) − 1 Σ 12 = 0 {\displaystyle (\Sigma _{11})^{-1}\Sigma _{12}=0} . Similarly, one can prove that μ 1 ( Σ 11 ) − 1 Σ 12 = 0 {\displaystyle \mu _{1}(\Sigma _{11})^{-1}\Sigma _{12}=0} and Σ 21 ( Σ 11 ) −