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Tuesday, December 24, 2024

Probabilistic programming

From Wikipedia, the free encyclopedia

Probabilistic programming (PP) is a programming paradigm in which probabilistic models are specified and inference for these models is performed automatically. It represents an attempt to unify probabilistic modeling and traditional general purpose programming in order to make the former easier and more widely applicable. It can be used to create systems that help make decisions in the face of uncertainty.

Programming languages used for probabilistic programming are referred to as "probabilistic programming languages" (PPLs).

Applications

Probabilistic reasoning has been used for a wide variety of tasks such as predicting stock prices, recommending movies, diagnosing computers, detecting cyber intrusions and image detection. However, until recently (partially due to limited computing power), probabilistic programming was limited in scope, and most inference algorithms had to be written manually for each task.

Nevertheless, in 2015, a 50-line probabilistic computer vision program was used to generate 3D models of human faces based on 2D images of those faces. The program used inverse graphics as the basis of its inference method, and was built using the Picture package in Julia. This made possible "in 50 lines of code what used to take thousands".

The Gen probabilistic programming library (also written in Julia) has been applied to vision and robotics tasks.

More recently, the probabilistic programming system Turing.jl has been applied in various pharmaceutical and economics applications.

Probabilistic programming in Julia has also been combined with differentiable programming by combining the Julia package Zygote.jl with Turing.jl. 

Probabilistic programming languages are also commonly used in Bayesian cognitive science to develop and evaluate models of cognition. 

Probabilistic programming languages

PPLs often extend from a basic language. For instance, Turing.jl is based on Julia, Infer.NET is based on .NET Framework, while PRISM extends from Prolog. However, some PPLs, such as WinBUGS, offer a self-contained language that maps closely to the mathematical representation of the statistical models, with no obvious origin in another programming language.

The language for WinBUGS was implemented to perform Bayesian computation using Gibbs Sampling and related algorithms. Although implemented in a relatively unknown programming language (Component Pascal), this language permits Bayesian inference for a wide variety of statistical models using a flexible computational approach. The same BUGS language may be used to specify Bayesian models for inference via different computational choices ("samplers") and conventions or defaults, using a standalone program WinBUGS (or related R packages, rbugs and r2winbugs) and JAGS (Just Another Gibbs Sampler, another standalone program with related R packages including rjags, R2jags, and runjags). More recently, other languages to support Bayesian model specification and inference allow different or more efficient choices for the underlying Bayesian computation, and are accessible from the R data analysis and programming environment, e.g.: Stan, NIMBLE and NUTS. The influence of the BUGS language is evident in these later languages, which even use the same syntax for some aspects of model specification.

Several PPLs are in active development, including some in beta test. Two popular tools are Stan and PyMC.

Relational

A probabilistic relational programming language (PRPL) is a PPL specially designed to describe and infer with probabilistic relational models (PRMs).

A PRM is usually developed with a set of algorithms for reducing, inference about and discovery of concerned distributions, which are embedded into the corresponding PRPL.

Probabilistic logic programming

Probabilistic logic programming is a programming paradigm that extends logic programming with probabilities.

Most approaches to probabilistic logic programming are based on the distribution semantics, which splits a program into a set of probabilistic facts and a logic program. It defines a probability distribution on interpretations of the Herbrand universe of the program.

List of probabilistic programming languages

This list summarises the variety of PPLs that are currently available, and clarifies their origins.

Name Extends from Host language
Analytica
C++
bayesloop Python Python
Bean Machine PyTorch Python
Venture Scheme C++
BayesDB SQLite, Python
PRISM B-Prolog
Infer.NET .NET Framework .NET Framework
diff-SAT Answer set programming, SAT (DIMACS CNF)
PSQL SQL
BUGS
Component Pascal
Dyna Prolog
Figaro Scala Scala
ProbLog Prolog Python
ProBT C++, Python
Stan BUGS C++
Hakaru Haskell Haskell
BAli-Phy (software) Haskell C++
ProbCog
Java, Python
PyMC Python Python
Rainier Scala Scala
greta TensorFlow R
pomegranate Python Python
Lea Python Python
WebPPL JavaScript JavaScript
Picture Julia Julia
Turing.jl Julia Julia
Gen Julia Julia
Edward TensorFlow Python
TensorFlow Probability TensorFlow Python
Edward2 TensorFlow Probability Python
Pyro PyTorch Python
NumPyro JAX Python
Birch
C++
PSI
D
Blang

MultiVerse Python Python
Anglican Clojure Clojure

Difficulty

  • Reasoning about variables as probability distributions causes difficulties for novice programmers, but these difficulties can be addressed through use of Bayesian network visualizations and graphs of variable distributions embedded within the source code editor.
  • As many PPLs rely on the specification of priors on the variables of interest, specifying informed priors is often difficult for novices. In some cases, libraries such as PyMC provide automated methods to find the parameterization of informed priors.
  • Bounded rationality

    From Wikipedia, the free encyclopedia
    https://en.wikipedia.org/wiki/Bounded_rationality

    Bounded rationality is the idea that rationality is limited when individuals make decisions, and under these limitations, rational individuals will select a decision that is satisfactory rather than optimal.

    Limitations include the difficulty of the problem requiring a decision, the cognitive capability of the mind, and the time available to make the decision. Decision-makers, in this view, act as satisficers, seeking a satisfactory solution, with everything that they have at the moment rather than an optimal solution. Therefore, humans do not undertake a full cost-benefit analysis to determine the optimal decision, but rather, choose an option that fulfills their adequacy criteria.

    Some models of human behavior in the social sciences assume that humans can be reasonably approximated or described as rational entities, as in rational choice theory or Downs' political agency model. The concept of bounded rationality complements the idea of rationality as optimization, which views decision-making as a fully rational process of finding an optimal choice given the information available. Therefore, bounded rationality can be said to address the discrepancy between the assumed perfect rationality of human behaviour (which is utilised by other economics theories), and the reality of human cognition. In short, bounded rationality revises notions of perfect rationality to account for the fact that perfectly rational decisions are often not feasible in practice because of the intractability of natural decision problems and the finite computational resources available for making them. The concept of bounded rationality continues to influence (and be debated in) different disciplines, including political science, economics, psychology, law, philosophy, and cognitive science.

    Background and motivation

    Bounded rationality was coined by Herbert A. Simon, where it was proposed as an alternative basis for the mathematical and neoclassical economic modelling of decision-making, as used in economics, political science, and related disciplines. Many economics models assume that agents are on average rational, and can in large quantities be approximated to act according to their preferences in order to maximise utility. With bounded rationality, Simon's goal was "to replace the global rationality of economic man with a kind of rational behavior that is compatible with the access to information and the computational capacities that are actually possessed by organisms, including man, in the kinds of environments in which such organisms exist." Soon after the term bounded rationality appeared, studies in the topic area began examining the issue in depth. A study completed by Allais in 1953 began to generate ideas of the irrationality of decision making as he found that given preferences, individuals will not always choose the most rational decision and therefore the concept of rationality was not always reliable in economic predictions.

    In Models of Man, Simon argues that most people are only partly rational, and are irrational in the remaining part of their actions. In another work, he states "boundedly rational agents experience limits in formulating and solving complex problems and in processing (receiving, storing, retrieving, transmitting) information". Simon used the analogy of a pair of scissors, where one blade represents "cognitive limitations" of actual humans and the other the "structures of the environment", illustrating how minds compensate for limited resources by exploiting known structural regularity in the environment.

    Simon describes a number of dimensions along which classical models of rationality can be made somewhat more realistic, while remaining within the vein of fairly rigorous formalization. These include:

    • limiting the types of utility functions
    • recognizing the costs of gathering and processing information
    • the possibility of having a vector or multi-valued utility function

    Simon suggests that economic agents use heuristics to make decisions rather than a strict rigid rule of optimization. They do this because of the complexity of the situation. An example of behaviour inhibited by heuristics can be seen when comparing the cognitive strategies utilised in simple situations (e.g. tic-tac-toe), in comparison to strategies utilised in difficult situations (e.g. chess). Both games, as defined by game theory economics, are finite games with perfect information, and therefore equivalent. However, within chess, mental capacities and abilities are a binding constraint, therefore optimal choices are not a possibility. Thus, in order to test the mental limits of agents, complex problems, such as those within chess, should be studied to test how individuals work around their cognitive limits, and what behaviours or heuristics are used to form solutions

    Anchoring and adjustment are types of heuristics that give some explanation to bounded rationality and why decision makers do not make rational decisions. A study undertaken by Zenko et al. showed that the amount of physical activity completed by decision makers was able to be influenced by anchoring and adjustment as most decision makers would typically be considered irrational and would unlikely do the amount of physical activity instructed and it was shown that these decision makers use anchoring and adjustment to decide how much exercise they will complete.

    Other heuristics that are closely related to the concept of bounded rationality include the availability heuristic and representativeness heuristic. The availability heuristic refers to how people tend to overestimate the likelihood of events that are easily brought to mind, such as vivid or recent experiences. This can lead to biased judgments based on incomplete or unrepresentative information. The representativeness heuristic states that people often judge the probability of an event based on how closely it resembles a typical or representative case, ignoring other relevant factors like base rates or sample size. These mental shortcuts and systematic errors in thinking demonstrate how people's decision-making abilities are limited and often deviate from perfect rationality.  

    Example

    An example of bounded rationality in individuals would be a customer who made a suboptimal decision to order some food at the restaurant because they felt rushed by the waiter who was waiting beside the table. Another example is a trader who would make a moderate and risky decision to trade their stock due to time pressure and imperfect information of the market at that time.

    In organisational context, a CEO cannot make fully rational decisions in an ad-hoc situation because their cognition was overwhelmed by a lot of information in that tense situation. The CEO also needs to take time to process all the information given to them, but due to the limited time and fast decision making needed, they will disregard some information in determining the decision.

    Bounded rationality can have significant effects on political decision-making, voter behavior, and policy outcomes. A prominent example of this is heuristic-based voting. According to the theory of bounded rationality, individuals have limited time, information, and cognitive resources to make decisions. In the context of voting, this means that most voters cannot realistically gather and process all available information about candidates, issues, and policies. Even if such information were available, the time and effort required to analyze it would be prohibitively high for many voters. As a result, voters often resort to heuristics, which allow voters to make decisions based on cues like party affiliation, candidate appearance, or single-issue positions, rather than engaging in a comprehensive evaluation of all relevant factors. For example, a voter who relies on the heuristic of party affiliation may vote for a candidate whose policies do not actually align with their interests, simply because the candidate belongs to their preferred party.  

    Model extensions

    As decision-makers have to make decisions about how and when to decide, Ariel Rubinstein proposed to model bounded rationality by explicitly specifying decision-making procedures as decision-makers with the same information are also not able to analyse the situation equally thus reach the same rational decision. Rubinstein argues that consistency in reaching final decision for the same level of information must factor in the decision making procedure itself. This puts the study of decision procedures on the research agenda.

    Gerd Gigerenzer stated that decision theorists, to some extent, have not adhered to Simon's original ideas. Rather, they have considered how decisions may be crippled by limitations to rationality, or have modeled how people might cope with their inability to optimize. Gigerenzer proposes and shows that simple heuristics often lead to better decisions than theoretically optimal procedures. Moreover, Gigerenzer claimed, agents react relative to their environment and use their cognitive processes to adapt accordingly.

    Huw Dixon later argued that it may not be necessary to analyze in detail the process of reasoning underlying bounded rationality. If we believe that agents will choose an action that gets them close to the optimum, then we can use the notion of epsilon-optimization, which means we choose our actions so that the payoff is within epsilon of the optimum. If we define the optimum (best possible) payoff as , then the set of epsilon-optimizing options S(ε) can be defined as all those options s such that:

    The notion of strict rationality is then a special case (ε=0). The advantage of this approach is that it avoids having to specify in detail the process of reasoning, but rather simply assumes that whatever the process is, it is good enough to get near to the optimum.

    From a computational point of view, decision procedures can be encoded in algorithms and heuristics. Edward Tsang argues that the effective rationality of an agent is determined by its computational intelligence. Everything else being equal, an agent that has better algorithms and heuristics could make more rational (closer to optimal) decisions than one that has poorer heuristics and algorithms.

    Tshilidzi Marwala and Evan Hurwitz in their study on bounded rationality observed that advances in technology (e.g. computer processing power because of Moore's law, artificial intelligence, and big data analytics) expand the bounds that define the feasible rationality space. Because of this expansion of the bounds of rationality, machine automated decision making makes markets more efficient.

    The model of bounded rationality also extends to bounded self-interest, in which humans are sometimes willing to forsake their own self-interests for the benefits of others due to incomplete information that the individuals have at the time being. This is something that had not been considered in earlier economic models.

    The theory of rational inattention, an extension of bounded rationality, studied by Christopher Sims, found that decisions may be chosen with incomplete information as opposed to affording the cost to receive complete information. This shows that decision makers choose to endure bounded rationality.

    On the other hand, another extension came from the notion of bounded rationality and was explained by Ulrich Hoffrage and Torsten Reimer in their studies of a "fast and frugal heuristic approach". The studies explained that complete information sometimes is not needed as there are easier and simpler ways to reach the same optimal outcome. However, this approach which is usually known as the gaze heuristic was explained to be the theory for non-complex decision making only.

    Behavioral Economics

    Bounded rationality attempts to address assumption points discussed within neoclassical economics theory during the 1950s. This theory assumes that the complex problem, the way in which the problem is presented, all alternative choices, and a utility function, are all provided to decision-makers in advance, where this may not be realistic. This was widely used and accepted for a number of decades, however economists realised some disadvantages exist in utilising this theory. This theory did not consider how problems are initially discovered by decision-makers, which could have an impact on the overall decision. Additionally, personal values, the way in which alternatives are discovered and created, and the environment surrounding the decision-making process are also not considered when using this theory. Alternatively, bounded rationality focuses on the cognitive ability of the decision-maker and the factors which may inhibit optimal decision-making. Additionally, placing a focus on organisations rather than focusing on markets as neoclassical economics theory does, bounded rationality is also the basis for many other economics theories (e.g. organisational theory) as it emphasises that the "...performance and success of an organisation is governed primarily by the psychological limitations of its members..." as stated by John D.W. Morecroft (1981).[27]

    One concept closely related to the idea of bounded rationality is nudging. The connection between nudging and bounded rationality lies in the fact that nudges are designed to help people overcome the cognitive limitations and biases that arise from their bounded rationality. Nudging involves designing choice architectures that guide people towards making better decisions without limiting their freedom of choice. The concept was popularized by Richard Thaler and Cass Sunstein in their 2008 book "Nudge: Improving Decisions About Health, Wealth, and Happiness." As nudging has become more popular in the last decade, governments around the world and nongovernmental organizations like the United Nations have established behavioral insights teams or incorporated nudging into their policy-making processes.

    One way nudges are used is with the aim of simplifying complex decisions by presenting information in a clear and easily understandable format, reducing the cognitive burden on individuals. Nudges can also be designed to counteract common heuristics and biases, such as the default bias (people's tendency to stick with the default option). For example, with adequate other policies in place, making posthumous organ donation the default option with an opt-out provision has been shown to increase actual donation rates. Moreover, in cases where the information needed to make an informed decision is incomplete, nudges can provide the relevant information. For instance, displaying the calorie content of menu items can help people make healthier food choices. Nudges can also guide people towards satisfactory options when they are unable or unwilling to invest the time and effort to find the optimal choice. For example, providing a limited set of well-designed investment options in a retirement plan can help people make better financial decisions.

    In economic models based on behavioral economics, implementing bounded rationality implies finding replacements for utility maximization and profit maximization as used in conventional general equilibrium models. Stock-flow consistent models (SFC) and agent-based models (ABM) often implement that agents follow a sequence of simple rule-of-thumb behavior instead of an optimization procedure. Other dynamic models interpret bounded rationality as “looking for the direction of improvement“ such that agents use a gradient climbing approach to increase their utility.

    Principles of Boundedness

    In addition to bounded rationality, bounded willpower and bounded selfishness are two other key concepts in behavioral economics that challenge the traditional neoclassical economic assumption of perfectly rational, self-interested, and self-disciplined individuals. 

    Bounded willpower refers to the idea that people often have difficulty following through on their long-term plans and intentions due to limited self-control and the tendency to prioritize short-term desires. This can lead to problems like procrastination, impulsive spending, and unhealthy lifestyle choices. The concept of bounded willpower is closely related to the idea of hyperbolic discounting, which describes how people tend to value immediate rewards more highly than future ones, leading to inconsistent preferences over time.

    While traditional economic models assume that people are primarily motivated by self-interest, bounded selfishness suggests that people also have social preferences and care about factors such as fairness, reciprocity, and the well-being of others. This concept helps explain phenomena like charitable giving, cooperation in social dilemmas, and the existence of social norms. However, people's concern for others is often bounded in the sense that it is limited in scope and can be influenced by factors such as in-group favoritism and emotional distance.

    Together, these three concepts form the core of behavioral economics and have been used to develop more realistic models of human decision-making and behavior. By recognizing the limitations and biases that people face in their daily lives, behavioral economists aim to design policies, institutions, and choice architectures that can help people make better decisions and achieve their long-term goals.

    In psychology

    The collaborative works of Daniel Kahneman and Amos Tversky expand upon Herbert A. Simon's ideas in the attempt to create a map of bounded rationality. The research attempted to explore the choices made by what was assumed as rational agents compared to the choices made by individuals optimal beliefs and their satisficing behaviour. Kahneman cites that the research contributes mainly to the school of psychology due to imprecision of psychological research to fit the formal economic models; however, the theories are useful to economic theory as a way to expand simple and precise models and cover diverse psychological phenomena. Three major topics covered by the works of Daniel Kahneman and Amos Tversky include heuristics of judgement, risky choice, and framing effect, which were a culmination of research that fit under what was defined by Herbert A. Simon as the psychology of bounded rationality. In contrast to the work of Simon; Kahneman and Tversky aimed to focus on the effects bounded rationality had on simple tasks which therefore placed more emphasis on errors in cognitive mechanisms irrespective of the situation. The study undertaken by Kahneman found that emotions and the psychology of economic decisions play a larger role in the economics field than originally thought. The study focused on the emotions behind decision making such as fear and personal likes and dislikes and found these to be significant factors in economic decision making.

    Bounded rationality is also shown to be useful in negotiation techniques as shown in research undertaken by Dehai et al. that negotiations done using bounded rationality techniques by labourers and companies when negotiating a higher wage for workers were able to find an equal solution for both parties.

    Influence on social network structure

    Recent research has shown that bounded rationality of individuals may influence the topology of the social networks that evolve among them. In particular, Kasthurirathna and Piraveenan have shown that in socio-ecological systems, the drive towards improved rationality on average might be an evolutionary reason for the emergence of scale-free properties. They did this by simulating a number of strategic games on an initially random network with distributed bounded rationality, then re-wiring the network so that the network on average converged towards Nash equilibria, despite the bounded rationality of nodes. They observed that this re-wiring process results in scale-free networks. Since scale-free networks are ubiquitous in social systems, the link between bounded rationality distributions and social structure is an important one in explaining social phenomena.

    Problem solving

    From Wikipedia, the free encyclopedia
    https://en.wikipedia.org/wiki/Problem_solving

    Problem solving
    is the process of achieving a goal by overcoming obstacles, a frequent part of most activities. Problems in need of solutions range from simple personal tasks (e.g. how to turn on an appliance) to complex issues in business and technical fields. The former is an example of simple problem solving (SPS) addressing one issue, whereas the latter is complex problem solving (CPS) with multiple interrelated obstacles. Another classification of problem-solving tasks is into well-defined problems with specific obstacles and goals, and ill-defined problems in which the current situation is troublesome but it is not clear what kind of resolution to aim for. Similarly, one may distinguish formal or fact-based problems requiring psychometric intelligence, versus socio-emotional problems which depend on the changeable emotions of individuals or groups, such as tactful behavior, fashion, or gift choices.

    Solutions require sufficient resources and knowledge to attain the goal. Professionals such as lawyers, doctors, programmers, and consultants are largely problem solvers for issues that require technical skills and knowledge beyond general competence. Many businesses have found profitable markets by recognizing a problem and creating a solution: the more widespread and inconvenient the problem, the greater the opportunity to develop a scalable solution.

    There are many specialized problem-solving techniques and methods in fields such as science, engineering, business, medicine, mathematics, computer science, philosophy, and social organization. The mental techniques to identify, analyze, and solve problems are studied in psychology and cognitive sciences. Also widely researched are the mental obstacles that prevent people from finding solutions; problem-solving impediments include confirmation bias, mental set, and functional fixedness.

    Definition

    The term problem solving has a slightly different meaning depending on the discipline. For instance, it is a mental process in psychology and a computerized process in computer science. There are two different types of problems: ill-defined and well-defined; different approaches are used for each. Well-defined problems have specific end goals and clearly expected solutions, while ill-defined problems do not. Well-defined problems allow for more initial planning than ill-defined problems. Solving problems sometimes involves dealing with pragmatics (the way that context contributes to meaning) and semantics (the interpretation of the problem). The ability to understand what the end goal of the problem is, and what rules could be applied, represents the key to solving the problem. Sometimes a problem requires abstract thinking or coming up with a creative solution.

    Problem solving has two major domains: mathematical problem solving and personal problem solving. Each concerns some difficulty or barrier that is encountered.

    Psychology

    Problem solving in psychology refers to the process of finding solutions to problems encountered in life. Solutions to these problems are usually situation- or context-specific. The process starts with problem finding and problem shaping, in which the problem is discovered and simplified. The next step is to generate possible solutions and evaluate them. Finally a solution is selected to be implemented and verified. Problems have an end goal to be reached; how you get there depends upon problem orientation (problem-solving coping style and skills) and systematic analysis.

    Mental health professionals study the human problem-solving processes using methods such as introspection, behaviorism, simulation, computer modeling, and experiment. Social psychologists look into the person-environment relationship aspect of the problem and independent and interdependent problem-solving methods. Problem solving has been defined as a higher-order cognitive process and intellectual function that requires the modulation and control of more routine or fundamental skills.

    Empirical research shows many different strategies and factors influence everyday problem solving. Rehabilitation psychologists studying people with frontal lobe injuries have found that deficits in emotional control and reasoning can be re-mediated with effective rehabilitation and could improve the capacity of injured persons to resolve everyday problems. Interpersonal everyday problem solving is dependent upon personal motivational and contextual components. One such component is the emotional valence of "real-world" problems, which can either impede or aid problem-solving performance. Researchers have focused on the role of emotions in problem solving, demonstrating that poor emotional control can disrupt focus on the target task, impede problem resolution, and lead to negative outcomes such as fatigue, depression, and inertia. In conceptualization,human problem solving consists of two related processes: problem orientation, and the motivational/attitudinal/affective approach to problematic situations and problem-solving skills. People's strategies cohere with their goals and stem from the process of comparing oneself with others.

    Cognitive sciences

    Among the first experimental psychologists to study problem solving were the Gestaltists in Germany, such as Karl Duncker in The Psychology of Productive Thinking (1935). Perhaps best known is the work of Allen Newell and Herbert A. Simon.

    Experiments in the 1960s and early 1970s asked participants to solve relatively simple, well-defined, but not previously seen laboratory tasks. These simple problems, such as the Tower of Hanoi, admitted optimal solutions that could be found quickly, allowing researchers to observe the full problem-solving process. Researchers assumed that these model problems would elicit the characteristic cognitive processes by which more complex "real world" problems are solved.

    An outstanding problem-solving technique found by this research is the principle of decomposition.

    Computer science

    Much of computer science and artificial intelligence involves designing automated systems to solve a specified type of problem: to accept input data and calculate a correct or adequate response, reasonably quickly. Algorithms are recipes or instructions that direct such systems, written into computer programs.

    Steps for designing such systems include problem determination, heuristics, root cause analysis, de-duplication, analysis, diagnosis, and repair. Analytic techniques include linear and nonlinear programming, queuing systems, and simulation. A large, perennial obstacle is to find and fix errors in computer programs: debugging.

    Logic

    Formal logic concerns issues like validity, truth, inference, argumentation, and proof. In a problem-solving context, it can be used to formally represent a problem as a theorem to be proved, and to represent the knowledge needed to solve the problem as the premises to be used in a proof that the problem has a solution.

    The use of computers to prove mathematical theorems using formal logic emerged as the field of automated theorem proving in the 1950s. It included the use of heuristic methods designed to simulate human problem solving, as in the Logic Theory Machine, developed by Allen Newell, Herbert A. Simon and J. C. Shaw, as well as algorithmic methods such as the resolution principle developed by John Alan Robinson.

    In addition to its use for finding proofs of mathematical theorems, automated theorem-proving has also been used for program verification in computer science. In 1958, John McCarthy proposed the advice taker, to represent information in formal logic and to derive answers to questions using automated theorem-proving. An important step in this direction was made by Cordell Green in 1969, who used a resolution theorem prover for question-answering and for such other applications in artificial intelligence as robot planning.

    The resolution theorem-prover used by Cordell Green bore little resemblance to human problem solving methods. In response to criticism of that approach from researchers at MIT, Robert Kowalski developed logic programming and SLD resolution, which solves problems by problem decomposition. He has advocated logic for both computer and human problem solving and computational logic to improve human thinking.

    Engineering

    When products or processes fail, problem solving techniques can be used to develop corrective actions that can be taken to prevent further failures. Such techniques can also be applied to a product or process prior to an actual failure event—to predict, analyze, and mitigate a potential problem in advance. Techniques such as failure mode and effects analysis can proactively reduce the likelihood of problems.

    In either the reactive or the proactive case, it is necessary to build a causal explanation through a process of diagnosis. In deriving an explanation of effects in terms of causes, abduction generates new ideas or hypotheses (asking "how?"); deduction evaluates and refines hypotheses based on other plausible premises (asking "why?"); and induction justifies a hypothesis with empirical data (asking "how much?"). The objective of abduction is to determine which hypothesis or proposition to test, not which one to adopt or assert. In the Peircean logical system, the logic of abduction and deduction contribute to our conceptual understanding of a phenomenon, while the logic of induction adds quantitative details (empirical substantiation) to our conceptual knowledge.

    Forensic engineering is an important technique of failure analysis that involves tracing product defects and flaws. Corrective action can then be taken to prevent further failures.

    Reverse engineering attempts to discover the original problem-solving logic used in developing a product by disassembling the product and developing a plausible pathway to creating and assembling its parts.

    Military science

    In military science, problem solving is linked to the concept of "end-states", the conditions or situations which are the aims of the strategy. Ability to solve problems is important at any military rank, but is essential at the command and control level. It results from deep qualitative and quantitative understanding of possible scenarios. Effectiveness in this context is an evaluation of results: to what extent the end states were accomplished. Planning is the process of determining how to effect those end states.

    Processes

    Some models of problem solving involve identifying a goal and then a sequence of subgoals towards achieving this goal. Andersson, who introduced the ACT-R model of cognition, modelled this collection of goals and subgoals as a goal stack in which the mind contains a stack of goals and subgoals to be completed, and a single task being carried out at any time.

    Knowledge of how to solve one problem can be applied to another problem, in a process known as transfer.

    Problem-solving strategies

    Problem-solving strategies are steps to overcoming the obstacles to achieving a goal. The iteration of such strategies over the course of solving a problem is the "problem-solving cycle".

    Common steps in this cycle include recognizing the problem, defining it, developing a strategy to fix it, organizing knowledge and resources available, monitoring progress, and evaluating the effectiveness of the solution. Once a solution is achieved, another problem usually arises, and the cycle starts again.

    Insight is the sudden aha! solution to a problem, the birth of a new idea to simplify a complex situation. Solutions found through insight are often more incisive than those from step-by-step analysis. A quick solution process requires insight to select productive moves at different stages of the problem-solving cycle. Unlike Newell and Simon's formal definition of a move problem, there is no consensus definition of an insight problem.

    Some problem-solving strategies include:

    Abstraction
    solving the problem in a tractable model system to gain insight into the real system
    Analogy
    adapting the solution to a previous problem which has similar features or mechanisms
    Brainstorming
    (especially among groups of people) suggesting a large number of solutions or ideas and combining and developing them until an optimum solution is found
    Bypasses
    transform the problem into another problem that is easier to solve, bypassing the barrier, then transform that solution back to a solution to the original problem.
    Critical thinking
    analysis of available evidence and arguments to form a judgement via rational, skeptical, and unbiased evaluation
    Divide and conquer
    breaking down a large, complex problem into smaller, solvable problems
    Help-seeking
    obtaining external assistance to deal with obstacles
    Hypothesis testing
    assuming a possible explanation to the problem and trying to prove (or, in some contexts, disprove) the assumption
    Lateral thinking
    approaching solutions indirectly and creatively
    Means-ends analysis
    choosing an action at each step to move closer to the goal
    Morphological analysis
    assessing the output and interactions of an entire system
    Observation / Question
    in the natural sciences an observation is an act or instance of noticing or perceiving and the acquisition of information from a primary source. A question is an utterance which serves as a request for information.
    Proof of impossibility
    try to prove that the problem cannot be solved. The point where the proof fails will be the starting point for solving it
    Reduction
    transforming the problem into another problem for which solutions exist
    Research
    employing existing ideas or adapting existing solutions to similar problems
    Root cause analysis
    identifying the cause of a problem
    Trial-and-error
    testing possible solutions until the right one is found

    Problem-solving methods

    Common barriers

    Common barriers to problem solving include mental constructs that impede an efficient search for solutions. Five of the most common identified by researchers are: confirmation bias, mental set, functional fixedness, unnecessary constraints, and irrelevant information.

    Confirmation bias

    Confirmation bias is an unintentional tendency to collect and use data which favors preconceived notions. Such notions may be incidental rather than motivated by important personal beliefs: the desire to be right may be sufficient motivation.

    Scientific and technical professionals also experience confirmation bias. One online experiment, for example, suggested that professionals within the field of psychological research are likely to view scientific studies that agree with their preconceived notions more favorably than clashing studies. According to Raymond Nickerson, one can see the consequences of confirmation bias in real-life situations, which range in severity from inefficient government policies to genocide. Nickerson argued that those who killed people accused of witchcraft demonstrated confirmation bias with motivation. Researcher Michael Allen found evidence for confirmation bias with motivation in school children who worked to manipulate their science experiments to produce favorable results.

    However, confirmation bias does not necessarily require motivation. In 1960, Peter Cathcart Wason conducted an experiment in which participants first viewed three numbers and then created a hypothesis in the form of a rule that could have been used to create that triplet of numbers. When testing their hypotheses, participants tended to only create additional triplets of numbers that would confirm their hypotheses, and tended not to create triplets that would negate or disprove their hypotheses.

    Mental set

    Mental set is the inclination to re-use a previously successful solution, rather than search for new and better solutions. It is a reliance on habit.

    It was first articulated by Abraham S. Luchins in the 1940s with his well-known water jug experiments. Participants were asked to fill one jug with a specific amount of water by using other jugs with different maximum capacities. After Luchins gave a set of jug problems that could all be solved by a single technique, he then introduced a problem that could be solved by the same technique, but also by a novel and simpler method. His participants tended to use the accustomed technique, oblivious of the simpler alternative. This was again demonstrated in Norman Maier's 1931 experiment, which challenged participants to solve a problem by using a familiar tool (pliers) in an unconventional manner. Participants were often unable to view the object in a way that strayed from its typical use, a type of mental set known as functional fixedness (see the following section).

    Rigidly clinging to a mental set is called fixation, which can deepen to an obsession or preoccupation with attempted strategies that are repeatedly unsuccessful. In the late 1990s, researcher Jennifer Wiley found that professional expertise in a field can create a mental set, perhaps leading to fixation.

    Groupthink, in which each individual takes on the mindset of the rest of the group, can produce and exacerbate mental set. Social pressure leads to everybody thinking the same thing and reaching the same conclusions.

    Functional fixedness

    Functional fixedness is the tendency to view an object as having only one function, and to be unable to conceive of any novel use, as in the Maier pliers experiment described above. Functional fixedness is a specific form of mental set, and is one of the most common forms of cognitive bias in daily life.

    As an example, imagine a man wants to kill a bug in his house, but the only thing at hand is a can of air freshener. He may start searching for something to kill the bug instead of squashing it with the can, thinking only of its main function of deodorizing.

    Tim German and Clark Barrett describe this barrier: "subjects become 'fixed' on the design function of the objects, and problem solving suffers relative to control conditions in which the object's function is not demonstrated." Their research found that young children's limited knowledge of an object's intended function reduces this barrier Research has also discovered functional fixedness in educational contexts, as an obstacle to understanding: "functional fixedness may be found in learning concepts as well as in solving chemistry problems."

    There are several hypotheses in regards to how functional fixedness relates to problem solving. It may waste time, delaying or entirely preventing the correct use of a tool.

    Unnecessary constraints

    Unnecessary constraints are arbitrary boundaries imposed unconsciously on the task at hand, which foreclose a productive avenue of solution. The solver may become fixated on only one type of solution, as if it were an inevitable requirement of the problem. Typically, this combines with mental set—clinging to a previously successful method.

    Visual problems can also produce mentally invented constraints. A famous example is the dot problem: nine dots arranged in a three-by-three grid pattern must be connected by drawing four straight line segments, without lifting pen from paper or backtracking along a line. The subject typically assumes the pen must stay within the outer square of dots, but the solution requires lines continuing beyond this frame, and researchers have found a 0% solution rate within a brief allotted time.

    This problem has produced the expression "think outside the box". Such problems are typically solved via a sudden insight which leaps over the mental barriers, often after long toil against them. This can be difficult depending on how the subject has structured the problem in their mind, how they draw on past experiences, and how well they juggle this information in their working memory. In the example, envisioning the dots connected outside the framing square requires visualizing an unconventional arrangement, which is a strain on working memory.

    Irrelevant information

    Irrelevant information is a specification or data presented in a problem that is unrelated to the solution. If the solver assumes that all information presented needs to be used, this often derails the problem solving process, making relatively simple problems much harder.

    For example: "Fifteen percent of the people in Topeka have unlisted telephone numbers. You select 200 names at random from the Topeka phone book. How many of these people have unlisted phone numbers?" The "obvious" answer is 15%, but in fact none of the unlisted people would be listed among the 200. This kind of "trick question" is often used in aptitude tests or cognitive evaluations. Though not inherently difficult, they require independent thinking that is not necessarily common. Mathematical word problems often include irrelevant qualitative or numerical information as an extra challenge.

    Avoiding barriers by changing problem representation

    The disruption caused by the above cognitive biases can depend on how the information is represented: visually, verbally, or mathematically. A classic example is the Buddhist monk problem:

    A Buddhist monk begins at dawn one day walking up a mountain, reaches the top at sunset, meditates at the top for several days until one dawn when he begins to walk back to the foot of the mountain, which he reaches at sunset. Making no assumptions about his starting or stopping or about his pace during the trips, prove that there is a place on the path which he occupies at the same hour of the day on the two separate journeys.

    The problem cannot be addressed in a verbal context, trying to describe the monk's progress on each day. It becomes much easier when the paragraph is represented mathematically by a function: one visualizes a graph whose horizontal axis is time of day, and whose vertical axis shows the monk's position (or altitude) on the path at each time. Superimposing the two journey curves, which traverse opposite diagonals of a rectangle, one sees they must cross each other somewhere. The visual representation by graphing has resolved the difficulty.

    Similar strategies can often improve problem solving on tests.

    Other barriers for individuals

    People who are engaged in problem solving tend to overlook subtractive changes, even those that are critical elements of efficient solutions. This tendency to solve by first, only, or mostly creating or adding elements, rather than by subtracting elements or processes is shown to intensify with higher cognitive loads such as information overload.

    Dreaming: problem solving without waking consciousness

    People can also solve problems while they are asleep. There are many reports of scientists and engineers who solved problems in their dreams. For example, Elias Howe, inventor of the sewing machine, figured out the structure of the bobbin from a dream.

    The chemist August Kekulé was considering how benzene arranged its six carbon and hydrogen atoms. Thinking about the problem, he dozed off, and dreamt of dancing atoms that fell into a snakelike pattern, which led him to discover the benzene ring. As Kekulé wrote in his diary,

    One of the snakes seized hold of its own tail, and the form whirled mockingly before my eyes. As if by a flash of lightning I awoke; and this time also I spent the rest of the night in working out the consequences of the hypothesis.

    There also are empirical studies of how people can think consciously about a problem before going to sleep, and then solve the problem with a dream image. Dream researcher William C. Dement told his undergraduate class of 500 students that he wanted them to think about an infinite series, whose first elements were OTTFF, to see if they could deduce the principle behind it and to say what the next elements of the series would be. He asked them to think about this problem every night for 15 minutes before going to sleep and to write down any dreams that they then had. They were instructed to think about the problem again for 15 minutes when they awakened in the morning.

    The sequence OTTFF is the first letters of the numbers: one, two, three, four, five. The next five elements of the series are SSENT (six, seven, eight, nine, ten). Some of the students solved the puzzle by reflecting on their dreams. One example was a student who reported the following dream:

    I was standing in an art gallery, looking at the paintings on the wall. As I walked down the hall, I began to count the paintings: one, two, three, four, five. As I came to the sixth and seventh, the paintings had been ripped from their frames. I stared at the empty frames with a peculiar feeling that some mystery was about to be solved. Suddenly I realized that the sixth and seventh spaces were the solution to the problem!

    With more than 500 undergraduate students, 87 dreams were judged to be related to the problems students were assigned (53 directly related and 34 indirectly related). Yet of the people who had dreams that apparently solved the problem, only seven were actually able to consciously know the solution. The rest (46 out of 53) thought they did not know the solution.

    Mark Blechner conducted this experiment and obtained results similar to Dement's. He found that while trying to solve the problem, people had dreams in which the solution appeared to be obvious from the dream, but it was rare for the dreamers to realize how their dreams had solved the puzzle. Coaxing or hints did not get them to realize it, although once they heard the solution, they recognized how their dream had solved it. For example, one person in that OTTFF experiment dreamed:

    There is a big clock. You can see the movement. The big hand of the clock was on the number six. You could see it move up, number by number, six, seven, eight, nine, ten, eleven, twelve. The dream focused on the small parts of the machinery. You could see the gears inside.

    In the dream, the person counted out the next elements of the series—six, seven, eight, nine, ten, eleven, twelve—yet he did not realize that this was the solution of the problem. His sleeping mindbrain solved the problem, but his waking mindbrain was not aware how.

    Albert Einstein believed that much problem solving goes on unconsciously, and the person must then figure out and formulate consciously what the mindbrain has already solved. He believed this was his process in formulating the theory of relativity: "The creator of the problem possesses the solution." Einstein said that he did his problem solving without words, mostly in images. "The words or the language, as they are written or spoken, do not seem to play any role in my mechanism of thought. The psychical entities which seem to serve as elements in thought are certain signs and more or less clear images which can be 'voluntarily' reproduced and combined."

    Cognitive sciences: two schools

    Problem-solving processes differ across knowledge domains and across levels of expertise. For this reason, cognitive sciences findings obtained in the laboratory cannot necessarily generalize to problem-solving situations outside the laboratory. This has led to a research emphasis on real-world problem solving, since the 1990s. This emphasis has been expressed quite differently in North America and Europe, however. Whereas North American research has typically concentrated on studying problem solving in separate, natural knowledge domains, much of the European research has focused on novel, complex problems, and has been performed with computerized scenarios.

    Europe

    In Europe, two main approaches have surfaced, one initiated by Donald Broadbent in the United Kingdom and the other one by Dietrich Dörner in Germany. The two approaches share an emphasis on relatively complex, semantically rich, computerized laboratory tasks, constructed to resemble real-life problems. The approaches differ somewhat in their theoretical goals and methodology. The tradition initiated by Broadbent emphasizes the distinction between cognitive problem-solving processes that operate under awareness versus outside of awareness, and typically employs mathematically well-defined computerized systems. The tradition initiated by Dörner, on the other hand, has an interest in the interplay of the cognitive, motivational, and social components of problem solving, and utilizes very complex computerized scenarios that contain up to 2,000 highly interconnected variables.

    North America

    In North America, initiated by the work of Herbert A. Simon on "learning by doing" in semantically rich domains, researchers began to investigate problem solving separately in different natural knowledge domains—such as physics, writing, or chess playing—rather than attempt to extract a global theory of problem solving. These researchers have focused on the development of problem solving within certain domains, that is on the development of expertise.

    Areas that have attracted rather intensive attention in North America include:

    • calculation
    • computer skills
    • game playing
    • lawyers' reasoning
    • managerial problem solving
    • mathematical problem solving
    • mechanical problem solving
    • personal problem solving
    • political decision making
    • problem solving in electronics
    • problem solving for innovations and inventions: TRIZ
    • reading
    • social problem solving
    • writing

    Characteristics of complex problems

    Complex problem solving (CPS) is distinguishable from simple problem solving (SPS). In SPS there is a singular and simple obstacle. In CPS there may be multiple simultaneous obstacles. For example, a surgeon at work has far more complex problems than an individual deciding what shoes to wear. As elucidated by Dietrich Dörner, and later expanded upon by Joachim Funke, complex problems have some typical characteristics, which include:

    • complexity (large numbers of items, interrelations, and decisions)
    • enumerability
    • heterogeneity
    • connectivity (hierarchy relation, communication relation, allocation relation)
    • dynamics (time considerations)
    • intransparency (lack of clarity of the situation)
      • commencement opacity
      • continuation opacity
    • polytely (multiple goals)
      • inexpressivenes
      • opposition
      • transience

    Collective problem solving

    People solve problems on many different levels—from the individual to the civilizational. Collective problem solving refers to problem solving performed collectively. Social issues and global issues can typically only be solved collectively.

    The complexity of contemporary problems exceeds the cognitive capacity of any individual and requires different but complementary varieties of expertise and collective problem solving ability.

    Collective intelligence is shared or group intelligence that emerges from the collaboration, collective efforts, and competition of many individuals.

    In collaborative problem solving people work together to solve real-world problems. Members of problem-solving groups share a common concern, a similar passion, and/or a commitment to their work. Members can ask questions, wonder, and try to understand common issues. They share expertise, experiences, tools, and methods. Groups may be fluid based on need, may only occur temporarily to finish an assigned task, or may be more permanent depending on the nature of the problems.

    For example, in the educational context, members of a group may all have input into the decision-making process and a role in the learning process. Members may be responsible for the thinking, teaching, and monitoring of all members in the group. Group work may be coordinated among members so that each member makes an equal contribution to the whole work. Members can identify and build on their individual strengths so that everyone can make a significant contribution to the task. Collaborative group work has the ability to promote critical thinking skills, problem solving skills, social skills, and self-esteem. By using collaboration and communication, members often learn from one another and construct meaningful knowledge that often leads to better learning outcomes than individual work.

    Collaborative groups require joint intellectual efforts between the members and involve social interactions to solve problems together. The knowledge shared during these interactions is acquired during communication, negotiation, and production of materials. Members actively seek information from others by asking questions. The capacity to use questions to acquire new information increases understanding and the ability to solve problems.

    In a 1962 research report, Douglas Engelbart linked collective intelligence to organizational effectiveness, and predicted that proactively "augmenting human intellect" would yield a multiplier effect in group problem solving: "Three people working together in this augmented mode [would] seem to be more than three times as effective in solving a complex problem as is one augmented person working alone".

    Henry Jenkins, a theorist of new media and media convergence, draws on the theory that collective intelligence can be attributed to media convergence and participatory culture. He criticizes contemporary education for failing to incorporate online trends of collective problem solving into the classroom, stating "whereas a collective intelligence community encourages ownership of work as a group, schools grade individuals". Jenkins argues that interaction within a knowledge community builds vital skills for young people, and teamwork through collective intelligence communities contributes to the development of such skills.

    Collective impact is the commitment of a group of actors from different sectors to a common agenda for solving a specific social problem, using a structured form of collaboration.

    After World War II the UN, the Bretton Woods organization, and the WTO were created. Collective problem solving on the international level crystallized around these three types of organization from the 1980s onward. As these global institutions remain state-like or state-centric it is unsurprising that they perpetuate state-like or state-centric approaches to collective problem solving rather than alternative ones.

    Crowdsourcing is a process of accumulating ideas, thoughts, or information from many independent participants, with aim of finding the best solution for a given challenge. Modern information technologies allow for many people to be involved and facilitate managing their suggestions in ways that provide good results. The Internet allows for a new capacity of collective (including planetary-scale) problem solving.

    Probabilistic programming

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