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What you will learn in Causal AI

by Robert Osazuwa Ness
Foreword by Lindsay Edwards

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Chapter by chapter

Concepts explained in this book

• structural causal modelA causal model is a statistical model built on assumptions about the causal process that generates data and supports reasoning about interventions.
• causal graphical modelA causal generative model is a generative probabilistic model whose structure represents causal mechanisms and supports intervention-based generation of distributions.
• assignment functionAn assignment function is a deterministic rule in a structural causal model that determines an endogenous variable from its causal parents and exogenous inputs.
• conditional independenceA conditional independence constraint is a requirement that specified variables be independent given specified values or events of other variables.
• exogenous variableAn exogenous cause is a causal input whose variation originates outside the endogenous system represented by a structural causal model.
• causal identificationCausal identifiability is the property that a causal quantity is uniquely determined by the available distribution and causal assumptions.
• causal inferenceA method for estimating or reasoning about the causal effects of interventions or exposures from data.
• causationCausal association is an association between variables that has a causal interpretation under an intervention and the accompanying causal assumptions.
• interventionAn intervention is an externally imposed change to a variable or process intended to alter what happens in a causal system.
• confounderA hidden confounder is an unobserved common cause of the treatment and outcome that can bias causal effect estimation.
• conditional probabilityConditional probability is the probability distribution of one or more variables given specified values or events for other variables.
• counterfactual reasoningCounterfactual reasoning is reasoning about hypothetical outcomes under conditions different from those that actually occurred.
• endogenous variableAn endogenous variable is a variable in a structural causal model whose value is determined by an assignment function from its causes and exogenous inputs.
• d-separationD-separation is a graphical criterion for determining whether conditioning on variables blocks all paths of statistical or causal association in a directed acyclic graph.
• counterfactual outcomeA counterfactual statement is a conditional hypothetical statement whose factual conditions overlap with a hypothetical condition or hypothetical outcome.
• language modelA large language model is a deep neural language model with billions of parameters trained on vast amounts of text and other data.
• causal Markov kernelA causal Markov kernel is a conditional probability factor associated with a node in a causal graph and interpreted as the mechanism determining that node from its parents.
• variational inferenceVariational inference approximates a target probability distribution by optimizing the parameters of a tractable distribution or stochastic process.
• variational autoencoderA variational autoencoder is a generative neural-network model that combines an encoder-defined approximating distribution with a decoder-defined generative distribution over observed data and latent variables.
• latent variableA latent variable is an unobserved variable in a statistical model, here represented as a lower-dimensional continuous encoding of image causes.
• backdoor adjustment estimandThe backdoor adjustment formula computes an interventional outcome distribution by summing or integrating conditional outcome probabilities over a valid adjustment set weighted by its marginal distribution.
• colliderA collider is a variable in a causal graph with directed paths entering it from at least two other variables, so those paths converge at that variable.
• encoderAn encoder is the component of a representation-learning model that maps observed inputs, optionally with labels or other conditioning variables, to a latent representation.
• data-generating processA data-generating process is the underlying mechanism or probability distribution that produces observed data and its causal relationships.
• decodeA neural decoder is a neural network component that maps a latent or compressed representation to an output representation.
• pyro.doAn intervention operator is a formal operation on a structural causal model that replaces a target variable's assignment function with a specified intervention value.
• parameter estimationParameter estimation is the process of inferring unknown model parameter values from observed data or substantive assumptions.
• statistical independence testA statistical independence test is a hypothesis-testing procedure that evaluates whether variables are statistically independent or conditionally independent from observed data.
• probability of necessityThe probability of necessity is the probability that an outcome would not have occurred without an observed treatment or cause among units that received the cause and experienced the outcome.
• graph surgeryGraph mutilation is a causal-graph transformation that removes or modifies edges to represent an intervention or altered causal mechanism.
• probabilistic reasoningProbabilistic reasoning is reasoning under uncertainty using probability distributions and probabilistic relationships.
• counterfactual notationCounterfactual notation represents the value or distribution of a variable under a hypothetical intervention or condition.
• causal assumptionA causal assumption is a constraint about how variables or events are causally related that guides causal inference.
• probability distributionA distribution specifies the probabilities or densities of the possible values of a random variable or set of variables.
• parameterA weight parameter is a learned numerical coefficient that controls the contribution of an input in a machine-learning function or model.
• fully bayes parameter estimationBayesian estimation combines a prior distribution with observed data to produce a posterior distribution or estimate of parameters.
• directed acyclic graphDAG is a graph class representing a directed acyclic graph and operations on its nodes and edges.
• parallel world graphA parallel world graph is a graph representation that places variables from multiple counterfactual worlds together without necessarily permitting the same separation relations as a counterfactual graph.
• canonical distributionA canonical distribution is a standard mathematically characterized probability distribution used as a primitive for probabilistic models.
• causal decision theoryA causal decision formulation selects actions by evaluating their effects under interventions in a causal model.
• probability of causationA probability of causation is a counterfactual probability describing whether a binary cause was necessary, sufficient, both, enabling, or disabling for a binary outcome.
• instrumental variableAn instrumental variable is a variable that affects an exposure or treatment and influences an outcome only through that exposure or treatment under specified causal assumptions.
• independence of mechanismIndependence of mechanism is the causal principle that a mechanism generating one variable is invariant to changes in other, distinct mechanisms or in the variable's cause distribution.
• guide functionA guide function is a Pyro stochastic function that defines the variational approximating process used for inference.
• average treatment effect estimationAn average treatment effect is the average difference in an outcome between treated and untreated potential outcomes across a population.
• causal machine learningA causal machine learning model combines machine learning with representations of causal relationships to answer intervention or counterfactual questions.
• counterfactual graphA counterfactual graph is a graph formed by collapsing equivalent nodes across the intervention worlds of a parallel world graph while retaining nodes relevant to a counterfactual query.
• adjustment setAn adjustment set is a set of variables that satisfies the backdoor criterion for estimating a causal effect between specified variables.
• random variableA random variable is a variable whose possible values and uncertainty are described by a probability distribution.
• linear regressionA linear model represents an outcome as a linear or affine function of predictor variables and model parameters, typically with an error term.
• causal mechanismA causal mechanism is the physical or probabilistic process through which causes generate effects in a system.
• stochastic interventionA stochastic intervention is an intervention that assigns or changes variables according to a probability distribution rather than fixing them to a single value.
• reward signalA reward is a scalar feedback signal that evaluates the desirability of an agent's outcome or action in reinforcement learning.
• monotonicityMonotonicity is the property of a function or relationship preserving direction of order, so that increasing an input does not reverse the ordered output relationship.
• probability of necessity and sufficiencyThe probability of necessity and sufficiency is the probability that an outcome would be absent under absence of a binary cause and present under presence of that cause.
• Markov kernelA Markov kernel is a conditional probability factor obtained by simplifying a probability factorization under a Markov assumption.
• maximum likelihood estimationMaximum likelihood estimation selects parameter values that maximize the probability or likelihood of the observed data.
• do functionA do-operator is a formal causal operator that represents externally setting a variable to a specified value while estimating the resulting distributions.
• stochastic variational inferenceStochastic variational inference is a scalable variational inference method that optimizes an approximating distribution using stochastic optimization, often with minibatches of data.
• conditional average treatment effect estimationA conditional average treatment effect is the average causal effect of an intervention within a specified subgroup or conditional covariate configuration.
• causal representation learningA causal representation is a learned representation that encodes causal relationships or mechanisms among variables or events.
• samplingA sampling process is a procedure that generates observed outcomes by drawing values from a probability distribution.
• hypothesis testingA hypothesis test is a statistical procedure for evaluating whether observed data provide sufficient evidence against a specified null hypothesis.
• probability of sufficiencyThe probability of sufficiency is the probability that an intervention would produce an outcome among units that did not receive the intervention and did not experience the outcome.
• partial identificationPartial identification is an inference setting in which observed data and assumptions restrict a causal quantity to a set or range rather than a unique value.
• TabularCPDTabularCPD is a pgmpy class for representing a conditional probability distribution in tabular form.
• causal hierarchy theoremA causal hierarchy is an ordering of information levels corresponding to increasingly strong causal queries, including association, intervention, and counterfactual reasoning.
• PyroPyro is an open-source probabilistic programming library for modeling and performing Bayesian and probabilistic inference with Python and PyTorch.
• causal questionA causal query is a formally specified question about causal effects, interventions, or other quantities defined by a causal model.
• effect of treatment on the treatedThe effect of treatment on the treated is the average causal effect among units that received treatment, comparing their treated outcomes with the outcomes they would have had under no treatment.
• causal abstractionCausal abstraction is the creation of high-level variables from lower-level variables while preserving causal assumptions about the data-generating process.
• causal discoveryCausal discovery is the identification of causal relationships from data, including determining what causes what.
• uniform distributionA uniform distribution assigns equal probability density throughout a specified interval or equal probability to each value in a finite set.
• mediatorA mediator is an intermediate variable through which part or all of the causal influence of an exposure on an outcome operates.
• factual conditionA factual condition is an observed fact, event, or piece of evidence that restricts the population or scope described by a conditional query.
• refutationRefutation analysis is a sensitivity-analysis technique that tests whether estimated causal conclusions are vulnerable to violations of their assumptions.
• semi-supervised learningSemi-supervision is a learning setting that combines labeled and unlabeled examples to train a model.
• chain rule of probabilityThe chain rule of probability factorizes a joint probability distribution into a product of conditional probabilities.
• probabilistic modelProbabilistic modeling is the construction of models that represent uncertainty and data-generating processes with probability distributions.
• observational joint distributionAn observational distribution is the probability distribution of variables generated under ordinary observation without externally intervening on the system.
• do-calculusDo-calculus is a formal system of rules for transforming causal and interventional probability expressions using graphical-model assumptions.
• Markov equivalence classA Markov equivalence class is a set of causal directed acyclic graphs that encode the same conditional independence relationships through d-separation.
• joint probability distributionA full joint distribution is the joint probability distribution over both observed and latent variables in a model.
• functional constraintA functional constraint is a testable restriction on an observed probability distribution implied by a causal graph, including one obtained after accounting for latent variables.
• expected utilityExpected utility is the probability-weighted average utility of possible outcomes under a specified action, policy, or intervention.
• coefficientA coefficient is a numerical parameter multiplying a predictor or parent variable in a linear function.
• front-door estimandA front-door adjustment estimand is a causal effect quantity identified through a mediator that satisfies front-door identification conditions.
• conditional hypotheticalA conditional hypothetical statement is a hypothetical statement that specifies one or more hypothetical conditions.
• generationA generative model is a model that represents a process for generating observations, typically by specifying a joint or conditional probability distribution.
• distribution functionA probability distribution function assigns probability values or probability densities to outcomes of a random variable.
• probability theoryProbability is a numerical measure between zero and one expressing the likelihood of an event or value under a specified model or evidence.
• causal consistencyCausal consistency is the principle that a unit's observed outcome under its received treatment equals its corresponding potential outcome.
• hypothetical conditionA hypothetical condition is an alternative value or circumstance imposed or considered in a counterfactual analysis.
• ignorabilityThe ignorability assumption states that a causal variable and its potential outcomes are conditionally independent given specified confounders.
• estimandAn estimand is the population quantity or probabilistic target that a statistical analysis seeks to determine.
• causal Markov propertyThe global Markov property states that d-separation in a causal graph implies conditional independence in its joint probability distribution.
• switch strategyA switch strategy is a game strategy in which the contestant abandons the initially selected door for the other unopened door.
• approximate distributionA variational distribution is a parameterized probability distribution chosen to approximate a target distribution in variational inference.
• counterfactual inference algorithmA counterfactual inference algorithm is a computational procedure for determining the outcome that would have occurred under an alternative condition using a causal model.
• populationA population is the complete set of individuals, units, or observations about which an analysis seeks to generalize.
• reparameterization trickA reparameterization trick rewrites a stochastic variable as a deterministic function of newly introduced exogenous random variables while preserving its marginal distribution.
• policyA policy is a rule or mapping that selects an action from observed variables, potentially through a stochastic process.
• d-connectionD-connection is the graphical relation in which a path between variables remains active under a conditioning set and indicates dependence in the modeled distribution.
• counterfactual identificationCounterfactual identification is the process of determining whether a counterfactual quantity can be uniquely computed from available distributions and causal assumptions.
• independent and identically distributedIndependent and identically distributed data are observations that are mutually independent and share the same probability distribution.
• independenceIndependence is a probabilistic relationship in which knowing the value of one variable does not change the distribution of another variable.
• digital imageAn image is a two-dimensional arrangement of visual measurements, such as pixel intensities, representing a scene, object, or symbol.
• single-world intervention graphA single-world intervention graph is a causal graph formed by splitting intervention nodes to represent counterfactual variables under a single-world assumption.
• potential outcomes frameworkA potential outcome is the outcome that would result for a unit under a specified treatment or condition, whether or not that outcome is observed.
• abductionAbduction is the inference step that uses observed facts to infer the latent circumstances or exogenous values that could have produced them.
• subpopulationA subpopulation is a defined subset of a broader population that shares specified characteristics or conditions.
• variable eliminationVariable elimination computes a target distribution by summing out selected variables from factors represented by a graphical model.
• DoWhyDoWhy is an open-source Python library for causal inference that structures causal analysis around modeling assumptions, identification, estimation, and refutation.
• parameter modularityParameter modularity is the property that parameters representing distinct causal mechanisms are separated so that updating one does not update the others.
• probabilistic programmingProbabilistic programming is the construction of executable probabilistic models by combining ordinary program logic with random-variable sampling and inference.
• conditional probability tableA conditional probability table is a tabular representation of the probability distribution of a variable for combinations of values of its parent variables.
• causal sufficiencyA causally sufficient set is a set of variables that contains all common causes of every pair of variables in the set.
• binary counterfactual queryA binary counterfactual is a counterfactual involving binary treatment conditions and binary potential outcomes.
• markov chain monte carloMarkov chain Monte Carlo is a family of sampling algorithms that generate a Markov chain whose limiting distribution is a target probability distribution.
• Monte Carlo simulationA Monte Carlo method is a randomized computational technique that estimates quantities by repeatedly sampling from a probability distribution.
• backdoor identificationBackdoor adjustment is a causal inference technique that estimates an intervention effect by conditioning on variables that block all noncausal backdoor paths between treatment and outcome.
• possible worldA possible world is a hypothetical state of affairs representing how a causal system might have unfolded under different conditions.
• associationThe association level, also called the associational level, is the first level of the causal hierarchy and concerns statistical dependence and observational questions.
• parallel worlds accountThe parallel worlds account is an approach to counterfactuals that represents an alternative outcome in a world differing from the observed world in a causal event.
• statistical sampleA random sample is a dataset formed by repeatedly selecting individuals from a population according to a random sampling mechanism.
• alternative hypothesisAn alternative hypothesis is a statistical claim that contrasts with the null hypothesis and represents the effect or pattern a test may support.
• null hypothesisA null hypothesis is a statistical claim, typically asserting the absence of an effect or association, that is evaluated by a hypothesis test.
• minimal d-separating setA minimal d-separating set is a d-separating set whose blocking property is not represented by a smaller relevant d-separating set.
• probabilistic graphical modelA probabilistic graphical model represents a joint probability distribution using a graph whose structure encodes relationships among variables.
• attributionAttribution is the assignment of a causal source and corresponding credit or blame for an outcome.
• randomizationRandom assignment, or randomization, allocates units or intervention values to conditions according to a random mechanism.
• collider biasCollider bias is bias caused by conditioning on a collider or its descendant, thereby inducing an association between its causes.
• bayesian statisticsBayesian inference is statistical inference that updates prior beliefs about unknown quantities using observed data and a likelihood model.
• randomnessRandomness is the absence of a deterministic specification of an outcome, represented in a model by stochastic variation or random variables.
• regretRegret is the difference between the utility or reward realized by an action and the utility or reward that would have been realized by an alternative action.
• single-world assumptionA single-world assumption is the assumption that a variable’s natural, unintervened value can be known and that an intervention does not alter the system in a way that changes that natural value.
• interpretabilityExplainable AI is the development of artificial intelligence systems whose predictions or decisions can be interpreted or explained to people.
• ChiRhoChiRho is a causal probabilistic programming library that extends Pyro with handlers and primitives for causal modeling and inference.
• probability densityA probability density is a nonnegative function describing the distribution of a continuous random variable, whose integral over all possible values is one.
• evidence lower boundThe evidence lower bound is a variational objective that lower-bounds the expected log-likelihood of observed data.
• Kullback–Leibler divergenceKullback–Leibler divergence quantifies the difference between two probability distributions.
• objective functionA loss function is an objective function that quantifies the cost or error associated with model parameters or predictions.
• causal frameA causal frame is a selection of the variables and factors considered relevant to answering a causal query.
• node-splitting operationA node-splitting operation replaces an intervention node in a causal graph with separate components representing its intervened and natural values.
• location parameterA location parameter is a distribution parameter that determines the position of a distribution along its variable's scale.
• shape parameterA shape parameter changes the form of a probability distribution without merely shifting or uniformly rescaling it.
• parametric assumptionA parametric assumption specifies a functional or distributional form that restricts the parameters or relationships in a causal model.
• transition probabilityA transition function in reinforcement learning specifies the probability distribution of the next state given the current state and action.
• estimatorAn estimator is a statistic or rule that produces an estimate of an unknown population quantity.
• additive structural causal modelAn additive structural causal model is a structural causal model in which each variable is assigned by a function of its causal parents plus an additive exogenous disturbance.
• determinismDeterminism is the philosophical view that a system's complete state and governing laws uniquely determine its future state.
• indicator functionAn indicator function returns one when a specified condition holds and zero otherwise.
• backdoor pathA backdoor path is a path between two variables that begins with an arrow into the putative cause and represents a potentially noncausal association through common causes.
• robustnessRobustness is the property of maintaining acceptable model behavior when deployment conditions differ from training conditions.
• anti-causal learningAnti-causal learning is a learning setting in which features are caused by the label or target variable.
• causal learningCausal learning is a learning setting in which features or inputs cause the label or target variable.
• proxy metricA proxy variable is an observed variable that indirectly encodes information about another attribute of interest.
• fairnessFairness is the property that an algorithmic or statistical system treats individuals or groups according to specified equity criteria.
• decision makingDecision-making is the process of selecting actions or policies among alternatives, including selection performed by automated systems.
• attentionAttention is a neural-network mechanism that assigns different weights to parts of an input when computing a representation or producing an output.
• inductive biasAn inductive bias is an assumption implicit in the choice of a machine learning algorithm that constrains how it generalizes from data.
• goodness-of-fit measureA goodness-of-fit statistic quantifies how well a model fits or accounts for observed data.
• prior probabilityA prior distribution is a probability distribution expressing beliefs about unknown quantities before observing the relevant data.
• latent variable modelA latent variable model is a statistical model that represents unobserved variables alongside observed variables.
• plate modelingPlate modeling is a probabilistic-model visualization technique that groups repeated variables into nested graphical substructures.
• plate notationPlate notation is a graphical notation that represents repeated variables or subgraphs with enclosing rectangles and repetition indices.
• observational dataObservational data are measurements collected without the investigator assigning the treatment or intervention being studied.
• Newcomb's paradoxNewcomb's paradox, also called Newcomb's problem, is a decision-theory thought experiment in which an agent chooses between boxes whose contents have been predicted in advance.
• backdoor criterionThe backdoor criterion is a graphical condition requiring a covariate set to block every backdoor path from treatment to outcome while excluding descendants of the treatment.
• stateA state is the representation of an agent's situation that determines the relevant decision and future dynamics in a sequential decision process.
• p-valueA p-value is the probability, under a null hypothesis, of obtaining a test statistic at least as extreme as the observed one.
• reinforcement learningA machine-learning paradigm in which an agent learns actions or policies by interacting with an environment and receiving rewards or penalties.
• categorical distributionCategorical is a Pyro distribution object representing a categorical probability distribution.
• BARTBART is a denoising sequence-to-sequence transformer language model developed for generating text from input text.
• fine-tuningFine-tuning is the process of adapting a pretrained model by continuing parameter training on task-specific or domain-specific data.
• GPT-2GPT-2 is an autoregressive transformer language model developed by OpenAI for generating and completing text.
• causal llmA causal LLM is a large language model embedded in a causal graphical-model structure so that its conditional text distributions represent causal mechanisms.
• encoder-decoder architectureA sequence-to-sequence model maps an input sequence to an output sequence, often for tasks such as translation, summarization, or conditional generation.
• uplift modelingUplift modeling is the estimation of individual or subgroup differences in outcomes caused by applying a treatment rather than withholding it.
• algorithmic recourseAlgorithmic recourse identifies an achievable or actionable change to an individual's inputs that would produce a preferred machine-learning decision.
• Bayesian modelBayesian modeling is the construction and analysis of probabilistic models using prior distributions, likelihoods, and posterior distributions.
• accept/reject algorithmAn accept/reject algorithm repeatedly samples from a proposal process and retains only outcomes satisfying a specified condition or acceptance rule.
• stay strategyA stay strategy is a game strategy in which the contestant retains the initially selected door.
• actionAn action is a decision or operation selected by an agent or performed by an individual in a given situation.
• intervention distributionAn intervention distribution is the probability distribution over variables induced by externally setting or changing one or more variables in a causal model.
• experimentationAn A/B test is an experiment that compares outcomes under two assigned variants, commonly labeled A and B.
• Y0y0 is a Python library for representing and manipulating causal graphs, including graphs for counterfactual and parallel-world reasoning.
• id*ID* is a graphical identification algorithm that identifies counterfactual queries from observational and experimental data.
• idcIDC is a graphical identification algorithm that extends ID to queries conditioned on evidence.
• Markov propertyA Markov assumption is the assumption that each variable in an ordered chain depends only on the variable immediately preceding it, conditional on that predecessor.
• estimator consistencyEstimator consistency is the property that an estimate approaches the true parameter value as the amount of data increases.
• cross-validationCross-validation is a model evaluation technique that assesses predictive performance on data withheld from training.
• likelihoodLikelihood is the probability of observed data viewed as a function of a candidate model parameter value.
• log-likelihoodLog-likelihood is the natural logarithm of the likelihood of observed data under candidate parameter values.
• negative log-likelihoodThe negative log-likelihood loss function is the negative logarithm of the likelihood and is minimized to estimate parameters.
• posterior probabilityA posterior distribution is the conditional probability distribution of unknown quantities given observed data and prior assumptions.
• regularizationRegularization is a modeling technique that adds a penalty or constraint to an optimization objective to discourage undesirable parameter values.
• weight decayL2 regularization is a regularization method that penalizes the sum of squared parameter values.
• causal reinforcement learning algorithmA reinforcement-learning paradigm that uses causal relationships and counterfactual reasoning to identify actions that produce rewards and assign credit to them.
• probability of disablementThe probability of disablement is the conditional probability that an outcome would be absent if a binary cause were absent, conditioned only on the outcome having occurred.
• probability of enablementThe probability of enablement is the conditional probability that an outcome would occur if a binary cause were present, conditioned only on the outcome having been absent.
• analysis of varianceAnalysis of variance is a statistical technique for testing whether the means of multiple groups differ.
• naive bayes classifierA naive Bayes classifier is a probabilistic classifier that applies Bayes’ rule while assuming conditional independence among features given the class.
• do-notationDo-notation represents the probability distribution resulting from externally setting one or more variables to specified values.
• decision ruleA decision rule is a rule that selects an action based on a utility, loss, or outcome criterion.
• causal structureA causal structure is the organization of variables and directed causal relationships represented by a causal model.
• propensity scoreA propensity score is a function of observed covariates representing the probability or treatment-predictive summary associated with receiving a treatment.
• alignment trainingAI alignment is the task of making an artificial intelligence system's behavior consistent with human goals and preferences.
• program inductionProgram induction is the task of inferring a program that explains observed input-output behavior.
• SCM equivalence classAn SCM equivalence class is a set of structural causal models that are consistent with the same causal DAG and joint probability distribution.
• chi-squared distributionA chi-squared distribution is a continuous probability distribution commonly associated with sums of squared standard normal variables and categorical test statistics.
• permutation testA permutation test constructs a null distribution by repeatedly rearranging observed data or labels instead of relying on a canonical parametric test distribution.
• Bernoulli distributionThe Bernoulli distribution is a discrete probability distribution over a binary outcome with a success-probability parameter.
• log probabilityA log probability is the natural logarithm of a probability value.
• discrete variableA discrete outcome is one of a finite or countable set of distinct values that a discrete random variable can take.
• actual worldAn actual world is the causal-model representation corresponding to the observed condition and its realized evidence.
• causal faithfulnessA faithfulness violation is a case in which a conditional independence in the joint distribution does not correspond to a d-separation in the ground-truth causal graph.
• causal mathematicsCausal mathematics is the formal probabilistic or expectation-based representation of a causal question.
• commodification of inferenceCommodification of inference is a software abstraction pattern in which a framework handles inference mechanics after a user specifies an estimation problem through assumptions and high-level inputs.
• conjugacyA conjugate prior is a prior distribution whose posterior distribution has the same mathematical form as the prior when combined with a specified likelihood.
• Dirichlet distributionThe Dirichlet distribution is a continuous probability distribution over vectors of nonnegative values that sum to one.
• biasStatistical bias is a systematic difference between an estimator's expected value or observed pattern and the relevant population quantity or target relationship.
• instrumental variable estimandAn instrumental variable estimand is a causal quantity identified using an instrument that affects treatment while satisfying the relevant exclusion and independence assumptions.
• marginal probabilityA marginal probability is the probability of a variable or event obtained without conditioning on other variables by summing or integrating them out.
• statistical modelingStatistical modeling represents data-generating relationships with probabilistic or deterministic mathematical models.
• causal factorA causal factor is a variable or property that contributes to producing an effect in a system.
• conditional expectationA conditional expectation is the expected value of a quantity given observed values or events.
• interventional expectationAn interventional expectation is the expected value of a quantity under an externally imposed intervention on one or more variables.
• decomposabilityDecomposability is the property of a model being separable into components that correspond to smaller causal or domain relationships.
• propensity score methodPropensity score matching is a causal adjustment technique that pairs treated and untreated individuals with similar propensity scores before comparing outcomes.
• action stepAn action step is the stage of counterfactual inference that represents a hypothetical condition as an ideal intervention in a modified causal graph.
• indicator variableA binary indicator is a variable that represents whether a specified condition holds using two possible values.
• BayesianNetworkBayesianNetwork is a pgmpy class for representing a Bayesian network and its probabilistic dependencies.
• categorical dataA categorical variable is a variable whose values are labels from a finite set of categories rather than numerical measurements.
• importance samplingImportance sampling estimates expectations under a target distribution using samples from another distribution and likelihood-based weights.
• uncertaintyUncertainty is the state of incomplete or imperfect knowledge about the value, state, or outcome of a variable or process.
• lost causeA lost cause is a person who would experience the undesirable outcome regardless of whether the promotion or treatment is applied.
• persuadableA persuadable is a person who would experience the favorable outcome if a promotion or treatment were applied but the unfavorable outcome otherwise.
• sleeping dogA sleeping dog is a person who would experience the favorable outcome without a promotion or treatment but the unfavorable outcome if it were applied.
• sure thingA sure thing is a person who would experience the favorable outcome regardless of whether the promotion or treatment is applied.
• Poisson distributionThe Poisson distribution gives the probabilities of the number of events occurring in a fixed interval under a constant-rate event process.
• protected attributeA protected attribute is a characteristic, such as a demographic trait, whose use or effects in automated decisions are subject to fairness requirements.
• Markov decision processA Markov decision process is a sequential decision model in which an agent selects actions in states and receives rewards, with future dynamics determined by the current state and action.
• causal invarianceCausal invariance is the stability of a causal relationship across changes in environments or data distributions.
• hallucinationA hallucination is confidently generated model output that contains false, unsupported, or incorrectly inferred content.
• customer churnChurn rate is the rate at which customers or subscribers discontinue a service during a specified period.
• probability weighting methodA probability weighting method estimates target-distribution quantities by assigning weights to samples according to their probabilities.
• identify_effectidentify_effect is a DoWhy method that derives candidate causal estimands from a causal model and its observed variables.
• constraint-based algorithmA constraint-based algorithm infers candidate causal graphs by testing conditional-independence constraints in observed data and comparing them with graphical separation constraints.
• testable implicationA testable implication is an observable statistical consequence that follows from a model's assumptions and can be checked against data.
• tokenA token is a sequence of characters treated as one unit of input or output by a language model.
• softmax decision ruleA softmax decision rule selects actions probabilistically according to a softmax transformation of their scores or utilities.
• consistency ruleThe consistency rule is a causal inference principle stating that a hypothetical outcome equals the observed outcome when the hypothetical condition matches the factual condition.
• function classA function class is a specified set of functions from which a model's assignment or prediction function may be selected.
• nonlinear additive noise modelA nonlinear additive noise model is an additive structural causal model with nonlinear assignment functions and additive exogenous noise.
• exogenous variable distributionAn exogenous variable distribution is a probability distribution specifying uncertainty about the value of an exogenous variable in a structural causal model.

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