STIT logic (from seeing to it that) is a family of modal and branching-time logics for reasoning about agency and choice. A typical STIT operator has the form [ i s t i t : φ ] {\displaystyle [i\ {\mathsf {stit}}:\varphi ]} , usually read as "agent i {\displaystyle i} sees to it that φ {\displaystyle \varphi } ", and is interpreted in models where agents choose between alternative possible futures. STIT logics are used in action theory, deontic logic, epistemic logic, and the theory of intelligent agents to formalise notions such as "could have done otherwise", responsibility, joint action, and strategic ability in an indeterministic world. == Etymology == The acronym STIT comes from the English phrase "seeing to it that", introduced in influential work by Nuel Belnap and Michael Perloff on the logical analysis of agentive expressions. In this tradition, "to see to it that φ {\displaystyle \varphi } " is treated as a primitive agency operator, rather than being reduced to ordinary modal necessity. == History == Modern STIT logic arose in the 1980s in the context of branching-time semantics and formal theories of agency. Belnap and Perloff's article "Seeing to it that: A canonical form for agentives" introduced the idea of treating expressions of the form "agent i sees to it that φ" as a primitive modal operator, and analysed such sentences using a branching tree of moments and histories. This approach was further developed in a series of papers on indeterminism and agency and provided the conceptual core for later STIT formalisms. In the 1990s the basic formal systems of STIT logic were worked out. Horty and Belnap's influential paper on the deliberative STIT operator distinguished between a "Chellas" STIT that merely records the result of an agent's present choice and a "deliberative" STIT that requires the agent's choice to make a difference, and connected STIT with issues of action, omission, ability and obligation. Around the same time, Ming Xu proved completeness and decidability results for basic STIT systems, including a single-agent logic with Kripke-style semantics and axiomatizations for multi-agent deliberative STIT, thereby establishing STIT as a well-behaved normal modal framework. This early work was systematised in Belnap, Perloff and Xu's monograph Facing the Future: Agents and Choices in Our Indeterminist World, which presents a general branching-time semantics for individual and group STIT operators, discusses independence-of-agents conditions and articulates the metaphysical picture of an indeterministic "tree" of moments. At roughly the same time, Horty's book Agency and Deontic Logic developed deontic STIT logics in which obligations are tied to agents' available choices rather than to static states of affairs, and used the resulting systems to analyse "ought implies can", contrary-to-duty obligations and deontic paradoxes. These works helped to position STIT at the intersection of action theory, temporal logic and deontic logic. From the late 1990s and 2000s onward, STIT logics were combined with epistemic, temporal and strategic modalities. Broersen introduced complete STIT logics for knowledge and action and deontic-epistemic STIT systems that distinguish different modes of mens rea, with applications to responsibility and the specification of multi-agent systems. Work on group and coalitional agency investigated axiomatisations and complexity results for group STIT logics, and related STIT-based analyses of agency to coalition logic and alternating-time temporal logic (ATL) by exhibiting formal embeddings between the frameworks. Explicit temporal operators were added to STIT in so-called temporal STIT logics. Lorini proposed a temporal STIT with "next" and "until" operators along histories and showed how it can be applied to normative reasoning about ongoing behaviour and commitments. Ciuni and Lorini compared different semantics for temporal STIT, clarifying the relationships between branching-time, game-based and epistemic approaches, while Boudou and Lorini gave a semantics for temporal STIT based on concurrent game structures, thus strengthening links with standard models of multi-agent interaction used for ATL and strategy logic. In parallel, complexity-theoretic work by Balbiani, Herzig and Troquard and by Schwarzentruber and co-authors investigated the satisfiability and model-checking problems for various STIT fragments, showing for instance that many expressive group STIT logics are undecidable or of high computational complexity. In the 2010s, STIT ideas were combined with justification logic, imagination operators and refined deontic notions. Justification STIT logics, developed by Olkhovikov and others, merge explicit justifications with STIT-style agency so that producing a proof can itself be treated as an action that brings about knowledge, and they come with completeness and decidability results. Olkhovikov and Wansing introduced STIT imagination logics, together with axiomatic systems and tableau calculi, to model acts of voluntary imagining and their role in doxastic control. Other authors have proposed STIT-based logics of responsibility, blameworthiness and intentionality for use in philosophical and AI settings. Xu's survey article "Combinations of STIT with Ought and Know" (2015) reviews many of these developments and emphasises the interplay between deontic and epistemic STIT logics. Current research on STIT focuses on proof theory, automated reasoning and richer expressive resources. Lyon and van Berkel, building on earlier work on labelled calculi for STIT, have developed cut-free sequent systems and proof-search algorithms that yield syntactic decision procedures for a range of deontic and non-deontic multi-agent STIT logics and support applications such as duty checking and compliance checking in autonomous systems. Sawasaki has proposed first-order cstit-based STIT logics that can distinguish de re and de dicto readings of agency statements and has proved strong completeness results for Hilbert systems over finite models, moving the STIT programme beyond the purely propositional level. Further work investigates interpreted-system and computationally grounded semantics for STIT and its extensions in order to model the behaviour of autonomous agents in multi-agent settings, and proposes STIT-based semantics for epistemic notions based on patterns of information disclosure in interactive systems. == Branching-time semantics == STIT logics are usually interpreted over branching-time models. A standard STIT frame consists of: a non-empty set of moments T {\displaystyle T} , partially ordered by < {\displaystyle <} so that ( T , < ) {\displaystyle (T,<)} forms a tree (every pair of moments with a common predecessor has a greatest lower bound); a set of histories, each history being a maximal linearly ordered subset of T {\displaystyle T} ; a non-empty set of agents A g {\displaystyle Ag} ; for each agent i ∈ A g {\displaystyle i\in Ag} and moment m {\displaystyle m} , a choice function c h o i c e i m {\displaystyle {\mathsf {choice}}_{i}^{m}} that partitions the set of histories passing through m {\displaystyle m} into choice cells. The idea is that a moment represents a time at which choices are made, and histories represent complete possible future courses of events. At each moment, each agent's choice corresponds to selecting one of the available cells of histories determined by their choice function. Formulas are evaluated at pairs ( m , h ) {\displaystyle (m,h)} of a moment and a history through that moment (sometimes written m / h {\displaystyle m/h} ). A valuation assigns truth-values to atomic propositions at such indices; Boolean connectives are interpreted pointwise as in Kripke-style modal logic. == Chellas and deliberative STIT operators == Several STIT operators have been distinguished in the literature. A common approach uses two closely related operators, often called Chellas STIT and deliberative STIT. Let H m {\displaystyle H_{m}} be the set of histories passing through a moment m {\displaystyle m} , and write H m {\displaystyle H_{m}} ⟦ φ ⟧ m = { h ∈ H m ∣ M , m / h ⊨ φ } {\displaystyle {\text{⟦}}\varphi {\text{⟧}}_{m}=\{h\in H_{m}\mid M,m/h\models \varphi \}} for the set of histories at m {\displaystyle m} where φ {\displaystyle \varphi } holds. The Chellas STIT operator, often written [ i c s t i t : φ ] {\displaystyle [i\ {\mathsf {cstit}}:\varphi ]} , is given by M , m / h ⊨ [ i c s t i t : φ ] iff c h o i c e i m ( h ) ⊆ ⟦ φ ⟧ m . {\displaystyle M,m/h\models [i\ {\mathsf {cstit}}:\varphi ]\quad {\text{iff}}\quad {\mathsf {choice}}_{i}^{m}(h)\subseteq {\text{⟦}}\varphi {\text{⟧}}_{m}.} Intuitively, agent i {\displaystyle i} sees to it that φ {\displaystyle \varphi } if φ {\displaystyle \varphi } holds at all histories compatible with their present choice. The deliberative STIT operator, [ i d s t i t : φ ] {\displaystyle [i\ {\mathsf {dstit}}:\varphi ]} , adds
Bayesian programming
Bayesian programming is a formalism and a methodology for having a technique to specify probabilistic models and solve problems when less than the necessary information is available. Edwin T. Jaynes proposed that probability could be considered as an alternative and an extension of logic for rational reasoning with incomplete and uncertain information. In his founding book Probability Theory: The Logic of Science he developed this theory and proposed what he called "the robot," which was not a physical device, but an inference engine to automate probabilistic reasoning—a kind of Prolog for probability instead of logic. Bayesian programming is a formal and concrete implementation of this "robot". Bayesian programming may also be seen as an algebraic formalism to specify graphical models such as, for instance, Bayesian networks, dynamic Bayesian networks, Kalman filters or hidden Markov models. Indeed, Bayesian programming is more general than Bayesian networks and has a power of expression equivalent to probabilistic factor graphs. == Formalism == A Bayesian program is a means of specifying a family of probability distributions. The constituent elements of a Bayesian program are presented below: Program { Description { Specification ( π ) { Variables Decomposition Forms Identification (based on δ ) Question {\displaystyle {\text{Program}}{\begin{cases}{\text{Description}}{\begin{cases}{\text{Specification}}(\pi ){\begin{cases}{\text{Variables}}\\{\text{Decomposition}}\\{\text{Forms}}\\\end{cases}}\\{\text{Identification (based on }}\delta )\end{cases}}\\{\text{Question}}\end{cases}}} A program is constructed from a description and a question. A description is constructed using some specification ( π {\displaystyle \pi } ) as given by the programmer and an identification or learning process for the parameters not completely specified by the specification, using a data set ( δ {\displaystyle \delta } ). A specification is constructed from a set of pertinent variables, a decomposition and a set of forms. Forms are either parametric forms or questions to other Bayesian programs. A question specifies which probability distribution has to be computed. === Description === The purpose of a description is to specify an effective method of computing a joint probability distribution on a set of variables { X 1 , X 2 , ⋯ , X N } {\displaystyle \left\{X_{1},X_{2},\cdots ,X_{N}\right\}} given a set of experimental data δ {\displaystyle \delta } and some specification π {\displaystyle \pi } . This joint distribution is denoted as: P ( X 1 ∧ X 2 ∧ ⋯ ∧ X N ∣ δ ∧ π ) {\displaystyle P\left(X_{1}\wedge X_{2}\wedge \cdots \wedge X_{N}\mid \delta \wedge \pi \right)} . To specify preliminary knowledge π {\displaystyle \pi } , the programmer must undertake the following: Define the set of relevant variables { X 1 , X 2 , ⋯ , X N } {\displaystyle \left\{X_{1},X_{2},\cdots ,X_{N}\right\}} on which the joint distribution is defined. Decompose the joint distribution (break it into relevant independent or conditional probabilities). Define the forms of each of the distributions (e.g., for each variable, one of the list of probability distributions). ==== Decomposition ==== Given a partition of { X 1 , X 2 , … , X N } {\displaystyle \left\{X_{1},X_{2},\ldots ,X_{N}\right\}} containing K {\displaystyle K} subsets, K {\displaystyle K} variables are defined L 1 , ⋯ , L K {\displaystyle L_{1},\cdots ,L_{K}} , each corresponding to one of these subsets. Each variable L k {\displaystyle L_{k}} is obtained as the conjunction of the variables { X k 1 , X k 2 , ⋯ } {\displaystyle \left\{X_{k_{1}},X_{k_{2}},\cdots \right\}} belonging to the k t h {\displaystyle k^{th}} subset. Recursive application of Bayes' theorem leads to: P ( X 1 ∧ X 2 ∧ ⋯ ∧ X N ∣ δ ∧ π ) = P ( L 1 ∧ ⋯ ∧ L K ∣ δ ∧ π ) = P ( L 1 ∣ δ ∧ π ) × P ( L 2 ∣ L 1 ∧ δ ∧ π ) × ⋯ × P ( L K ∣ L K − 1 ∧ ⋯ ∧ L 1 ∧ δ ∧ π ) {\displaystyle {\begin{aligned}&P\left(X_{1}\wedge X_{2}\wedge \cdots \wedge X_{N}\mid \delta \wedge \pi \right)\\={}&P\left(L_{1}\wedge \cdots \wedge L_{K}\mid \delta \wedge \pi \right)\\={}&P\left(L_{1}\mid \delta \wedge \pi \right)\times P\left(L_{2}\mid L_{1}\wedge \delta \wedge \pi \right)\times \cdots \times P\left(L_{K}\mid L_{K-1}\wedge \cdots \wedge L_{1}\wedge \delta \wedge \pi \right)\end{aligned}}} Conditional independence hypotheses then allow further simplifications. A conditional independence hypothesis for variable L k {\displaystyle L_{k}} is defined by choosing some variable X n {\displaystyle X_{n}} among the variables appearing in the conjunction L k − 1 ∧ ⋯ ∧ L 2 ∧ L 1 {\displaystyle L_{k-1}\wedge \cdots \wedge L_{2}\wedge L_{1}} , labelling R k {\displaystyle R_{k}} as the conjunction of these chosen variables and setting: P ( L k ∣ L k − 1 ∧ ⋯ ∧ L 1 ∧ δ ∧ π ) = P ( L k ∣ R k ∧ δ ∧ π ) {\displaystyle P\left(L_{k}\mid L_{k-1}\wedge \cdots \wedge L_{1}\wedge \delta \wedge \pi \right)=P\left(L_{k}\mid R_{k}\wedge \delta \wedge \pi \right)} We then obtain: P ( X 1 ∧ X 2 ∧ ⋯ ∧ X N ∣ δ ∧ π ) = P ( L 1 ∣ δ ∧ π ) × P ( L 2 ∣ R 2 ∧ δ ∧ π ) × ⋯ × P ( L K ∣ R K ∧ δ ∧ π ) {\displaystyle {\begin{aligned}&P\left(X_{1}\wedge X_{2}\wedge \cdots \wedge X_{N}\mid \delta \wedge \pi \right)\\={}&P\left(L_{1}\mid \delta \wedge \pi \right)\times P\left(L_{2}\mid R_{2}\wedge \delta \wedge \pi \right)\times \cdots \times P\left(L_{K}\mid R_{K}\wedge \delta \wedge \pi \right)\end{aligned}}} Such a simplification of the joint distribution as a product of simpler distributions is called a decomposition, derived using the chain rule. This ensures that each variable appears at the most once on the left of a conditioning bar, which is the necessary and sufficient condition to write mathematically valid decompositions. ==== Forms ==== Each distribution P ( L k ∣ R k ∧ δ ∧ π ) {\displaystyle P\left(L_{k}\mid R_{k}\wedge \delta \wedge \pi \right)} appearing in the product is then associated with either a parametric form (i.e., a function f μ ( L k ) {\displaystyle f_{\mu }\left(L_{k}\right)} ) or a question to another Bayesian program P ( L k ∣ R k ∧ δ ∧ π ) = P ( L ∣ R ∧ δ ^ ∧ π ^ ) {\displaystyle P\left(L_{k}\mid R_{k}\wedge \delta \wedge \pi \right)=P\left(L\mid R\wedge {\widehat {\delta }}\wedge {\widehat {\pi }}\right)} . When it is a form f μ ( L k ) {\displaystyle f_{\mu }\left(L_{k}\right)} , in general, μ {\displaystyle \mu } is a vector of parameters that may depend on R k {\displaystyle R_{k}} or δ {\displaystyle \delta } or both. Learning takes place when some of these parameters are computed using the data set δ {\displaystyle \delta } . An important feature of Bayesian programming is this capacity to use questions to other Bayesian programs as components of the definition of a new Bayesian program. P ( L k ∣ R k ∧ δ ∧ π ) {\displaystyle P\left(L_{k}\mid R_{k}\wedge \delta \wedge \pi \right)} is obtained by some inferences done by another Bayesian program defined by the specifications π ^ {\displaystyle {\widehat {\pi }}} and the data δ ^ {\displaystyle {\widehat {\delta }}} . This is similar to calling a subroutine in classical programming and provides an easy way to build hierarchical models. === Question === Given a description (i.e., P ( X 1 ∧ X 2 ∧ ⋯ ∧ X N ∣ δ ∧ π ) {\displaystyle P\left(X_{1}\wedge X_{2}\wedge \cdots \wedge X_{N}\mid \delta \wedge \pi \right)} ), a question is obtained by partitioning { X 1 , X 2 , ⋯ , X N } {\displaystyle \left\{X_{1},X_{2},\cdots ,X_{N}\right\}} into three sets: the searched variables, the known variables and the free variables. The 3 variables S e a r c h e d {\displaystyle Searched} , K n o w n {\displaystyle Known} and F r e e {\displaystyle Free} are defined as the conjunction of the variables belonging to these sets. A question is defined as the set of distributions: P ( S e a r c h e d ∣ Known ∧ δ ∧ π ) {\displaystyle P\left(Searched\mid {\text{Known}}\wedge \delta \wedge \pi \right)} made of many "instantiated questions" as the cardinal of K n o w n {\displaystyle Known} , each instantiated question being the distribution: P ( Searched ∣ Known ∧ δ ∧ π ) {\displaystyle P\left({\text{Searched}}\mid {\text{Known}}\wedge \delta \wedge \pi \right)} === Inference === Given the joint distribution P ( X 1 ∧ X 2 ∧ ⋯ ∧ X N ∣ δ ∧ π ) {\displaystyle P\left(X_{1}\wedge X_{2}\wedge \cdots \wedge X_{N}\mid \delta \wedge \pi \right)} , it is always possible to compute any possible question using the following general inference: P ( Searched ∣ Known ∧ δ ∧ π ) = ∑ Free [ P ( Searched ∧ Free ∣ Known ∧ δ ∧ π ) ] = ∑ Free [ P ( Searched ∧ Free ∧ Known ∣ δ ∧ π ) ] P ( Known ∣ δ ∧ π ) = ∑ Free [ P ( Searched ∧ Free ∧ Known ∣ δ ∧ π ) ] ∑ Free ∧ Searched [ P ( Searched ∧ Free ∧ Known ∣ δ ∧ π ) ] = 1 Z × ∑ Free [ P ( Searched ∧ Free ∧ Known ∣ δ ∧ π ) ] {\displaystyle {\begin{aligned}&P\left({\text{Searched}}\mid {\text{Known}}\wedge \delta \wedge \pi \right)\\={}&\sum _{\text{Free}}\left[P\left({\text{Searched}}\wedge {\text{Free}}\mid {\text{Known}}\wedge \delta \wedge \
Targeted maximum likelihood estimation
Targeted Maximum Likelihood Estimation (TMLE) (also more accurately referred to as Targeted Minimum Loss-Based Estimation) is a general statistical estimation framework for causal inference and semiparametric models. TMLE combines ideas from maximum likelihood estimation, semiparametric efficiency theory, and machine learning. It was introduced by Mark J. van der Laan and colleagues in the mid-2000s as a method that yields asymptotically efficient plug-in estimators while allowing the use of flexible, data-adaptive algorithms such as ensemble machine learning for nuisance parameter estimation. TMLE is used in epidemiology, biostatistics, and the social sciences to estimate causal effects in observational and experimental studies. Applications of TMLE include Longitudinal TMLE (LTMLE) for time-varying treatments and confounders. Variations in how the targeting step in TMLE is carried out have resulted in various versions of TMLE such as Collaborative TMLE (CTMLE) and Adaptive TMLE for improved finite-sample performance and automated variable selection. == History == The TMLE framework was first described by van der Laan and Rubin (2006) as a general approach for the construction of efficient plug-in estimators of smooth features of the data density. It was demonstrated in the context of causal inference and missing data problems. It was developed to address limitations of traditional doubly robust methods, such as Augmented Inverse Probability Weighting (AIPW), by respecting the plug-in principle in the sense that it respects that the target parameter is a function of the data density that is an element of the statistical model. TMLE estimates the data density or relevant parts of it with machine learning and targets these machine learning fits before it is plugged in the target parameter mapping. In this manner, a TMLE always respects global knowledge and satisfies known bounds such as that the target parameter is a probability . Since its introduction, TMLE has been developed in a series of theoretical and applied papers, culminating in book-length treatments of the method and its applications to survival analysis, adaptive designs, and longitudinal data. == Methodology == At its core, TMLE is a two-step estimation procedure: Initial estimation: Machine learning methods (such as the Super Learner ensemble) are used to obtain flexible estimates of nuisance parameters, such as outcome regressions and propensity scores. Targeting step: The initial estimate is updated by solving a score equation (the efficient influence function) so that the final estimator is consistent, asymptotically normal, and efficient under mild regularity conditions. The targeted machine learning fit is then mapped into the corresponding estimator of the target parameter by simply plugging it in the target parameter mapping. This approach balances the bias–variance trade-off by combining data-adaptive estimation with semiparametric efficiency theory. TMLE is doubly robust, meaning it remains consistent if either the outcome model or the treatment model is consistently estimated. === Formula === Here we explain the TMLE of the average treatment effect of a binary treatment on an outcome adjusting for baseline covariates. Consider i.i.d. observations O i = ( W i , A i , Y i ) {\displaystyle O_{i}=(W_{i},A_{i},Y_{i})} from a distribution P 0 {\displaystyle P_{0}} , where W {\displaystyle W} are baseline covariates, A {\displaystyle A} is a binary treatment, and Y {\displaystyle Y} is an outcome. Let Q ¯ ( a , w ) = E [ Y ∣ A = a , W = w ] {\displaystyle {\bar {Q}}(a,w)=\mathbb {E} [Y\mid A=a,W=w]} represent the outcome model and g ( a ∣ w ) = P ( A = a ∣ W = w ) {\displaystyle g(a\mid w)=P(A=a\mid W=w)} represent the propensity score. The average treatment effect (ATE) is given by ψ 0 = E { Q ¯ ( 1 , W ) − Q ¯ ( 0 , W ) } . {\displaystyle \psi _{0}=\mathbb {E} \{{\bar {Q}}(1,W)-{\bar {Q}}(0,W)\}.} A basic TMLE for the ATE proceeds as follows: Step 1: Estimate initial models. Obtain estimates Q ¯ ^ ( a , w ) {\displaystyle {\hat {\bar {Q}}}(a,w)} and g ^ ( a ∣ w ) {\displaystyle {\hat {g}}(a\mid w)} , often using flexible methods such as Super Learner. Step 2: Compute the clever covariate. Define: H ( A , W ) = A g ^ ( 1 ∣ W ) − 1 − A g ^ ( 0 ∣ W ) . {\displaystyle H(A,W)={\frac {A}{{\hat {g}}(1\mid W)}}-{\frac {1-A}{{\hat {g}}(0\mid W)}}.} Step 3: Estimate the fluctuation parameter. Fit a logistic regression of Y {\displaystyle Y} on H ( A , W ) {\displaystyle H(A,W)} with logit ( Q ¯ ^ ( A , W ) ) {\displaystyle \operatorname {logit} ({\hat {\bar {Q}}}(A,W))} as offset. This yields ε ^ {\displaystyle {\hat {\varepsilon }}} , the MLE that solves the score equation: 1 n ∑ i = 1 n H ( A i , W i ) { Y i − Q ¯ ^ ε ( A i , W i ) } = 0. {\displaystyle {\frac {1}{n}}\sum _{i=1}^{n}H(A_{i},W_{i}){\big \{}Y_{i}-{\hat {\bar {Q}}}^{\varepsilon }(A_{i},W_{i}){\big \}}=0.} Step 4: Update the initial estimate. Apply the "blip" to obtain the targeted estimate: Q ¯ ^ ∗ ( A , W ) = expit ( logit ( Q ¯ ^ ( A , W ) ) + ε ^ H ( A , W ) ) . {\displaystyle {\hat {\bar {Q}}}^{}(A,W)=\operatorname {expit} {\Big (}\operatorname {logit} {\big (}{\hat {\bar {Q}}}(A,W){\big )}+{\hat {\varepsilon }}\,H(A,W){\Big )}.} Step 5: Compute the TMLE. The ATE estimate is: ψ ^ TMLE = 1 n ∑ i = 1 n [ Q ¯ ^ ∗ ( 1 , W i ) − Q ¯ ^ ∗ ( 0 , W i ) ] . {\displaystyle {\hat {\psi }}_{\text{TMLE}}={\frac {1}{n}}\sum _{i=1}^{n}{\big [}{\hat {\bar {Q}}}^{}(1,W_{i})-{\hat {\bar {Q}}}^{}(0,W_{i}){\big ]}.} Inference. The efficient influence function (EIF) for the ATE is: D ∗ ( O ) = H ( A , W ) { Y − Q ¯ ∗ ( A , W ) } + Q ¯ ∗ ( 1 , W ) − Q ¯ ∗ ( 0 , W ) − ψ . {\displaystyle D^{}(O)=H(A,W)\{Y-{\bar {Q}}^{}(A,W)\}+{\bar {Q}}^{}(1,W)-{\bar {Q}}^{}(0,W)-\psi .} The variance is estimated by σ ^ 2 = n − 1 ∑ i = 1 n ( D ∗ ( O i ) ) 2 {\displaystyle {\hat {\sigma }}^{2}=n^{-1}\sum _{i=1}^{n}{\big (}D^{}(O_{i}){\big )}^{2}} , yielding Wald-type confidence intervals ψ ^ TMLE ± z 1 − α / 2 σ ^ / n {\displaystyle {\hat {\psi }}_{\text{TMLE}}\pm z_{1-\alpha /2}\,{\hat {\sigma }}/{\sqrt {n}}} . Remark. For continuous outcomes, a linear fluctuation Q ¯ ^ ∗ = Q ¯ ^ + ε ^ H {\displaystyle {\hat {\bar {Q}}}^{}={\hat {\bar {Q}}}+{\hat {\varepsilon }}\,H} may be used instead. For bounded continuous outcomes, the logistic fluctuation (after rescaling Y {\displaystyle Y} to [ 0 , 1 ] {\displaystyle [0,1]} ) is often preferred for improved finite-sample performance. == Applications == TMLE has been applied in: Epidemiology: Estimating causal effects of exposures and interventions in observational cohort studies. Clinical trials and real-world evidence: The Targeted Learning roadmap provides a structured framework for generating and validating real-world evidence (RWE), bridging randomized trials and observational data using TMLE and related estimation techniques. This approach enables transparency, sensitivity analysis, and stronger causal inference for regulatory and clinical trial contexts. High-dimensional settings: Integration with ensemble methods for causal effect estimation. TMLE has been successfully applied in pharmacoepidemiology where a large number of covariates are automatically selected to adjust for confounding. In a study of post–myocardial infarction statin use and 1-year mortality, TMLE demonstrated robust performance relative to inverse probability weighting in scenarios with hundreds of potential confounders. == Derivatives and extensions == Longitudinal TMLE (LTMLE): A methodological extension of TMLE for longitudinal data with time-varying treatments, confounders, and censoring. It allows the estimation of dynamic treatment regimes and intervention-specific causal effects over time. This framework was originally introduced by van der Laan & Gruber (2012). Collaborative TMLE (CTMLE): Enhances finite-sample performance and variable selection by collaboratively fitting the treatment mechanism in conjunction with the target parameter. == Software == Several R packages implement TMLE and related methods: tmle: Functions for binary, categorical, and continuous outcomes. ltmle: Implementation for longitudinal data with time-varying treatments and outcomes. ctmle: Algorithms for collaborative TMLE and adaptive variable selection. SuperLearner: A theoretically grounded, cross-validated ensemble learning method that combines predictions from multiple algorithms to minimize predictive risk. Widely used in TMLE for estimating nuisance parameters. The original implementation is available as the R package SuperLearner. Recent machine learning platforms like H2O AutoML implement similar ensemble strategies, combining diverse learners in parallel and leveraging stacking and blending techniques, effectively functioning as a large-scale Super Learner.
Ilastik
ilastik is free open source software for image classification and segmentation. No previous experience in image processing is required to run the software. Since 2018 ilastik is further developed and maintained by Anna Kreshuk's group at European Molecular Biology Laboratory. == Features == ilastik allows user to annotate an arbitrary number of classes in images with a mouse interface. Using these user annotations and the generic (nonlinear) image features, the user can train a random forest classifier. Trained ilastik classifiers can be applied new data not included in the training set in ilastik via its batch processing functionality, or without using the graphical user interface, in headless mode. ilastik can be integrated into various related tools: Pre-trained workflows can be executed directly from ImageJ/Fiji using the ilastik-ImageJ plugin. Pre-trained ilastik Pixel Classification workflows can be run directly in Python with the ilastik Python package, which is available via conda. ilastik has a CellProfiler module to use ilastik classifiers to process images within a CellProfiler framework. == History == ilastik was first released in 2011 by scientists at the Heidelberg Collaboratory for Image Processing (HCI), University of Heidelberg. == Application == The Interactive Learning and Segmentation Toolkit Carving Cell classification and neuron classification Synapse detection Cell tracking Neural Network Classification == Resources == ilastik project is hosted on GitHub. It is a collaborative project, any contributions such as comments, bug reports, bug fixes or code contributions are welcome. The ilastik team can be contacted for user support on the image.sc forum.
BrownBoost
BrownBoost is a boosting algorithm that may be robust to noisy datasets. BrownBoost is an adaptive version of the boost by majority algorithm. As is the case for all boosting algorithms, BrownBoost is used in conjunction with other machine learning methods. BrownBoost was introduced by Yoav Freund in 2001. == Motivation == AdaBoost performs well on a variety of datasets; however, it can be shown that AdaBoost does not perform well on noisy data sets. This is a result of AdaBoost's focus on examples that are repeatedly misclassified. In contrast, BrownBoost effectively "gives up" on examples that are repeatedly misclassified. The core assumption of BrownBoost is that noisy examples will be repeatedly mislabeled by the weak hypotheses and non-noisy examples will be correctly labeled frequently enough to not be "given up on." Thus only noisy examples will be "given up on," whereas non-noisy examples will contribute to the final classifier. In turn, if the final classifier is learned from the non-noisy examples, the generalization error of the final classifier may be much better than if learned from noisy and non-noisy examples. The user of the algorithm can set the amount of error to be tolerated in the training set. Thus, if the training set is noisy (say 10% of all examples are assumed to be mislabeled), the booster can be told to accept a 10% error rate. Since the noisy examples may be ignored, only the true examples will contribute to the learning process. == Algorithm description == BrownBoost uses a non-convex potential loss function, thus it does not fit into the AdaBoost framework. The non-convex optimization provides a method to avoid overfitting noisy data sets. However, in contrast to boosting algorithms that analytically minimize a convex loss function (e.g. AdaBoost and LogitBoost), BrownBoost solves a system of two equations and two unknowns using standard numerical methods. The only parameter of BrownBoost ( c {\displaystyle c} in the algorithm) is the "time" the algorithm runs. The theory of BrownBoost states that each hypothesis takes a variable amount of time ( t {\displaystyle t} in the algorithm) which is directly related to the weight given to the hypothesis α {\displaystyle \alpha } . The time parameter in BrownBoost is analogous to the number of iterations T {\displaystyle T} in AdaBoost. A larger value of c {\displaystyle c} means that BrownBoost will treat the data as if it were less noisy and therefore will give up on fewer examples. Conversely, a smaller value of c {\displaystyle c} means that BrownBoost will treat the data as more noisy and give up on more examples. During each iteration of the algorithm, a hypothesis is selected with some advantage over random guessing. The weight of this hypothesis α {\displaystyle \alpha } and the "amount of time passed" t {\displaystyle t} during the iteration are simultaneously solved in a system of two non-linear equations ( 1. uncorrelated hypothesis w.r.t example weights and 2. hold the potential constant) with two unknowns (weight of hypothesis α {\displaystyle \alpha } and time passed t {\displaystyle t} ). This can be solved by bisection (as implemented in the JBoost software package) or Newton's method (as described in the original paper by Freund). Once these equations are solved, the margins of each example ( r i ( x j ) {\displaystyle r_{i}(x_{j})} in the algorithm) and the amount of time remaining s {\displaystyle s} are updated appropriately. This process is repeated until there is no time remaining. The initial potential is defined to be 1 m ∑ j = 1 m 1 − erf ( c ) = 1 − erf ( c ) {\displaystyle {\frac {1}{m}}\sum _{j=1}^{m}1-{\mbox{erf}}({\sqrt {c}})=1-{\mbox{erf}}({\sqrt {c}})} . Since a constraint of each iteration is that the potential be held constant, the final potential is 1 m ∑ j = 1 m 1 − erf ( r i ( x j ) / c ) = 1 − erf ( c ) {\displaystyle {\frac {1}{m}}\sum _{j=1}^{m}1-{\mbox{erf}}(r_{i}(x_{j})/{\sqrt {c}})=1-{\mbox{erf}}({\sqrt {c}})} . Thus the final error is likely to be near 1 − erf ( c ) {\displaystyle 1-{\mbox{erf}}({\sqrt {c}})} . However, the final potential function is not the 0–1 loss error function. For the final error to be exactly 1 − erf ( c ) {\displaystyle 1-{\mbox{erf}}({\sqrt {c}})} , the variance of the loss function must decrease linearly w.r.t. time to form the 0–1 loss function at the end of boosting iterations. This is not yet discussed in the literature and is not in the definition of the algorithm below. The final classifier is a linear combination of weak hypotheses and is evaluated in the same manner as most other boosting algorithms. == BrownBoost learning algorithm definition == Input: m {\displaystyle m} training examples ( x 1 , y 1 ) , … , ( x m , y m ) {\displaystyle (x_{1},y_{1}),\ldots ,(x_{m},y_{m})} where x j ∈ X , y j ∈ Y = { − 1 , + 1 } {\displaystyle x_{j}\in X,\,y_{j}\in Y=\{-1,+1\}} The parameter c {\displaystyle c} Initialise: s = c {\displaystyle s=c} . (The value of s {\displaystyle s} is the amount of time remaining in the game) r i ( x j ) = 0 {\displaystyle r_{i}(x_{j})=0} ∀ j {\displaystyle \forall j} . The value of r i ( x j ) {\displaystyle r_{i}(x_{j})} is the margin at iteration i {\displaystyle i} for example x j {\displaystyle x_{j}} . While s > 0 {\displaystyle s>0} : Set the weights of each example: W i ( x j ) = e − ( r i ( x j ) + s ) 2 c {\displaystyle W_{i}(x_{j})=e^{-{\frac {(r_{i}(x_{j})+s)^{2}}{c}}}} , where r i ( x j ) {\displaystyle r_{i}(x_{j})} is the margin of example x j {\displaystyle x_{j}} Find a classifier h i : X → { − 1 , + 1 } {\displaystyle h_{i}:X\to \{-1,+1\}} such that ∑ j W i ( x j ) h i ( x j ) y j > 0 {\displaystyle \sum _{j}W_{i}(x_{j})h_{i}(x_{j})y_{j}>0} Find values α , t {\displaystyle \alpha ,t} that satisfy the equation: ∑ j h i ( x j ) y j e − ( r i ( x j ) + α h i ( x j ) y j + s − t ) 2 c = 0 {\displaystyle \sum _{j}h_{i}(x_{j})y_{j}e^{-{\frac {(r_{i}(x_{j})+\alpha h_{i}(x_{j})y_{j}+s-t)^{2}}{c}}}=0} . (Note this is similar to the condition E W i + 1 [ h i ( x j ) y j ] = 0 {\displaystyle E_{W_{i+1}}[h_{i}(x_{j})y_{j}]=0} set forth by Schapire and Singer. In this setting, we are numerically finding the W i + 1 = exp ( ⋯ ⋯ ) {\displaystyle W_{i+1}=\exp \left({\frac {\cdots }{\cdots }}\right)} such that E W i + 1 [ h i ( x j ) y j ] = 0 {\displaystyle E_{W_{i+1}}[h_{i}(x_{j})y_{j}]=0} .) This update is subject to the constraint ∑ ( Φ ( r i ( x j ) + α h ( x j ) y j + s − t ) − Φ ( r i ( x j ) + s ) ) = 0 {\displaystyle \sum \left(\Phi \left(r_{i}(x_{j})+\alpha h(x_{j})y_{j}+s-t\right)-\Phi \left(r_{i}(x_{j})+s\right)\right)=0} , where Φ ( z ) = 1 − erf ( z / c ) {\displaystyle \Phi (z)=1-{\mbox{erf}}(z/{\sqrt {c}})} is the potential loss for a point with margin r i ( x j ) {\displaystyle r_{i}(x_{j})} Update the margins for each example: r i + 1 ( x j ) = r i ( x j ) + α h ( x j ) y j {\displaystyle r_{i+1}(x_{j})=r_{i}(x_{j})+\alpha h(x_{j})y_{j}} Update the time remaining: s = s − t {\displaystyle s=s-t} Output: H ( x ) = sign ( ∑ i α i h i ( x ) ) {\displaystyle H(x)={\textrm {sign}}\left(\sum _{i}\alpha _{i}h_{i}(x)\right)} == Empirical results == In preliminary experimental results with noisy datasets, BrownBoost outperformed AdaBoost's generalization error; however, LogitBoost performed as well as BrownBoost. An implementation of BrownBoost can be found in the open source software JBoost.
Aldus PhotoStyler
Aldus PhotoStyler was a graphics software program developed by the Taiwanese company Ulead. Released in June 1991 as the first 24 bit image editor for Windows, it was bought the same year by the Aldus Prepress group. Its main competition was Adobe Photoshop. Version 2.0 (late 1993) introduced a new user interface and improved color calibration. PhotoStyler SE - lacking some features of the version 2.0 - was bundled with scanners like HP ScanJet. The product disappeared from the Adobe product line after Adobe acquired Aldus in 1994.
List of datasets for machine-learning research
These datasets are used in machine learning (ML) research and have been cited in peer-reviewed academic journals. Datasets are an integral part of the field of machine learning. Major advances in this field can result from advances in learning algorithms (such as deep learning), computer hardware, and, less intuitively, the availability of high-quality training datasets. High-quality labeled training datasets for supervised and semi-supervised machine-learning algorithms are usually difficult and expensive to produce because of the large amount of time needed to label the data. Although they do not need to be labeled, high-quality unlabeled datasets for unsupervised learning can also be difficult and costly to produce. Many organizations, including governments, publish and share their datasets, often using common metadata formats (such as Croissant). The datasets are classified, based on the licenses, into two groups: open data and non-open data. The datasets from various governmental-bodies are presented in List of open government data sites. The datasets are ported on open data portals. They are made available for searching, depositing and accessing through interfaces like Open API. The datasets are made available as various sorted types and subtypes. == List of sorting used for datasets == The data portal is classified based on its type of license. The open source license based data portals are known as open data portals which are used by many government organizations and academic institutions. == List of open data portals == == List of portals suitable for multiple types of applications == The data portal sometimes lists a wide variety of subtypes of datasets pertaining to many machine learning applications. == List of portals suitable for a specific subtype of applications == The data portals which are suitable for a specific subtype of machine learning application are listed in the subsequent sections. == Image data == == Text data == These datasets consist primarily of text for tasks such as natural language processing, sentiment analysis, translation, and cluster analysis. === Reviews === === News articles === === Messages === === Twitter and tweets === === Dialogues === === Legal === === Other text === == Sound data == These datasets consist of sounds and sound features used for tasks such as speech recognition and speech synthesis. === Speech === === Music === === Other sounds === == Signal data == Datasets containing electric signal information requiring some sort of signal processing for further analysis. === Electrical === === Motion-tracking === === Other signals === == Chemical data == Datasets from physical systems. === Chemical Reactions with transition states (TS) === === OpenReACT-CHON-EFH === OpenReACT-CHON-EFH (Open Reaction Dataset of Atomic ConfiguraTions comprising C, H, O and N with Energies, Forces and Hessians) is a 2025 open-access benchmark for machine-learning interatomic potentials. RTP set – 35,087 stationary-point geometries (reactant, transition state and product) drawn from 11,961 elementary reactions, each labeled with density-functional energies, atomic forces and full Hessian matrices at the ωB97X-D/6-31G(d) level. IRC set – 34,248 structures along 600 minimum-energy reaction paths, used to test extrapolation beyond trained stationary points. NMS set – 62,527 off-equilibrium geometries generated by normal-mode sampling to probe model robustness under thermal perturbations. The collection underpins the study Does Hessian Data Improve the Performance of Machine Learning Potentials? and was used to train and benchmark the machine-learning interatomic potentials reported therein. The dataset itself is distributed under a CC licence via Figshare. == Physical data == Datasets from physical systems. === High-energy physics === === Systems === === Astronomy === === Earth science === === Other physical === == Biological data == Datasets from biological systems. === Human === === Animal === === Fungi === === Plant === === Microbe === === Drug discovery === == Anomaly data == == Question answering data == This section includes datasets that deals with structured data. == Dialog or instruction prompted data == This section includes datasets that contains multi-turn text with at least two actors, a "user" and an "agent". The user makes requests for the agent, which performs the request. == Cybersecurity == == Climate and sustainability == == Code data == == Multivariate data == === Financial === === Weather === === Census === === Transit === === Internet === === Games === === Other multivariate === == Curated repositories of datasets == As datasets come in myriad formats and can sometimes be difficult to use, there has been considerable work put into curating and standardizing the format of datasets to make them easier to use for machine learning research. OpenML: Web platform with Python, R, Java, and other APIs for downloading hundreds of machine learning datasets, evaluating algorithms on datasets, and benchmarking algorithm performance against dozens of other algorithms. PMLB: A large, curated repository of benchmark datasets for evaluating supervised machine learning algorithms. Provides classification and regression datasets in a standardized format that are accessible through a Python API. Metatext NLP: https://metatext.io/datasets web repository maintained by community, containing nearly 1000 benchmark datasets, and counting. Provides many tasks from classification to QA, and various languages from English, Portuguese to Arabic. Appen: Off The Shelf and Open Source Datasets hosted and maintained by the company. These biological, image, physical, question answering, signal, sound, text, and video resources number over 250 and can be applied to over 25 different use cases.