Conditional random fields (CRFs) are a class of statistical modeling methods often applied in pattern recognition and machine learning and used for structured prediction. Whereas a classifier predicts a label for a single sample without considering "neighbouring" samples, a CRF can take context into account. To do so, the predictions are modelled as a graphical model, which represents the presence of dependencies between the predictions. The kind of graph used depends on the application. For example, in natural language processing, "linear chain" CRFs are popular, for which each prediction is dependent only on its immediate neighbours. In image processing, the graph typically connects locations to nearby and/or similar locations to enforce that they receive similar predictions. Other examples where CRFs are used are: labeling or parsing of sequential data for natural language processing or biological sequences, part-of-speech tagging, shallow parsing, named entity recognition, gene finding, peptide critical functional region finding, and object recognition and image segmentation in computer vision. == Description == CRFs are a type of discriminative undirected probabilistic graphical model. Lafferty, McCallum and Pereira define a CRF on observations X {\displaystyle {\boldsymbol {X}}} and random variables Y {\displaystyle {\boldsymbol {Y}}} as follows: Let G = ( V , E ) {\displaystyle G=(V,E)} be a graph such that Y = ( Y v ) v ∈ V {\displaystyle {\boldsymbol {Y}}=({\boldsymbol {Y}}_{v})_{v\in V}} , so that Y {\displaystyle {\boldsymbol {Y}}} is indexed by the vertices of G {\displaystyle G} . Then ( X , Y ) {\displaystyle ({\boldsymbol {X}},{\boldsymbol {Y}})} is a conditional random field when each random variable Y v {\displaystyle {\boldsymbol {Y}}_{v}} , conditioned on X {\displaystyle {\boldsymbol {X}}} , obeys the Markov property with respect to the graph; that is, its probability is dependent only on its neighbours in G and not its past states: P ( Y v | X , { Y w : w ≠ v } ) = P ( Y v | X , { Y w : w ∼ v } ) {\displaystyle P({\boldsymbol {Y}}_{v}|{\boldsymbol {X}},\{{\boldsymbol {Y}}_{w}:w\neq v\})=P({\boldsymbol {Y}}_{v}|{\boldsymbol {X}},\{{\boldsymbol {Y}}_{w}:w\sim v\})} , where w ∼ v {\displaystyle {\mathit {w}}\sim v} means that w {\displaystyle w} and v {\displaystyle v} are neighbors in G {\displaystyle G} . What this means is that a CRF is an undirected graphical model whose nodes can be divided into exactly two disjoint sets X {\displaystyle {\boldsymbol {X}}} and Y {\displaystyle {\boldsymbol {Y}}} , the observed and output variables, respectively; the conditional distribution p ( Y | X ) {\displaystyle p({\boldsymbol {Y}}|{\boldsymbol {X}})} is then modeled. === Inference === For general graphs, the problem of exact inference in CRFs is intractable. The inference problem for a CRF is basically the same as for an MRF and the same arguments hold. However, there exist special cases for which exact inference is feasible: If the graph is a chain or a tree, message passing algorithms yield exact solutions. The algorithms used in these cases are analogous to the forward-backward and Viterbi algorithm for the case of HMMs. If the CRF only contains pair-wise potentials and the energy is submodular, combinatorial min cut/max flow algorithms yield exact solutions. If exact inference is impossible, several algorithms can be used to obtain approximate solutions. These include: Loopy belief propagation Alpha expansion Mean field inference Linear programming relaxations === Parameter learning === Learning the parameters θ {\displaystyle \theta } is usually done by maximum likelihood learning for p ( Y i | X i ; θ ) {\displaystyle p(Y_{i}|X_{i};\theta )} . If all nodes have exponential family distributions and all nodes are observed during training, this optimization is convex. It can be solved for example using gradient descent algorithms, or Quasi-Newton methods such as the L-BFGS algorithm. On the other hand, if some variables are unobserved, the inference problem has to be solved for these variables. Exact inference is intractable in general graphs, so approximations have to be used. === Examples === In sequence modeling, the graph of interest is usually a chain graph. An input sequence of observed variables X {\displaystyle X} represents a sequence of observations and Y {\displaystyle Y} represents a hidden (or unknown) state variable that needs to be inferred given the observations. The Y i {\displaystyle Y_{i}} are structured to form a chain, with an edge between each Y i − 1 {\displaystyle Y_{i-1}} and Y i {\displaystyle Y_{i}} . As well as having a simple interpretation of the Y i {\displaystyle Y_{i}} as "labels" for each element in the input sequence, this layout admits efficient algorithms for: model training, learning the conditional distributions between the Y i {\displaystyle Y_{i}} and feature functions from some corpus of training data. decoding, determining the probability of a given label sequence Y {\displaystyle Y} given X {\displaystyle X} . inference, determining the most likely label sequence Y {\displaystyle Y} given X {\displaystyle X} . The conditional dependency of each Y i {\displaystyle Y_{i}} on X {\displaystyle X} is defined through a fixed set of feature functions of the form f ( i , Y i − 1 , Y i , X ) {\displaystyle f(i,Y_{i-1},Y_{i},X)} , which can be thought of as measurements on the input sequence that partially determine the likelihood of each possible value for Y i {\displaystyle Y_{i}} . The model assigns each feature a numerical weight and combines them to determine the probability of a certain value for Y i {\displaystyle Y_{i}} . Linear-chain CRFs have many of the same applications as conceptually simpler hidden Markov models (HMMs), but relax certain assumptions about the input and output sequence distributions. An HMM can loosely be understood as a CRF with very specific feature functions that use constant probabilities to model state transitions and emissions. Conversely, a CRF can loosely be understood as a generalization of an HMM that makes the constant transition probabilities into arbitrary functions that vary across the positions in the sequence of hidden states, depending on the input sequence. Notably, in contrast to HMMs, CRFs can contain any number of feature functions, the feature functions can inspect the entire input sequence X {\displaystyle X} at any point during inference, and the range of the feature functions need not have a probabilistic interpretation. == Variants == === Higher-order CRFs and semi-Markov CRFs === CRFs can be extended into higher order models by making each Y i {\displaystyle Y_{i}} dependent on a fixed number k {\displaystyle k} of previous variables Y i − k , . . . , Y i − 1 {\displaystyle Y_{i-k},...,Y_{i-1}} . In conventional formulations of higher order CRFs, training and inference are only practical for small values of k {\displaystyle k} (such as k ≤ 5), since their computational cost increases exponentially with k {\displaystyle k} . However, another recent advance has managed to ameliorate these issues by leveraging concepts and tools from the field of Bayesian nonparametrics. Specifically, the CRF-infinity approach constitutes a CRF-type model that is capable of learning infinitely-long temporal dynamics in a scalable fashion. This is effected by introducing a novel potential function for CRFs that is based on the Sequence Memoizer (SM), a nonparametric Bayesian model for learning infinitely-long dynamics in sequential observations. To render such a model computationally tractable, CRF-infinity employs a mean-field approximation of the postulated novel potential functions (which are driven by an SM). This allows for devising efficient approximate training and inference algorithms for the model, without undermining its capability to capture and model temporal dependencies of arbitrary length. There exists another generalization of CRFs, the semi-Markov conditional random field (semi-CRF), which models variable-length segmentations of the label sequence Y {\displaystyle Y} . This provides much of the power of higher-order CRFs to model long-range dependencies of the Y i {\displaystyle Y_{i}} , at a reasonable computational cost. Finally, large-margin models for structured prediction, such as the structured Support Vector Machine can be seen as an alternative training procedure to CRFs. === Latent-dynamic conditional random field === Latent-dynamic conditional random fields (LDCRF) or discriminative probabilistic latent variable models (DPLVM) are a type of CRFs for sequence tagging tasks. They are latent variable models that are trained discriminatively. In an LDCRF, like in any sequence tagging task, given a sequence of observations x = x 1 , … , x n {\displaystyle x_{1},\dots ,x_{n}} , the main problem the model must solve is how to assign a sequence of labels y = y 1 , … , y n {\displaystyle y_{1},\dots ,y_{n}} from one finite set
Eager learning
In artificial intelligence, eager learning is a learning method in which the system tries to construct a general, input-independent target function during training of the system, as opposed to lazy learning, where generalization beyond the training data is delayed until a query is made to the system. The main advantage gained in employing an eager learning method, such as an artificial neural network, is that the target function will be approximated globally during training, thus requiring much less space than using a lazy learning system. Eager learning systems also deal much better with noise in the training data. Eager learning is an example of offline learning, in which post-training queries to the system have no effect on the system itself, and thus the same query to the system will always produce the same result. The main disadvantage with eager learning is that it is generally unable to provide good local approximations in the target function.
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Hierarchical Risk Parity
Hierarchical Risk Parity (HRP) is an advanced investment portfolio optimization framework developed in 2016 by Marcos López de Prado at Guggenheim Partners and Cornell University. HRP is a probabilistic graph-based alternative to the prevailing mean-variance optimization (MVO) framework developed by Harry Markowitz in 1952, and for which he received the Nobel Prize in economic sciences. HRP algorithms apply discrete mathematics and machine learning techniques to create diversified and robust investment portfolios that outperform MVO methods out-of-sample. HRP aims to address the limitations of traditional portfolio construction methods, particularly when dealing with highly correlated assets. Following its publication, HRP has been implemented in numerous open-source libraries, and received multiple extensions. == Key features == HRP portfolios have been proposed as a robust alternative to traditional quadratic optimization methods, including the Critical Line Algorithm (CLA) of Markowitz. HRP addresses three central issues commonly associated with quadratic optimizers: numerical instability, excessive concentration in a small number of assets, and poor out-of-sample performance. HRP leverages techniques from graph theory and machine learning to construct diversified portfolios using only the information embedded in the covariance matrix. Unlike quadratic programming methods, HRP does not require the covariance matrix to be invertible. Consequently, HRP remains applicable even in cases where the covariance matrix is ill-conditioned or singular—conditions under which standard optimizers fail. Monte Carlo simulations indicate that HRP achieves lower out-of-sample variance than CLA, despite the fact that minimizing variance is the explicit optimization objective of CLA. Furthermore, HRP portfolios exhibit lower realized risk compared to those generated by traditional risk parity methodologies. Empirical backtests have demonstrated that HRP would have historically outperformed conventional portfolio construction techniques. Algorithms within the HRP framework are characterized by the following features: Machine Learning Approach: HRP employs hierarchical clustering, a machine learning technique, to group similar assets based on their correlations. This allows the algorithm to identify the underlying hierarchical structure of the portfolio, and avoid that errors spread through the entire network. Risk-Based Allocation: The algorithm allocates capital based on risk, ensuring that assets only compete with similar assets for representation in the portfolio. This approach leads to better diversification across different risk sources, while avoiding the instability associated with noisy returns estimates. Covariance Matrix Handling: Unlike traditional methods like Mean-Variance Optimization, HRP does not require inverting the covariance matrix. This makes it more stable and applicable to portfolios with a large number of assets, particularly when the covariance matrix's condition number is high. == The problem: Markowitz's Curse == Portfolio construction is perhaps the most recurrent financial problem. On a daily basis, investment managers must build portfolios that incorporate their views and forecasts on risks and returns. Despite the theoretical elegance of Markowitz's mean-variance framework, its practical implementation is hindered by several limitations that undermine the reliability of solutions derived from the Critical Line Algorithm (CLA). A principal concern is the high sensitivity of optimal portfolios to small perturbations in expected returns: even minor forecasting errors can result in significantly different allocations (Michaud, 1998). Given the inherent difficulty of producing accurate return forecasts, numerous researchers have advocated for approaches that forgo expected returns entirely and instead rely solely on the covariance structure of asset returns. This has given rise to risk-based allocation methods, among which risk parity is a widely cited example (Jurczenko, 2015). While eliminating return forecasts mitigates some instability, it does not eliminate it. Quadratic programming techniques employed in portfolio optimization require the inversion of a positive-definite covariance matrix, meaning all eigenvalues must be strictly positive. When the matrix is numerically ill-conditioned—that is, when the ratio of its largest to smallest eigenvalue (its condition number) is large—matrix inversion becomes unreliable and prone to significant numerical errors (Bailey and López de Prado, 2012). The condition number of a covariance, correlation, or any symmetric (and thus diagonalizable) matrix is defined as the absolute value of the ratio between its largest and smallest eigenvalues in modulus. The figure on the right presents the sorted eigenvalues of several correlation matrices; the condition number is represented by the ratio of the first to last eigenvalues in each sequence. A diagonal correlation matrix, which is equal to its own inverse, exhibits the minimum possible condition number. As the number of correlated (or multicollinear) assets in a portfolio increases, the condition number rises. At high levels, this leads to severe numerical instability, whereby slight modifications in any matrix entry may result in drastically different inverses. This phenomenon, often referred to as Markowitz’s curse, encapsulates the paradox wherein increased correlation among assets heightens the theoretical need for diversification, yet simultaneously increases the likelihood of unstable optimization outcomes. Consequently, the potential benefits of diversification are frequently overshadowed by estimation errors. These problems are exacerbated as the dimensionality of the covariance matrix increases. The estimation of each covariance term consumes degrees of freedom, and in general, a minimum of 1 2 N ( N + 1 ) {\displaystyle {\frac {1}{2}}N(N+1)} independent and identically distributed (IID) observations is required to estimate a non-singular covariance matrix of dimension N {\displaystyle N} . For example, constructing an invertible covariance matrix of dimension 50 necessitates at least five years of daily IID observations. However, empirical evidence suggests that the correlation structure of financial assets is highly unstable over such extended periods. These difficulties are highlighted by the observation that even naïve allocation strategies—such as equally weighted portfolios—have frequently outperformed both mean-variance and risk-based optimizations in out-of-sample tests (De Miguel et al., 2009). == The solution: Hierarchical Risk Parity == The HRP algorithm addresses Markowitz's curse in three steps: Hierarchical Clustering: Assets are grouped into clusters based on their correlations, forming a hierarchical tree structure. Quasi-Diagonalization: The correlation matrix is reordered based on the clustering results, revealing a block diagonal structure. Recursive Bisection: Weights are assigned to assets through a top-down approach, splitting the portfolio into smaller sub-portfolios and allocating capital based on inverse variance. === Step 1: Hierarchical clustering === Given a T × N {\displaystyle T\times N} matrix of asset returns X {\displaystyle X} , where each column represents a time series of returns for one of N {\displaystyle N} assets over T {\displaystyle T} time periods, a hierarchical clustering process can be used to construct a tree-based representation of asset relationships. First, we compute the N × N {\displaystyle N\times N} correlation matrix ρ = ρ i , j i , j = 1 . . . N {\displaystyle \rho ={\rho _{i,j}}\;{i,j=1\;...\;N}} , where ρ i , j = c o r r ( X i , X j ) {\displaystyle \rho _{i,j}=\mathrm {corr} (X_{i},X_{j})} . From this, a pairwise distance matrix D = d i , j {\displaystyle D={d_{i,j}}} is defined using the transformation: d i , j = 1 2 ( 1 − ρ i , j ) {\displaystyle d_{i,j}={\sqrt {{\frac {1}{2}}(1-\rho _{i,j})}}} This distance function defines a proper metric space, satisfying non-negativity, identity of indiscernibles, symmetry, and the triangle inequality. Next, a secondary distance matrix D ~ = d ~ i , j {\displaystyle {\tilde {D}}={{\tilde {d}}_{i,j}}} is computed, where each entry measures the Euclidean distance between the distance profiles of two assets: d ~ i , j = ∑ n = 1 N ( d n , i − d n , j ) 2 {\displaystyle {\tilde {d}}_{i,j}={\sqrt {\sum _{n=1}^{N}(d_{n,i}-d_{n,j})^{2}}}} While d i , j {\displaystyle d_{i,j}} reflects correlation-based proximity between two assets, d ~ i , j {\displaystyle {\tilde {d}}_{i,j}} quantifies dissimilarity across the entire system, as it depends on all pairwise distances. Hierarchical clustering proceeds by identifying the pair ( i , j ) {\displaystyle (i,j)} with the smallest value of d ~ i , j {\displaystyle {\tilde {d}}_{i,j}} (for i ≠ j {\displaystyle i\neq j} ), and forming a new cluster u [ 1 ] = ( i , j ) {\displaystyle u[1]=(i,j)} .
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