# Named Entity Recognition without Labelled Data:
A Weak Supervision Approach

Pierre Lison  Aliaksandr Hubin
Norwegian Computing Center
Oslo, Norway
<EMAIL_ADDRESS>
&Jeremy Barnes  Samia Touileb
Language Technology Group
University of Oslo, Norway
<EMAIL_ADDRESS>
###### Abstract

Named Entity Recognition (NER) performance often degrades rapidly when applied
to target domains that differ from the texts observed during training. When
in-domain labelled data is available, transfer learning techniques can be used
to adapt existing NER models to the target domain. But what should one do when
there is no hand-labelled data for the target domain? This paper presents a
simple but powerful approach to learn NER models in the absence of labelled
data through weak supervision. The approach relies on a broad spectrum of
labelling functions to automatically annotate texts from the target domain.
These annotations are then merged together using a hidden Markov model which
captures the varying accuracies and confusions of the labelling functions. A
sequence labelling model can finally be trained on the basis of this unified
annotation. We evaluate the approach on two English datasets (CoNLL 2003 and
news articles from Reuters and Bloomberg) and demonstrate an improvement of
about 7 percentage points in entity-level $F_{1}$ scores compared to an out-
of-domain neural NER model.

##  1 Introduction

Named Entity Recognition (NER) constitutes a core component in many NLP
pipelines and is employed in a broad range of applications such as information
extraction Raiman and Raiman (2018), question answering Mollá et al. (2006),
document de-identification Stubbs et al. (2015), machine translation Ugawa et
al. (2018) and even conversational models Ghazvininejad et al. (2018). Given a
document, the goal of NER is to identify and classify spans referring to an
entity belonging to pre-specified categories such as persons, organisations or
geographical locations.

NER models often rely on convolutional or recurrent neural architectures,
sometimes completed by a CRF layer Chiu and Nichols (2016); Lample et al.
(2016); Yadav and Bethard (2018). More recently, deep contextualised
representations relying on bidirectional LSTMS Peters et al. (2018),
transformers Devlin et al. (2019); Yan et al. (2019) or contextual string
embeddings Akbik et al. (2019) have also been shown to achieve state-of-the-
art performance on NER tasks.

These neural architectures require large corpora annotated with named
entities, such as Ontonotes Weischedel et al. (2011) or ConLL 2003 Tjong Kim
Sang and De Meulder (2003). When only modest amounts of training data are
available, transfer learning approaches can transfer the knowledge acquired
from related tasks into the target domain, using techniques such as simple
transfer Rodriguez et al. (2018), discriminative fine-tuning Howard and Ruder
(2018), adversarial transfer Zhou et al. (2019) or layer-wise domain
adaptation approaches Yang et al. (2017); Lin and Lu (2018).

However, in many practical settings, we wish to apply NER to domains where we
have no labelled data, making such transfer learning methods difficult to
apply. This paper presents an alternative approach using weak supervision to
bootstrap named entity recognition models without requiring any labelled data
from the target domain. The approach relies on labelling functions that
automatically annotate documents with named-entity labels. A hidden Markov
model (HMM) is then trained to unify the noisy labelling functions into a
single (probabilistic) annotation, taking into account the accuracy and
confusions of each labelling function. Finally, a sequence labelling model is
trained using a cross-entropy loss on this unified annotation.

As in other weak supervision frameworks, the labelling functions allow us to
inject expert knowledge into the sequence labelling model, which is often
critical when data is scarce or non-existent Hu et al. (2016); Wang and Poon
(2018). New labelling functions can be easily inserted to leverage the
knowledge sources at our disposal for a given textual domain. Furthermore,
labelling functions can often be ported across domains, which is not the case
for manual annotations that must be reiterated for every target domain.

The contributions of this paper are as follows:

  1. 1.

A broad collection of labelling functions for NER, including neural models
trained on various textual domains, gazetteers, heuristic functions, and
document-level constraints.

  2. 2.

A novel weak supervision model suited for sequence labelling tasks and able to
include probabilistic labelling predictions.

  3. 3.

An open-source implementation of these labelling functions and aggregation
model that can scale to large datasets
111https://github.com/NorskRegnesentral/weak-supervision-for-NER..

##  2 Related Work

##### Unsupervised domain adaptation:

Unsupervised domain adaptation attempts to adapt knowledge from a source
domain to predict new instances in a target domain which often has
substantially different characteristics. Earlier approaches often try to adapt
the feature space using pivots Blitzer et al. (2006, 2007); Ziser and Reichart
(2017) to create domain-invariant representations of predictive features.
Others learn low-dimensional transformation features of the data Guo et al.
(2009); Glorot et al. (2011); Chen et al. (2012); Yu and Jiang (2016); Barnes
et al. (2018). Finally, some approaches divide the feature space into general
and domain-dependent features Daumé III (2007). Multi-task learning can also
improve cross-domain performance Peng and Dredze (2017).

Recently, Han and Eisenstein (2019) proposed domain-adaptive fine-tuning,
where contextualised embeddings are first fine-tuned to both the source and
target domains with a language modelling loss and subsequently fine-tuned to
source domain labelled data. This approach outperforms several strong
baselines trained on the target domain of the WNUT 2016 NER task Strauss et
al. (2016).

##### Aggregation of annotations:

Approaches that aggregate annotations from multiples sources have largely
concentrated on noisy data from crowd sourced annotations, with some
annotators possibly being adversarial. The Bayesian Classifier Combination
approach of Kim and Ghahramani (2012) combines multiple independent
classifiers using a linear combination of predictions. Hovy et al. (2013)
learn a generative model able to aggregate crowd-sourced annotations and
estimate the trustworthiness of annotators. Rodrigues et al. (2014) present an
approach based on Conditional Random Fields (CRFs) whose model parameters are
learned jointly using EM. Nguyen et al. (2017b) propose a Hidden Markov Model
to aggregate crowd-sourced sequence annotations and find that explicitly
modelling the annotator leads to improvements for POS-tagging and NER.
Finally, Simpson and Gurevych (2019) proposed a fully Bayesian approach to the
problem of aggregating multiple sequential annotations, using variational EM
to compute posterior distributions over the model parameters.

##### Weak supervision:

The aim of weakly supervised modelling is to reduce the need for hand-
annotated data in supervised training. A particular instance of weak
supervision is distant supervision, which relies on external resources such as
knowledge bases to automatically label documents with entities that are known
to belong to a particular category Mintz et al. (2009); Ritter et al. (2013);
Shang et al. (2018). Ratner et al. (2017, 2019) generalised this approach with
the Snorkel framework which combines various supervision sources using a
generative model to estimate the accuracy (and possible correlations) of each
source. These aggregated supervision sources are then employed to train a
discriminative model. Current frameworks are, however, not easily adaptable to
sequence labelling tasks, as they typically require data points to be
independent. One exception is the work of Wang and Poon (2018), which relies
on deep probabilistic logic to perform joint inference on the full dataset.
Finally, Fries et al. (2017) presented a weak supervision approach to NER in
the biomedical domain. However, unlike the model proposed in this paper, their
approach relies on an ad-hoc mechanism for generating candidate spans to
classify.

The approach most closely related to this paper is Safranchik et al. (2020),
which describe a similar weak supervision framework for sequence labelling
based on an extension of HMMs called linked hidden Markov models. The authors
introduce a new type of noisy rules, called linking rules, to determine how
sequence elements should be grouped into spans of same tag. The main
differences between their approach and this paper are the linking rules, which
are not employed here, and the choice of labelling functions, in particular
the document-level relations detailed in Section 3.1.

##### Ensemble learning:

The proposed approach is also loosely related to ensemble methods such
bagging, boosting and random forests Sagi and Rokach (2018). These methods
rely on multiple classifiers run simultaneously and whose outputs are combined
at prediction time. In contrast, our approach (as in other weak supervision
frameworks) only requires labelling functions to be aggregated once, as an
intermediary step to create training data for the final model. This is a non-
trivial difference as running all labelling functions at prediction time is
computationally costly due to the need to run multiple neural models along
with gazetteers extracted from large knowledge bases.

##  3 Approach

The proposed model collects weak supervision from multiple labelling
functions. Each labelling function takes a text document as input and outputs
a series of spans associated with NER labels. These outputs are then
aggregated using a hidden Markov model (HMM) with multiple emissions (one per
labelling function) whose parameters are estimated in an unsupervised manner.
Finally, the aggregated labels are employed to learn a sequence labelling
model. Figure 1 illustrates this process. The process is performed on
documents from the target domain, e.g. a corpus of financial news.

Figure 1: Illustration of the weak supervision approach.

Labelling functions are typically specialised to detect only a subset of
possible labels. For instance, a gazetteer based on Wikipedia will only detect
mentions of persons, organisations and geographical locations and ignore
entities such as dates or percents. This marks a departure from existing
aggregation methods, which are originally designed for crowd-sourced data and
where annotators are supposed to make use of the full label set. In addition,
unlike previous weak supervision approaches, we allow labelling functions to
produce probabilistic predictions instead of deterministic values. The
aggregation model described in Section 3.2 directly captures these properties
in the emission model associated with each labelling function.

We first briefly describe the labelling functions integrated into the current
system. We review in Section 3.2 the aggregation model employed to combine the
labelling predictions. The final labelling model is presented in Section 3.3.
The complete list of 52 labelling functions employed in the experiments is
available in Appendix A.

###  3.1 Labelling functions

##### Out-of-domain NER models

The first set of labelling functions are sequence labelling models trained in
domains from which labelled data is available. In the experiments detailed in
Section 4, we use four such models, respectively trained on Ontonotes
Weischedel et al. (2011), CoNLL 2003 Tjong Kim Sang and De Meulder
(2003)222The ConLL 2003 NER model is of course deactivated for the
experimental evaluation on ConLL 2003., the Broad Twitter Corpus Derczynski et
al. (2016) and a NER-annotated corpus of SEC filings Salinas Alvarado et al.
(2015).

For the experiments in this paper, all aforementioned models rely on a
transition-based NER model Lample et al. (2016) which extracts features with a
stack of four convolutional layers with filter size of three and residual
connections. The model uses attention features and a multi-layer perceptron to
select the next transition. It is initialised with GloVe embeddings Pennington
et al. (2014) and implemented in Spacy Honnibal and Montani (2017). However,
the proposed approach does not impose any constraints on the model
architecture and alternative approaches based on e.g. contextualised
embeddings can also be employed.

##### Gazetteers

As in distant supervision approaches, we include a number of gazetteers from
large knowledge bases to identify named entities. Concretely, we use resources
from Wikipedia Geiß et al. (2018), Geonames Wick (2015), the Crunchbase Open
Data Map, DBPedia Lehmann et al. (2015) along with lists of countries,
languages, nationalities and religious or political groups.

To efficiently search for occurrences of these entities in large text
collections, we first convert each knowledge base into a trie data structure.
Prefix search is then applied to extract matches (using both case-sensitive
and case-insensitive mode, as they have distinct precision-recall trade-offs).

##### Heuristic functions

We also include various heuristic functions, each specialised in the
recognition of specific types of named entities. Several functions are
dedicated to the recognition of proper names based on casing, part-of-speech
tags or dependency relations. In addition, we integrate a variety of
handcrafted functions relying on regular expressions to detect occurrences of
various entities (see Appendix A for details). A probabilistic parser
specialised in the recognition of dates, times, money amounts, percents, and
cardinal/ordinal values Braun et al. (2017) is also incorporated.

##### Document-level relations

All labelling functions described above rely on local decisions on tokens or
phrases. However, texts are not loose collections of words, but exhibit a high
degree of internal coherence Grosz and Sidner (1986); Grosz et al. (1995)
which can be exploited to further improve the annotations.

We introduce one labelling function to capture label consistency constraints
in a document. As noted in Krishnan and Manning (2006); Wang et al. (2018),
named entities occurring multiple times through a document have a high
probability of belonging to the same category. For instance, while Komatsu may
both refer to a Japanese town or a multinational corporation, a text including
this mention will either be about the town or the company, but rarely both at
the same time. To capture these non-local dependencies, we define the
following label consistency model: given a text span $e$ occurring in a given
document, we look for all spans $Z_{e}$ in the document that contain the same
string as $e$. The (probabilistic) output of the labelling function then
corresponds to the relative frequency of each label $l$ for that string in the
document:

$P_{{\textsf{doc\\_majority}(e)}}(l)=\frac{\sum_{z\in
Z_{e}}P_{{\textsf{label}(z)}}(l)}{|Z_{e}|}$ (1)

The above formula depends on a distribution $P_{{\textsf{label}(z)}}$, which
can be defined on the basis of other labelling functions. Alternatively, a
two-stage model similar to Krishnan and Manning (2006) could be employed to
first aggregate local labelling functions and subsequently apply document-
level functions on aggregated predictions.

Another insight from Grosz and Sidner (1986) is the importance of the
attentional structure. When introduced for the first time, named entities are
often referred to in an explicit and univocal manner, while subsequent
mentions (once the entity is a part of the focus structure) frequently rely on
shorter references. The first mention of a person in a given text is for
instance likely to include the person’s full name, and is often shortened to
the person’s last name in subsequent mentions. As in Ratinov and Roth (2009),
we determine whether a proper name is a substring of another entity mentioned
earlier in the text. If so, the labelling function replicates the label
distribution of the first entity.

###  3.2 Aggregation model

The outputs of these labelling functions are then aggregated into a single
layer of annotation through an aggregation model. As we do not have access to
labelled data for the target domain, this model is estimated in a fully
unsupervised manner.

##### Model

We assume a list of $J$ labelling functions $\\{\lambda_{1},...\lambda_{J}\\}$
and a list of $S$ mutually exclusive NER labels $\\{l_{1},...l_{S}\\}$. The
aggregation model is represented as an HMM, in which the states correspond to
the true underlying labels. This model has multiple emissions (one per
labelling function) assumed to be mutually independent conditional on the
latent underlying label.

Formally, for each token $i\in\\{1,...,n\\}$ and labelling function $j$, we
assume a Dirichlet distribution for the probability labels
$\boldsymbol{P_{ij}}$. The parameters of this Dirichlet are separate vectors
$\boldsymbol{\alpha_{j}^{s_{i}}}\in\mathcal{R}^{S}_{[0,1]}$, for each of the
latent states $s_{i}\in\\{1,...,S\\}$. The latent states are assumed to have a
Markovian dependence structure between the tokens $\\{1,...,n\\}$. This
results in the HMM represented by a dependent mixtures of Dirichlet model:

$\displaystyle\boldsymbol{P_{ij}}|\boldsymbol{\alpha^{s_{i}}_{j}}\quad\overset{ind}{\sim}\quad\text{Dirichlet}\left(\boldsymbol{\alpha_{j}^{s_{i}}}\right),$
(2) $\displaystyle
p(s_{i}|s_{i-1})\quad=\quad\text{logit}^{-1}\left(\omega^{(s_{i},s_{i-1})}\right),$
(3)
$\displaystyle\text{logit}^{-1}\left(\omega^{(s_{i},s_{i-1})}\right)=\tfrac{e^{\omega^{(s_{i},s_{i-1})}}}{1+e^{\omega^{(s_{i},s_{i-1})}}}.$
(4)

Here, $\omega^{(s_{i},s_{i-1})}\in\mathcal{R}$ are the parameters of the
transition probability matrix controlling for a given state $s_{i-1}$ the
probability of transition to state $s_{i}$. Figure 2 illustrates the model
structure.

Thepluggedwellshave…$s_{i-1}$$\ s_{i\ }\
$$s_{i+1}$$s_{i+2}$…$\boldsymbol{\alpha}_{j}^{s_{i}}$$\boldsymbol{P}_{ij}$Labelling
function $j\in\\{1,...J\\}$ Figure 2: Aggregation model using a hidden Markov
model with multiple probabilistic emissions.

##### Parameter estimation

The learnable parameters of this HMM are (a) the transition matrix between
states and (b) the $\boldsymbol{\alpha}$ vectors of the Dirichlet distribution
associated with each labelling function. The transition matrix is of size
$|S|\times|S|$, while we have $|S|\times|J|$ $\boldsymbol{\alpha}$ vectors,
each of size $|S|$. The parameters are estimated with the Baum-Welch
algorithm, which is a variant of EM algorithm that relies on the forward-
backward algorithm to compute the statistics for the expectation step.

To ensure faster convergence, we introduce a new constraint to the likelihood
function: for each token position $i$, the corresponding latent label $s_{i}$
must have a non-zero probability in at least one labelling function (the
likelihood of this label is otherwise set to zero for that position). In other
words, the aggregation model will only predict a particular label if this
label is produced by least one labelling function. This simple constraint
facilitates EM convergence as it restricts the state space to a few possible
labels at every time-step.

##### Prior distributions

The HMM described above can be provided with informative priors. In
particular, the initial distribution for the latent states can be defined as a
Dirichlet based on counts $\delta$ for the most reliable labelling
function333The most reliable labelling function was found in our experiments
to be the NER model trained on Ontonotes 5.0.:

$\displaystyle p(s_{i})\overset{d}{=}\text{Dirichlet}(\boldsymbol{\delta}).$
(5)

The prior for each row $k$ of the transition probabilities matrix is also a
Dirichlet based on the frequencies of transitions between the observed classes
for the most reliable labelling function $\boldsymbol{\kappa}_{k}$:

$\displaystyle
p(s_{i}|s_{i-1}=k)\overset{d}{=}\text{Dirichlet}({\boldsymbol{\kappa}_{k}}).$
(6)

Finally, to facilitate convergence of the EM algorithm, informative starting
values can be specified for the emission model of each labelling function.
Assuming we can provide rough estimates of the recall $r_{jk}$ and precision
$\rho_{jk}$ for the labelling function $j$ on label $k$, the initial values
for the parameters of the emission model are expressed as:

$\displaystyle\alpha_{jk}^{s_{i}}\propto\begin{cases}r_{jk},\text{ if
}s_{i}=k,\\\
\left(1-r_{s_{i}k}\right)\left(1-\rho_{jk}\right)\delta_{k},\text{ if
}s_{i}\neq k.\end{cases}$

The probability of observing a given label $k$ emitted by the labelling
function $j$ is thus proportional to its recall if the true label is indeed
$k$. Otherwise (i.e. if the labelling function made an error), the probability
of emitting $k$ is inversely proportional to the precision of the labelling
function $j$.

##### Decoding

Once the parameters of the HMM model are estimated, the forward-backward
algorithm can be employed to associate each token marginally with a posterior
probability distribution over possible NER labels Rabiner (1990).

###  3.3 Sequence labelling model

Once the labelling functions are aggregated on documents from the target
domain, we can train a sequence labelling model on the unified annotations,
without imposing any constraints on the type of model to use. To take
advantage of the posterior marginal distribution ${\tilde{p}_{s}}$ over the
latent labels, the optimisation should seek to minimise the expected loss with
respect to ${\tilde{p}_{s}}$:

$\hat{\theta}=\operatorname*{arg\,min}_{\theta}\sum_{i}^{n}\mathbb{E}_{y\sim{\tilde{p}_{s}}}\left[loss(h_{\theta}(x_{i}),y)\right]$
(7)

where $h_{\theta}(\cdot)$ is the output of the sequence labelling model. This
is equivalent to minimising the cross-entropy error between the outputs of the
neural model and the probabilistic labels produced by the aggregation model.

##  4 Evaluation

We evaluate the proposed approach on two English-language datasets, namely the
CoNLL 2003 dataset and a collection of sentences from Reuters and Bloomberg
news articles annotated with named entities by crowd-sourcing. We include a
second dataset in order to evaluate the approach with a more fine-grained set
of NER labels than the ones in CoNLL 2003. As the objective of this paper is
to compare approaches to unsupervised domain adaptation, we do not rely on any
labelled data from these two target domains.

###  4.1  Data

##### CoNLL 2003

The CoNLL 2003 dataset Tjong Kim Sang and De Meulder (2003) consists of 1163
documents, including a total of 35089 entities spread over 4 labels: ORG, PER,
LOC and MISC.

##### Reuters & Bloomberg

We additionally crowd annotate 1054 sentences from Reuters and Bloomberg news
articles from Ding et al. (2014). We instructed the annotators to tag
sentences with the following 9 Ontonotes-inspired labels: PERSON, NORP, ORG,
LOC, PRODUCT, DATETIME, PERCENT, MONEY, QUANTITY. Note that the DATE and TIME
labels from Ontonotes are merged into DATETIME, and the LOC and GPE labels are
similarly merged into LOC. Each sentence was annotated by at least two
annotators, and a qualifying test with gold-annotated questions was conducted
for quality control. Cohen’s $\kappa$ for sentences with two annotators is
0.39, while Krippendorff’s $\alpha$ for three annotators is 0.44. We had to
remove QUANTITY labels from the annotations as the crowd results for this
particular label were highly inconsistent.

###  4.2 Baselines

##### Ontonotes-trained NER

The first baseline corresponds to a neural sequence labelling model trained on
the Ontonotes 5.0 corpus. We use here the same model from Section 3.1, which
is the single best-performing labelling function (that is, without aggregating
multiple predictions).

We also experimented with other neural architectures but these performed
similar or worse than the transition-based model, presumably because they are
more prone to overfitting on the source domain.

##### Majority voting (MV)

The simplest method for aggregating outputs is majority voting, i.e.
outputting the most frequent label among the ones predicted by each labelling
function. However, specialised labelling functions will output O for most
tokens, which means that the majority label is typically O. To mitigate this
problem, we first look at tokens that are marked with a non-O label by at
least $T$ labelling functions (where $T$ is a hyper-parameter tuned
experimentally), and then apply majority voting on this set of non-O labels.

##### Snorkel model

The Snorkel framework Ratner et al. (2017) does not directly support sequence
labelling tasks as data points are required to be independent. However,
heuristics can be used to extract named-entity candidates and then apply
labelling functions to infer their most likely labels Fries et al. (2017). For
this baseline, we use the three functions nnp_detector, proper_detector and
compound_detector (see Appendix A) to generate candidate spans. We then create
a matrix expressing the prediction of each labelling function for each span
(including a specific ”abstain” value to denote the absence of predictions)
and run the matrix-completion-style approach of Ratner et al. (2019) to
aggregate the predictions.

##### mSDA

is a strong domain adaptation baseline Chen et al. (2012) which augments the
feature space of a model with intermediate representations learned using
stacked denoising autoencoders. In our case, we learn the mSDA representations
on the unlabeled source and target domain data. These 800 dimensional vectors
are concatenated to 300 dimensional word embeddings and fed as input to a two-
layer LSTM with a skip connection. Finally, we train the LSTM on the labeled
source data and test on the target domain.

##### AdaptaBERT

This baseline corresponds to a state-of-the-art unsupervised domain adaptation
approach (AdaptaBERT) Han and Eisenstein (2019). The approach first uses
unlabeled data from both the source and target domains to domain-tune a
pretrained BERT model. The model is finally task-tuned in a supervised fashion
on the source domain labelled data (Ontonotes). At inference time, the model
is able to make use of the pretraining and domain tuning to predict entities
in the target domain. In our experiments, we use the cased-version of the base
BERT model (trained on Wikipedia and news text) and perform three fine-tuning
epochs for both domain-tuning and task-tuning. We additionally include an
ensemble model, which averages the predictions of five BERT models fine-tuned
with different random seeds.

#### Mixtures of multinomials

Following the notation from Section 3.2, we define
$Y_{i,j,k}=\text{I}(P_{i,j,k}=\max_{k^{\prime}\in\\{1,...,S\\}}P_{i,j,k^{\prime}})$
to be the most probable label for word $i$ by source $j$. One can model
$\boldsymbol{Y_{ij}}$ with a Multinomial probability distribution. The first
four baselines (the fifth one assumes Markovian dependence between the latent
states) listed below use the following independent, i.e.
$p(s_{i},s_{i-1})=p(s_{i})p(s_{i-1})$, mixtures of Multinomials model for
$\boldsymbol{Y_{ij}}$:

$\displaystyle\boldsymbol{Y_{ij}}|\boldsymbol{p^{s_{i}}_{j}}$
$\displaystyle\quad\overset{ind}{\sim}\quad\text{Multinomial}(\boldsymbol{p_{j}^{s_{i}}}),$
$\displaystyle\boldsymbol{s}_{i}$
$\displaystyle\quad\overset{ind}{\sim}\quad\text{Multinomial}(\boldsymbol{\sigma}).$

##### Accuracy model (ACC)

Rodrigues et al. (2014) assumes the following constraints on
$\boldsymbol{p^{s_{i}}_{j}}$:

$\displaystyle p_{jk}^{s_{i}}=\begin{cases}\pi_{j},\text{ if }s_{i}=k,\\\
\tfrac{1-\pi_{j}}{J-1}s_{i}\neq k.\end{cases}$

Here, for each labelling function it is assumed to have the same accuracy
$\pi_{j}$ for all of the tokens.

##### Confusion vector (CV)

Nguyen et al. (2017a) extends ACC by relying on separate success probabilities
for each token label:

$\displaystyle p_{jk}^{s_{i}}=\begin{cases}\pi_{jk},\text{ if }s_{i}=k,\\\
\tfrac{1-\pi_{jk}}{J-1}s_{i}\neq k.\end{cases}$

##### Confusion matrix (CM)

Dawid and Skene (1979) allows for distinct accuracies conditional on the
latent states, which results in:

$\displaystyle p_{jk}^{s_{i}}=\pi_{jk}^{s_{i}}.$ (8)

##### Sequential Confusion Matrix (SEQ)

extends the CM model of Simpson and Gurevych (2019), where an ”auto-
regressive” component is included in the observed part of the model. We assume
dependence on a covariate indicating that the label has not changed for a
given source, i.e.:

$\displaystyle
p_{jk}^{s_{i}}=\text{logit}^{-1}(\mu_{jk}^{s_{i}}+\text{I}(Y_{i-1,j,k}^{T}=Y_{i,j,k}^{T}){\beta}_{jk}^{s_{i}}).$

##### Dependent confusion matrix (DCM)

combines the CM-distinct accuracies conditional on the latent states of (8)
and the Markovian dependence of (3).

###  4.3 Results

| Token-level | Entity-level
---|---|---
Model: | P | R | $F_{1}$ | CEE | P | R | $F_{1}$
Ontonotes-trained NER | 0.719 | 0.706 | 0.712 | 2.671 | 0.694 | 0.620 | 0.654
Majority voting (MV) | 0.815 | 0.675 | 0.738 | 2.047 | 0.751 | 0.619 | 0.678
Confusion Matrix (CM) | 0.786 | 0.746 | 0.766 | 1.964 | 0.713 | 0.700 | 0.706
Sequential Confusion Matrix (SEQ) | 0.736 | 0.716 | 0.726 | 2.254 | 0.642 | 0.668 | 0.654
Dependent Confusion Matrix (DCM) | 0.785 | 0.744 | 0.764 | 1.983 | 0.710 | 0.698 | 0.704
Snorkel-aggregated labels | 0.710 | 0.661 | 0.684 | 2.264 | 0.714 | 0.621 | 0.664
mSDA (OntoNotes) | 0.640 | 0.569 | 0.603 | 2.813 | 0.560 | 0.562 | 0.561
AdaptaBERT (OntoNotes) | 0.693 | 0.733 | 0.712 | 2.280 | 0.652 | 0.736 | 0.691
AdaptaBERT (Ensemble) | 0.704 | 0.754 | 0.729 | 2.103 | 0.684 | 0.743 | 0.712
HMM-aggregated labels (only NER models) | 0.658 | 0.720 | 0.688 | 2.653 | 0.642 | 0.599 | 0.620
HMM-aggregated labels (only gazetteers) | 0.759 | 0.394 | 0.518 | 3.678 | 0.687 | 0.367 | 0.478
HMM-aggregated labels (only heuristics) | 0.722 | 0.771 | 0.746 | 1.989 | 0.718 | 0.683 | 0.700
HMM-aggregated labels (all but doc-level) | 0.714 | 0.778 | 0.744 | 1.878 | 0.713 | 0.693 | 0.702
HMM-aggregated labels (all functions) | 0.719 | 0.794 | 0.754 | 1.812 | 0.721 | 0.713 | 0.716
Neural net trained on HMM-agg. labels | 0.712 | 0.790 | 0.748 | 2.282 | 0.715 | 0.707 | 0.710
Table 1: Evaluation results on CoNLL 2003. MV=Majority Voting, P=Precision, R=Recall, CEE=Cross-entropy Error (lower is better). The results are micro-averaged on all labels (PER, ORG, LOC and MISC). | Token-level | Entity-level
---|---|---
Model: | P | R | $F_{1}$ | CEE | P | R | $F_{1}$
OntoNotes-trained NER | 0.793 | 0.791 | 0.792 | 2.648 | 0.694 | 0.635 | 0.664
Majority voting (MV) | 0.832 | 0.713 | 0.768 | 2.454 | 0.699 | 0.644 | 0.670
Confusion Matrix (CM) | 0.816 | 0.702 | 0.754 | 2.708 | 0.667 | 0.636 | 0.652
Sequential Confusion Matrix (SEQ) | 0.741 | 0.630 | 0.682 | 3.261 | 0.535 | 0.547 | 0.540
Dependent Confusion Matrix (DCM) | 0.819 | 0.706 | 0.758 | 2.702 | 0.673 | 0.641 | 0.656
mSDA (OntoNotes) | 0.749 | 0.751 | 0.750 | 2.501 | 0.618 | 0.684 | 0.649
AdaptaBERT (OntoNotes) | 0.799 | 0.801 | 0.800 | 2.351 | 0.668 | 0.734 | 0.699
AdaptaBERT (Ensemble) | 0.813 | 0.815 | 0.814 | 2.265 | 0.682 | 0.748 | 0.713
HMM-aggregated labels (all functions) | 0.804 | 0.823 | 0.814 | 2.219 | 0.749 | 0.697 | 0.722
Neural net trained on HMM-agg. labels | 0.805 | 0.827 | 0.816 | 2.448 | 0.749 | 0.701 | 0.724
Table 2: Evaluation results on 1094 crowd-annotated sentences from Reuters and
Bloomberg news articles. The results are micro-averaged on 8 labels (PERSON,
NORP, ORG, LOC, PRODUCT, DATE, PERCENT, and MONEY).

The evaluation results are shown in Tables 1 and 2, respectively for the CoNLL
2003 data and the sentences extracted from Reuters and Bloomberg. The metrics
are the (micro-averaged) precision, recall and $F_{1}$ scores at both the
token-level and entity-level. In addition, we indicate the token-level cross-
entropy error (in log-scale). As the labelling functions are defined on a
richer annotation scheme than the four labels of ConLL 2003, we map GPE to LOC
and EVENT, FAC, LANGUAGE, LAW, NORP, PRODUCT and WORK_OF_ART to MISC.

The results for the ACC and CV baselines are not included in the two tables as
the parameter estimation did not converge (and thus did not provide reliable
estimates of the parameters).

Table 1 further details the results obtained using only a subset of labelling
functions. Of particular interest is the positive contribution of document-
level functions, boosting the entity-level $F_{1}$ from 0.702 to 0.716. This
highlights the importance of document-level relations in NER.

The last line of the two tables reports the performance of the neural sequence
labelling model (described in Section 3.3) trained on the basis of the
aggregated labels. We observe that the performance of this neural model
remains close to the performance of the HMM-aggregated labels. This result
shows that the knowledge from the labelling functions can be injected into a
standard neural model without substantial loss.

###  4.4 Discussion

Although not shown in the results due to space constraints, we also analysed
whether the informative priors described in Section 3.2 influenced the
performance of the aggregation model. We found informative and non-informative
priors to yield similar performance for CoNLL 2003. However, the performance
of non-informative priors was very poor on the Reuters and Bloomberg sentences
($F_{1}$ at 0.12), thereby demonstrating the usefulness of informative priors
for small datasets.

We provide in Figure 3 an example with a few selected labelling functions. In
particular, we can observe that the Ontonotes-trained NER model mistakenly
labels ”Heidrun” as a product. This erroneous label, however, is counter-
balanced by other labelling functions, notably a document-level function
looking at the global label frequency of this string through the document. We
do, however, notice a few remaining errors, e.g. the labelling of ”Status
Weekly” as an organisation.

Figure 3: Extended example showing the outputs of 6 labelling functions, along
with the HMM-aggregated model. Figure 4: Pairwise agreement (left) and
disagreement (right) between the labelling functions on the CoNLL 2003 data
with labels PER, ORG, LOC, MISC, normalized by total number of labelled
examples.

Figure 4 illustrates the pairwise agreement and disagreement between labelling
functions on the CoNLL 2003 dataset. If both labelling functions make the same
prediction on a given token, we count this as an agreement, whereas
conflicting predictions (ignoring O labels), are seen as disagreement. Large
differences may exist between these functions for specific labels, especially
MISC. The functions with the highest overlap are those making predictions on
all labels, while labelling functions specialised to few labels (such as
legal_detector) often have less overlap. We also observe that the two
gazetteers from Crunchbase and Geonames disagree in about 15% of cases,
presumably due to company names that are also geographical locations, as in
the earlier Komatsu example.

In terms of computational efficiency, the estimation of HMM parameters is
relatively fast, requiring less than 30 mins on the entire CoNLL 2003 data.
Once the aggregation model is estimated, it can be directly applied to new
texts with a single forward-backward pass, and can therefore scale to datasets
with hundreds of thousands of documents. This runtime performance is an
important advantage compared to approaches such as AdaptaBERT Han and
Eisenstein (2019) which are relatively slow at inference time. The proposed
approach can also be ported to other languages than English, although
heuristic functions and gazetteers will need to be adapted to the target
language.

##  5 Conclusion

This paper presented a weak supervision model for sequence labelling tasks
such as Named Entity Recognition. To leverage all possible knowledge sources
available for the task, the approach uses a broad spectrum of labelling
functions, including data-driven NER models, gazetteers, heuristic functions,
and document-level relations between entities. Labelling functions may be
specialised to recognise specific labels while ignoring others. Furthermore,
unlike previous weak supervision approaches, labelling functions may produce
probabilistic predictions. The outputs of these labelling functions are then
merged together using a hidden Markov model whose parameters are estimated
with the Baum-Welch algorithm. A neural sequence labelling model can finally
be learned on the basis of these unified predictions.

Evaluation results on two datasets (CoNLL 2003 and news articles from Reuters
and Bloomberg) show that the method can boost NER performance by about 7
percentage points on entity-level $F_{1}$. In particular, the proposed model
outperforms the unsupervised domain adaptation approach through contextualised
embeddings of Han and Eisenstein (2019). Of specific linguistic interest is
the contribution of document-level labelling functions, which take advantage
of the internal coherence and narrative structure of the texts.

Future work will investigate how to take into account potential correlations
between labelling functions in the aggregation model, as done in e.g. Bach et
al. (2017). Furthermore, some of the labelling functions can be rather noisy
and model selection of the optimal subset of the labelling functions might
well improve the performance of our model. Model selection approaches that can
be adapted are discussed in Adams and Beling (2019); Hubin (2019). We also
wish to evaluate the approach on other types of sequence labelling tasks
beyond Named Entity Recognition.

## Acknowledgements

The research presented in this paper was conducted as part of the innovation
project ”FinAI: Artificial Intelligence tool to monitor global financial
markets” in collaboration with Exabel AS444www.exabel.com. This collaboration
is supported through the funding programme for ”User-driven Research based
Innovation” of the Research Council of Norway.

Additionally, this work is supported by the SANT project (Sentiment Analysis
for Norwegian Text), funded by the Research Council of Norway (grant number
270908)

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##  Appendix A Labelling functions

Group | Function name |  Description
---|---|---
Neural NER models | BTC |  Model trained on the Broad Twitter Corpus
BTC+c |  Model trained on the Broad Twitter Corpus + postprocessing
SEC |  Model trained on SEC-filings
SEC+c |  Model trained on SEC-filings + postprocessing
conll2003 |  Model trained on CoNLL 2003
conll2003+c |  Model trained on CoNLL 2003 + postprocessing
core_web_md |  Model trained on Ontonotes 5.0
core_web_md+c |  Model trained on Ontonotes 5.0 + postprocessing
|  |
Gazetteers | wiki_cased |  Gazetteer (case-sensitive) using Wikipedia entries
multitoken_wiki_cased |  Same as above, but restricted to multitoken entities
wiki_uncased |  Gazetteer (case-insensitive) using Wikipedia entries
multitoken_wiki_uncased |  Same as above, but restricted to multitoken entities
wiki_small_cased |  Gazetteer (case-sensitive) using Wikipedia entries with non-empty description
multitoken_wiki_small_cased |  Same as above, but restricted to multitoken entities
wiki_small_uncased |  Gazetteer (case-insensitive) using Wikipedia entries with non-empty description
multitoken_wiki_small_uncased |  Same as above, but restricted to multitoken entities
company_cased |  Gazetteer (case-sensitive) using a large list of company names
multitoken_company_cased |  Same as above, but restricted to multitoken entities
company_uncased |  Gazetteer from a large list of company names (case-insensitive)
multitoken_company_uncased |  Same as above, but restricted to multitoken entities
crunchbase_cased |  Gazetteer (case-sensitive) using the Crunchbase Open Data Map
multitoken_crunchbase_cased |  Same as above, but restricted to multitoken entities
crunchbase_uncased |  Gazetteer (case-insensitive) using the Crunchbase Open Data Map
multitoken_crunchbase_uncased |  Same as above, but restricted to multitoken entities
geo_cased |  Gazetteer (case-sensitive) using the Geonames database
multitoken_geo_cased |  Same as above, but restricted to multitoken entities
geo_uncased |  Gazetteer (case-insensitive) using the Geonames database
multitoken_geo_uncased |  Same as above, but restricted to multitoken entities
product_cased |  Gazetteer (case-sensitive) using products extracted from DBPedia
multitoken_product_cased |  Same as above, but restricted to multitoken entities
product_uncased |  Gazetteer (case-insensitive) using products extracted from DBPedia
multitoken_product_uncased |  Same as above, but restricted to multitoken entities
|  |
Heuristic functions | date_detector |  Detection of entities of type DATE
time_detector |  Detection of entities of type TIME
money_detector |  Detection of entities of type MONEY
number_detector |  Detection of entities CARDINAL, ORDINAL, PERCENT and QUANTITY
legal_detector |  Detection of entities of type LAW
misc_detector |  Detection of entities of type NORP, LANGUAGE, FAC or EVENT
full_name_detector |  Heuristic function to detect full person names
company_type_detector |  Detection of companies with a legal type suffix
nnp_detector |  Detection of sequences of tokens with NNP as POS-tag
infrequent_nnp_detector |  Detection of sequences of tokens with NNP as POS-tag
|  \+ including at least one infrequent token (rank $>$ 15000 in vocabulary)
proper_detector |  Detection of proper names based on casing
infrequent_proper_detector |  Detection of proper names based on casing
|  \+ including at least one infrequent token
proper2_detector |  Detection of proper names based on casing
infrequent_proper2_detector |  Detection of proper names based on casing
|  \+ including at least one infrequent token
| compound_detector |  Detection of proper noun phrases with compound dependency relations
| infrequent_compound_detector |  Detection of proper noun phrases with compound dependency relations
|  |  \+ including at least one infrequent token
| snips |  Probabilistic parser specialised in the recognition of dates, times, money amounts, percents, and cardinal/ordinal values
|  |
Document-level functions | doc_history |  Entity classification based on already introduced entities in the document
doc_majority_cased |  Entity classification based on majority labels in document (case-sensitive)
doc_majority_uncased |  Entity classification based on majority labels in document (case-insensitive)
Table 3: Full list of labelling functions employed in the experiments. The
neural NER models are provided in two versions: one that directly outputs the
raw model predictions, and one that runs a shallow postprocessing step on the
model predictions to correct known recognition errors (for instance, ensuring
that a numeric amount that is either preceded or followed by a currency symbol
is always classified as an entity of type MONEY).

##  Appendix B Label matching problem

The baseline models relying on mixtures of multinomials have to address the
so-called label matching problem, which needs some extra care.

The following approach was employed in the experiments from Section 4:

  * •

First, we put strong initial values to the probabilities $\boldsymbol{\sigma}$
of individual classes based on the frequency of appearance of these classes in
the most reliable labelling function. This is expected to increase the
probability of EM exploring the mode around the initialised values.

  * •

Second, we perform post-processing and set the labels to the states
corresponding to the labels with the highest pairwise correlations to the
latent labels from one of the three options:

    1. 1.

the most reliable labelling function (Ontonotes-trained NER);

    2. 2.

the majority voting labelling function;

    3. 3.

the suggested Dirichlet dependent mixture model.

Additionally, if this highest correlation is below the threshold of $0.1$ the
O label is assigned to the corresponding state. We empirically observed that
the label matching technique that performed best was to map the states to the
labels produced by the majority voter (based on the pairwise correlations).

##  Appendix C Detailed results

In Table 4, we provide the detailed results distributed by NER label for the
CoNLL data 2003 which were presented in micro-averaged form in Table 1 of the
main paper.

Label | Proportion | Model | Token-level | Entity-level
---|---|---|---|---
|  |  | P | R | $F_{1}$ | P | R | $F_{1}$
LOC | 30.3 % | Ontonotes-trained NER | 0.767 | 0.812 | 0.788 | 0.764 | 0.800 | 0.782
|  | Majority voting (MV) | 0.740 | 0.839 | 0.786 | 0.739 | 0.828 | 0.780
|  | Confusion Matrix | 0.721 | 0.895 | 0.798 | 0.714 | 0.890 | 0.792
|  | Sequential Confusion Matrix | 0.681 | 0.856 | 0.758 | 0.664 | 0.848 | 0.744
|  | Dependent Confusion Matrix | 0.718 | 0.890 | 0.794 | 0.710 | 0.886 | 0.788
|  | Snorkel-aggregated labels | 0.634 | 0.855 | 0.728 | 0.676 | 0.747 | 0.710
|  | HMM (only NER models) | 0.601 | 0.825 | 0.696 | 0.650 | 0.733 | 0.690
|  | HMM (only gazetteers) | 0.707 | 0.632 | 0.668 | 0.694 | 0.630 | 0.660
|  | HMM (heuristics) | 0.715 | 0.870 | 0.784 | 0.745 | 0.832 | 0.786
|  | HMM (all but doc-level) | 0.701 | 0.862 | 0.774 | 0.724 | 0.838 | 0.776
|  | HMM (all functions) | 0.726 | 0.859 | 0.786 | 0.738 | 0.839 | 0.786
|  | NN trained on HMM | 0.736 | 0.851 | 0.790 | 0.734 | 0.850 | 0.788
PER | 28.7 % | Ontonotes-trained NER | 0.850 | 0.833 | 0.842 | 0.787 | 0.741 | 0.764
|  | Majority voting (MV) | 0.915 | 0.871 | 0.892 | 0.831 | 0.775 | 0.802
|  | Confusion Matrix | 0.891 | 0.921 | 0.906 | 0.806 | 0.834 | 0.820
|  | Sequential Confusion Matrix | 0.849 | 0.879 | 0.864 | 0.730 | 0.789 | 0.758
|  | Dependent Confusion Matrix | 0.892 | 0.920 | 0.906 | 0.806 | 0.834 | 0.820
|  | Snorkel-aggregated labels | 0.816 | 0.903 | 0.858 | 0.769 | 0.717 | 0.742
|  | HMM (only NER models) | 0.837 | 0.860 | 0.848 | 0.770 | 0.744 | 0.756
|  | HMM (only gazetteers) | 0.917 | 0.452 | 0.606 | 0.835 | 0.391 | 0.532
|  | HMM (heuristics) | 0.836 | 0.933 | 0.882 | 0.791 | 0.799 | 0.794
|  | HMM (all but doc-level) | 0.859 | 0.917 | 0.888 | 0.814 | 0.782 | 0.798
|  | HMM (all functions) | 0.857 | 0.947 | 0.900 | 0.820 | 0.826 | 0.822
|  | NN trained on HMM | 0.856 | 0.946 | 0.898 | 0.814 | 0.824 | 0.818
ORG | 26.6 % | Ontonotes-trained NER | 0.536 | 0.517 | 0.526 | 0.437 | 0.306 | 0.360
|  | Majority voting (MV) | 0.725 | 0.512 | 0.600 | 0.610 | 0.434 | 0.508
|  | Confusion Matrix | 0.698 | 0.613 | 0.652 | 0.571 | 0.537 | 0.554
|  | Sequential Confusion Matrix | 0.632 | 0.590 | 0.610 | 0.485 | 0.515 | 0.500
|  | Dependent Confusion Matrix | 0.696 | 0.613 | 0.652 | 0.567 | 0.536 | 0.552
|  | Snorkel-aggregated labels | 0.512 | 0.639 | 0.568 | 0.519 | 0.496 | 0.508
|  | HMM (only NER models) | 0.516 | 0.549 | 0.532 | 0.425 | 0.333 | 0.374
|  | HMM (only gazetteers) | 0.648 | 0.304 | 0.414 | 0.512 | 0.235 | 0.322
|  | HMM (heuristics) | 0.566 | 0.625 | 0.594 | 0.549 | 0.501 | 0.524
|  | HMM (all but doc-level) | 0.565 | 0.631 | 0.596 | 0.551 | 0.494 | 0.520
|  | HMM (all functions) | 0.542 | 0.665 | 0.598 | 0.545 | 0.527 | 0.536
|  | NN trained on HMM | 0.539 | 0.665 | 0.596 | 0.537 | 0.519 | 0.528
MISC | 14.4 % | Ontonotes-trained NER | 0.676 | 0.599 | 0.636 | 0.702 | 0.583 | 0.636
|  | Majority voting (MV) | 0.861 | 0.187 | 0.308 | 0.809 | 0.193 | 0.312
|  | Confusion Matrix | 0.895 | 0.319 | 0.470 | 0.850 | 0.332 | 0.478
|  | Sequential Confusion Matrix | 0.850 | 0.320 | 0.464 | 0.791 | 0.333 | 0.468
|  | Dependent Confusion Matrix | 0.893 | 0.318 | 0.468 | 0.844 | 0.330 | 0.474
|  | Snorkel-aggregated labels | 0.852 | 0.398 | 0.542 | 0.863 | 0.400 | 0.546
|  | HMM (only NER models) | 0.667 | 0.544 | 0.600 | 0.708 | 0.518 | 0.598
|  | HMM (only gazetteers) | 0.745 | 0.011 | 0.022 | 0.594 | 0.008 | 0.016
|  | HMM (heuristics) | 0.842 | 0.499 | 0.626 | 0.850 | 0.478 | 0.612
|  | HMM (all but doc-level) | 0.714 | 0.596 | 0.650 | 0.781 | 0.575 | 0.662
|  | HMM (all functions) | 0.814 | 0.571 | 0.672 | 0.830 | 0.565 | 0.672
|  | NN trained on HMM | 0.852 | 0.577 | 0.688 | 0.866 | 0.583 | 0.696
Table 4: Detailed evaluation results on the CoNLL2003 dataset, depending on
NER labels.


# Universal Enveloping Algebras of Braided m-Lie Algebras

Shouchuan Zhang, Jieqiong He
Department of Mathematics, Hunan University
Changsha 410082, P.R. China

###### Abstract

Universal enveloping algebras of braided m-Lie algebras and PBW theorem are
obtained by means of combinatorics on words.

2000 Mathematics Subject Classification: 16W30, 16G10

keywords: Braided Lie algebra, Universal enveloping algebras.

##  0 Introduction

The theory of Lie superalgebras has been developed systematically, which
includes the representation theory and classifications of simple Lie
superalgebras and their varieties [8] [3]. In many physical applications or in
pure mathematical interest, one has to consider not only ${\bf Z}_{2}$\- or
${\bf Z}$\- grading but also $G$-grading of Lie algebras, where $G$ is an
abelian group equipped with a skew symmetric bilinear form given by a
2-cocycle. Lie algebras in symmetric and more general categories were
discussed in [6] and [5]. A sophisticated multilinear version of the Lie
bracket was considered in [9] [15]. Various generalized Lie algebras have
already appeared under different names, e.g. Lie color algebras, $\epsilon$
Lie algebras [14], quantum and braided Lie algebras, generalized Lie algebras
[2] and $H$-Lie algebras [1].

In [12], Majid introduced braided Lie algebras from geometrical point of view,
which have attracted attention in mathematics and mathematical physics (see
e.g. [13] and references therein).

In paper [17], braided m-Lie algebras was introduced, which generalize Lie
algebras, Lie color algebras and quantum Lie algebras. Two classes of braided
m-Lie algebras are given, which are generalized matrix braided m-Lie algebras
and braided m-Lie subalgebras of $End_{F}M$, where $M$ is a Yetter-Drinfeld
module over $B$ with dim $B<\infty$ . In particular, generalized classical
braided m-Lie algebras $sl_{q,f}(GM_{G}(A),F)$ and $osp_{q,t}(GM_{G}(A),M,F)$
of generalized matrix algebra $GM_{G}(A)$ are constructed and their connection
with special generalized matrix Lie superalgebra $sl_{s,f}(GM_{{\bf
Z}_{2}}(A^{s}),F)$ and orthosymplectic generalized matrix Lie super algebra
$osp_{s,t}(GM_{{\bf Z}_{2}}(A^{s}),M^{s},F)$ are established. The relationship
between representations of braided m-Lie algebras and their associated
algebras are established. In this paper we follow paper [17] and obtain
universal enveloping algebras of braided m-Lie algebras and PBW theorem by
means of combinatorics on words (see [10]).

Throughout, $F$ is a field,

##  1 Braided m-Lie Algebras

We recalled two concepts.

######  Definition 1.1.

(See [17]) Let $(L,[\ \ ])$ be an object in the braided tensor category
$({\cal C},C)$ with morphism $[\ \ ]:L\otimes L\rightarrow L$. If there exists
an algebra $(A,m)$ in $({\cal C},C)$ and monomorphism $\phi:L\rightarrow A$
such that $\phi[\ \ ]=m(\phi\otimes\phi)-m(\phi\otimes\phi)C_{L,L},$ then
$(L,[\ \ ])$ is called a braided m-Lie algebra in $({\cal C},C)$ induced by
multiplication of $A$ through $\phi$. Algebra $(A,m)$ is called an algebra
associated to $(L,[\ \ ])$.

A Lie algebra is a braided m-Lie algebra in the category of ordinary vector
spaces, a Lie color algebra is a braided m-Lie algebra in symmetric braided
tensor category $({\cal M}^{FG},C^{r})$ since the canonical map
$\sigma:L\rightarrow U(L)$ is injective (see [14, Proposition 4.1]), a quantum
Lie algebra is a braided m-Lie algebra in the Yetter-Drinfeld category
$(^{B}_{B}{\cal YD},C)$ by [4, Definition 2.1 and Lemma 2.2]), and a “good”
braided Lie algebra is a braided m-Lie algebra in the Yetter-Drinfeld category
$(^{B}_{B}{\cal YD},C)$ by [4, Definition 3.6 and Lemma 3.7]). For a
cotriangular Hopf algebra $(H,r)$, the $(H,r)$-Lie algebra defined in [1, 4.1]
is a braided m-Lie algebra in the braided tensor category $({}^{H}{\cal
M},C^{r})$. Therefore, the braided m-Lie algebras generalize most known
generalized Lie algebras.

For an algebra $(A,m)$ in $({\cal C},C)$, obviously $L=A$ is a braided m-Lie
algebra under operation $[\ \ ]=m-mC_{L,L}$, which is induced by $A$ through
$id_{A}$. This braided m-Lie algebra is written as $A^{-}$.

######  Definition 1.2.

(see [19]) Let H be a Hopf algebra, $(V,\alpha)$ and $(V,\delta)$ be a left
$H$-module and a left $H$-comodule, respectively. If

$\displaystyle\delta(\alpha(h\otimes v))=\delta(h\cdot v)=\sum
h_{(1)}v_{(-1)}S(h_{(3)})\otimes h_{(2)}.v_{0}$ (1.1)

$\forall v\in V$, $h\in G$, then $(V,\alpha,\delta)$ is called a Yetter-
Drinfeld module over $H$, or a $H$\- YD module in short. All of $H$\- YD
module construct a braided tensor category, called the Yetter-Drinfeld module
category, denoted as $(^{H}_{H}{\cal YD},C)$, where $C$ is the braiding.

If $H=FG$ is a group algebra and $(V,\alpha,\delta)$ is an $FG$\- YD module,
then $V$ becomes a $G$-graded space $V=\oplus_{g\in G}V_{g}$ and the condition
(1.1) becomes

$\displaystyle\delta(\alpha(h\otimes v))=\delta(h\cdot v)=\sum hgh^{-1}\otimes
h\cdot v$ (1.2)

for any $h,g\in G,$ $v\in V_{g}.$

Let $G$ be a group and $\chi$ a bicharacter of $G$, i.e. $\chi$ is a map from
$G\times G$ to $F$ satisfying $\chi(ab,c)=\chi(a,c)\chi(b,c)$,
$\chi(a,bc)=\chi(a,b)\chi(a,c)$ and $\chi(a,e)=1=\chi(e,a)$ for any $a,b,c\in
G$, where $e$ is the unit element of $G$. The braiding of $FG$-YD module
$(V,\alpha,\delta)$ is determined by bicharacter $\chi$ if $h\cdot
x=\chi(h,g)x$ for any $h,g\in G,$ $x\in V_{g}.$ In this case, $C(x\otimes
y)=\chi(g,h)y\otimes x$ for any homogeneous elements $x\in V_{g},$ $y\in
V_{h}$. Obviously, if the braiding of $FG$-YD module $(V,\alpha,\delta)$ is
determined by bicharacter $\chi$, then the braiding of $(V,\alpha,\delta)$ is
diagonal. Conversely, if the braiding of a braided vector space $V$ is
diagonal, then $V$ can becomes an $F\mathbb{Z}[I]$-YD module, which braiding
is determined by a bicharacter (see [7]). If $G$ is a finite abelian group and
$V$ is a $kG$-YD module, then the braiding of $V$ is diagonal (see [18]). In
this paper we only consider the braiding determined by bicharacter $\chi.$

##  2 Jacobi identity

######  Lemma 2.1.

(See [9]) If $L$ is braided m-Lie algebra, then Jacobi identity holds:

$\displaystyle[[ab]c]-[a[bc]]+\chi(a,b)b[ac]-\chi(b,c)[ac]b=0$ (2.1)

for any homogeneous elements $a,b,c\in L$, where $\chi(a,b)$ denotes
$\chi(g,h)$ for $a\in V_{g}$, $b\in V_{h}.$

Proof.

the left side $\displaystyle=$ $\displaystyle
abc-\chi(a,b)bac-\chi(ab,c)cab+\chi(ab,c)\chi(a,b)cba$ $\displaystyle-
abc+\chi(b,c)acb+\chi(a,bc)bca-\chi(a,bc)\chi(b,c)cba$
$\displaystyle+\chi(a,b)bac-\chi(a,b)\chi(a,c)bca-\chi(b,c)acb+\chi(b,c)\chi(a,c)cab$
$\displaystyle=$ $\displaystyle 0.\Box$

##  3  Universal enveloping algebras of braided m-Lie algebras and PBW theorem

Let $E$ be a homogeneous basis of braided m-Lie algebra $L$ and $B$ a set. Let
$B^{*}$ denote the set of all words (see [10]) on $B$ and $\varphi$ a
bijective map from $E$ to $B$. Define $[bc]=\varphi([ef])$ for any
$b=\varphi(e)$, $c=\varphi(f)$, $e,f\in E$. Let $\prec$ be an order of $B$ and
$P=:\\{b_{1}b_{2}\cdots b_{n}\ |\ b_{i}\in B,b_{n}\prec
b_{n-1}\prec\cdots\prec b_{1},n\in\mathbb{N}\\}$.

For any $w\in B^{*}$, let $\nu(w)$ denote the number of elements in set
$\\{(r,s,t)\ |\ w=rasbt;a,b\in B,r,s,t\in B^{*},a\prec b,\\}$ and $\nu(w)$ is
called the index of $w$. Obviously, we have

$v(ubav)=v(uabv)-1$

for any $a,b\in B,u,v\in B^{*},a\prec b$. We also have that $\nu(w)=0$ if and
only if $w\in F$.

For any a set $X$, let $FX$ denote the vector space spanned by $X$ with basis
$X.$ It is clear that $FB^{*}$ is the free algebra on $B$. Meantime $FB^{*}$
also is the tensor algebra $T(FB)$ over $FB$.

######  Lemma 3.1.

There exists $\lambda:FB^{*}\rightarrow FP$ such that

(i) $\lambda(f)=f,$ $f\in P$;

(ii) $\lambda(ubcv)=\chi(b,c)\lambda(ucbv)+\lambda(u[bc]v),u,v\in
B^{\ast},b,c\in B$;

(iii) $\lambda(uv)=\lambda(\lambda(u)v)=\lambda(u\lambda(v)),u,v\in FB^{*}$.

Proof. For $w\in B^{*}$, we define $\lambda(w)$ using an induction first on
the length and second on the index. If $w\in B$, define $\lambda(w)=w$. Let
the length of $w$ be larger than 1 and define
$\lambda(w)=:\chi(b,c)\lambda(ucbv)+\lambda(u[bc]v)$ for $w=ubcv$ with $b,c\in
B$, $u,v\in B^{*}$. Now we show that the definition is well-defined. For
$w=ubcv=u^{\prime}b^{\prime}c^{\prime}v^{\prime}$ with
$b,c,b^{\prime},c^{\prime}\in B$, $u,v,u^{\prime},v^{\prime}\in B^{*}$, we
only need show that

$\displaystyle\chi(b,c)\lambda(ucbv)+\lambda(u[bc]v)=\chi(b^{\prime},c^{\prime})\lambda(u^{\prime}c^{\prime}b^{\prime}v^{\prime})+\lambda(u^{\prime}[b^{\prime}c^{\prime}]v^{\prime}).$
(3.1)

We show this by following two steps.

($1^{\circ}$) If $|u|\leq|u^{\prime}|-2$, then
$u^{\prime}=ubct,v=tb^{\prime}c^{\prime}v^{\prime},t\in B^{*}$. By induction
hypothesis we have

the left side $\displaystyle=$
$\displaystyle\chi(b,c)\chi(b^{\prime},c^{\prime})\lambda(ucbtc^{\prime}b^{\prime}v^{\prime})+\chi(b,c)\lambda(ucbt[b^{\prime}c^{\prime}]v^{\prime})$
$\displaystyle+\chi(b^{\prime},c^{\prime})\lambda(u[bc]tc^{\prime}b^{\prime}v^{\prime})+\lambda(u[bc]t[b^{\prime}c^{\prime}]v^{\prime})$

and

the right side $\displaystyle=$
$\displaystyle\chi(b,c)\chi(b^{\prime},c^{\prime})\lambda(ucbtc^{\prime}b^{\prime}v^{\prime})+\chi(b,c)\lambda(ucbt[b^{\prime}c^{\prime}]v^{\prime})$
$\displaystyle+\chi(b^{\prime},c^{\prime})\lambda(u[bc]tc^{\prime}b^{\prime}v^{\prime})+\lambda(u[bc]t[b^{\prime}c^{\prime}]v^{\prime}).$

Thus (3.1) holds.

($2^{\circ}$) If $|u|=|u^{\prime}|-1$ then
$u^{\prime}=ub,c=b^{\prime},v=c^{\prime}v^{\prime}$. We only need show
$\chi(a,b)\lambda(rbacs)+\lambda(r[ab]cs)=\chi(b,c)\lambda(racbs)+\lambda(ra[bc]s)$.
By induction hypothesis we have

the left side $\displaystyle=$
$\displaystyle\chi(a,b)\\{\chi(a,c)\lambda(rbcas)+\lambda(rb[ac]s)\\}+\lambda(r[ab]cs)$
$\displaystyle=$
$\displaystyle\chi(a,b)\chi(a,c)\lambda(rbcas)+\chi(a,b)\lambda(rb[ac]s)+\lambda(r[ab]cs)$
$\displaystyle=$
$\displaystyle\chi(a,b)\chi(a,c)\\{chi(b,c)\lambda(rcbas)+\lambda(r[bc]as)\\}+\chi(a,b)\lambda(rb[ac]s)+\lambda(r[ab]cs)$
$\displaystyle=$
$\displaystyle\chi(a,b)\chi(a,c)chi(b,c)\lambda(rcbas)+\chi(a,b)\chi(a,c)\lambda(r[bc]as)+\chi(a,b)\lambda(rb[ac]s)$
$\displaystyle+\lambda(r[ab]cs)$

and

the right side $\displaystyle=$
$\displaystyle\chi(b,c)\\{\chi(a,c)\lambda(rcabs)+\lambda(r[ac]bs)\\}+\lambda(ra[bc]s)$
$\displaystyle=$
$\displaystyle\chi(b,c)\chi(a,c)\lambda(rcabs)+\chi(b,c)\lambda(r[ac]bs)+\lambda(ra[bc]s)$
$\displaystyle=$
$\displaystyle\chi(b,c)\chi(a,c)\\{\chi(a,b)\lambda(rcbas)+\lambda(rc[ab]s)\\}+\chi(b,c)\lambda(r[ac]bs)+\lambda(ra[bc]s)$
$\displaystyle=$
$\displaystyle\chi(b,c)\chi(a,c)\chi(a,b)\lambda(rcbas)+\chi(b,c)\chi(a,c)\lambda(rc[ab]s)+\chi(b,c)\lambda(r[ac]bs)$
$\displaystyle+\lambda(ra[bc]s).$

Thus

$\displaystyle\mbox{the left side }-\mbox{the right side }$ $\displaystyle=$
$\displaystyle\\{\chi(a,b)\chi(a,c)\lambda(r[bc]as)-\lambda(ra[bc]s)\\}+\\{\chi(a,b)\lambda(rb[ac]s)-\chi(b,c)\lambda(r[ac]bs)\\}$
$\displaystyle+\\{\lambda(r[ab]cs)-\chi(b,c)\chi(a,c)\lambda(rc[ab]s)\\}$
$\displaystyle=$
$\displaystyle-\lambda(r[a[bc]]s)+\lambda(\chi(a,b)rb[ac]s-\chi(b,c)r[ac]bs)+\lambda(r[[ab]c]s)$
$\displaystyle=$ $\displaystyle 0\ \ \ \ {\mbox{(by Jacobi identity)}}.$

For (iii), we use an induction first on the length and second on the index.
Assume $|w_{1}|\neq|w|$ and $w=w_{1}w_{2}$. If $w_{1}=ubct$, $b,c\in B$,
$u,t\in B^{*}$, then

$\displaystyle\lambda(w)$
$\displaystyle=\chi(b,c)\lambda(ucbtw_{2})+\lambda(u[bc]tw_{2})$
$\displaystyle=\chi(b,c)\lambda(\lambda(ucbt)w_{2})+\lambda(\lambda(u[bc]t)w_{2})$
$\displaystyle=\lambda(\lambda(w_{1})w_{2})$

If $w_{1}=b$, $w_{2}=cv$, $b,c\in B$, $v\in B^{*}$, then
$\lambda(w)=\lambda(\lambda(w_{1})w_{2}).$ $\Box$

######  Definition 3.2.

Suppose that ${L}$ is a braided m-Lie algebra in $({\cal C},C)$ and $U$ is a
algebra with Lie algebra homomorphism $i:L\rightarrow U^{-}$. $(U,i)$ is
called the universal algebra of braided m -Lie algebra $L$, if the following
condition holds: If for any an algebra $W$ in $({\cal C},C)$ with a Lie
algebra homomorphism $\psi:L\rightarrow W^{-}$ in $({\cal C},C)$, there exists
the unique algebra homomorphism $\bar{\psi}:U\rightarrow W$ in $({\cal C},C)$
such that the following is commutative:

$\begin{array}[]{lcccr}{}&\varphi&{}\hfil\\\ L&\longrightarrow&U\\\
&\psi\searrow&\downarrow\bar{\psi}\\\ &&W&.\end{array}.$

Obviously, $\varphi$ in section above is a Lie algebra monomorphism from $L$
to $FP$ in ${}^{FG}_{FG}{\mathcal{Y}D}$.

Let $U(L)=:FP$. Define the multiplication of $U(L)$ as follows:
$u*v=\lambda(uv)$ for any $u,v\in P.$ By Lemma 3.1 (iii), $U(L)$ is an
associative algebra: $u\ast(v\ast
w)=\lambda(u\lambda(vw))=\lambda(uvw)=\lambda(\lambda(uv)w)=(u\ast v)\ast w$
for any $u,v,w\in P.$ Obviously, $\lambda$ is an algebra homomorphism.

######  Lemma 3.3.

If $(V,\alpha,\delta)$ is an $FG$-YD module, then tensor algebra $T(V)$ over
$V$ is an $FG$-YD module.

Proof. By the universal property of tensor algebra, we can construct the
module operation $\alpha^{(T(V))}$ and comodule operation $\delta^{(T(V))}$of
$T(V)$ as follows:

i)

$\begin{array}[]{lcccr}{}&\delta^{(T(V))}&{}\hfil\\\
T(V)&\longrightarrow&FG\otimes T(V)\\\ i\uparrow&\nearrow({\rm id}\otimes
i)\delta^{(V)}&\uparrow{\rm id}\otimes i\\\ V&\longrightarrow&FG\otimes V\\\
&\delta^{(V)}&\ \ \ \ \ \ \ \ \ .\end{array}$

ii)

$\begin{array}[]{lcccr}{}&\alpha_{g}^{(V)}&{}\hfil\\\ V&\longrightarrow&V\\\
i\downarrow&\searrow i\alpha_{g}^{(V)}&i\downarrow\\\
T(V)&\longrightarrow&T(V)\\\ &\alpha_{g}^{(T(V))}&\ \ \ \ \ \ \ \ \
,\end{array}$

where $\alpha^{(V)}_{g}(v)=:\alpha(g\otimes v)=g\cdot v$ for any $v\in V,$
$g\in G.$

iii) For $\forall g\in G,$ $x_{j}\in V_{g_{j}}$, $1\leq j\leq r$, See that

$\displaystyle\delta(g\cdot(x_{1}\cdot\cdot\cdot x_{r}))$ $\displaystyle=$
$\displaystyle\delta(\alpha_{g}(x_{1}\cdot\cdot\cdot x_{r}))$ $\displaystyle=$
$\displaystyle\delta((g\cdot x_{1})\cdot\cdot\cdot(g\cdot
x_{r}))=\delta(g\cdot x_{1})\cdot\cdot\cdot\delta(g\cdot x_{r})$
$\displaystyle=$ $\displaystyle(gg_{1}g^{-1}\otimes(g\cdot
x_{1}))\cdot\cdot\cdot(gg_{r}g^{-1}\otimes(g\cdot x_{r}))$ $\displaystyle=$
$\displaystyle(gg_{1}g^{-1})\cdot\cdot\cdot(gg_{r}g^{-1})\otimes
x_{1}\cdot\cdot\cdot x_{r}$ $\displaystyle=$ $\displaystyle
g(g_{1}\cdot\cdot\cdot g_{r})g^{-1}\otimes x_{1}\cdot\cdot\cdot x_{r}.$

Thus $(T(V),\alpha,\delta)$ is an $FG$-YD module. Furthermore, considering (i)
and (ii), we have that $T(V)$ is an algebra in ${}^{FG}_{FG}{\mathcal{Y}D}$.
$\Box$

######  Lemma 3.4.

(i) $FB^{*}$ is an $FG$-YD module.

(ii) $FP$ is an $FG$-YD sub-module of $FB^{*}.$

(iii) $FP$ is an algebra in ${}^{FG}_{FG}{\cal YD}$.

Proof. (i) It follows from Lemma 3.3.

(ii) and (iii) are clear. $\Box$

######  Theorem 3.5.

(PBW). $(U(L),\varphi$) is the universal enveloping algebra of braided m-Lie
algebra $L$.

Proof. For any an algebra $W$ in ${}^{FG}_{FG}{\mathcal{Y}D}$ with a Lie
algebra homomorphism $\psi:L\rightarrow W^{-}$ in
${}^{FG}_{FG}{\mathcal{Y}D}$, define $\bar{\psi}:FB^{*}\rightarrow FP$ such
that $\bar{\psi}\varphi=\psi$ and $\theta=:\bar{\psi}\mid_{FP}$, the
restriction of $\bar{\psi}$ on $FP.$ It is clear that the following is
commutative.

$\begin{array}[]{lcccr}{}&\varphi&{}\hfil&\lambda&{}\hfil\\\
L&\longrightarrow&FB^{*}&\longrightarrow&FP\\\
&\psi\searrow&\bar{\psi}\downarrow&\swarrow\theta\\\ &&W&\ \ \ \ \ \
.\end{array}$

Now we show that $\theta$ is an algebra homomorphism, i.e.

$\displaystyle\theta(r\ast s)$ $\displaystyle=$
$\displaystyle\theta(r)\theta(s)$

for any $r,s\in B^{*}$. We show this using induction by following several
steps.

$(1^{\circ})$ If $rs\in P,$ then $\theta(r\ast
s)=\theta(\lambda(rs))=\theta(rs)=\theta(r)\theta(s)$.

$(2^{\circ})$ $r,s\in B$ and $r\prec s$. See that

$\displaystyle\theta(r\ast s)$ $\displaystyle=$
$\displaystyle\theta(\lambda(rs))$ $\displaystyle=$
$\displaystyle\theta(\lambda(sr\chi(r,s)+[rs]))$ $\displaystyle=$
$\displaystyle\theta(\lambda(sr))\chi(r,s)+\theta(\lambda([rs]))$
$\displaystyle=$ $\displaystyle\theta(\lambda(sr))\chi(r,s)+\theta([rs])\ \ \
(\mbox{ since the length of }[rs]<2\mbox{ and }\nu(sr)<\nu(rs))$
$\displaystyle=$ $\displaystyle\theta(\lambda(sr))\chi(r,s)+\theta([rs])$
$\displaystyle=$ $\displaystyle\theta(sr)\chi(r,s)+\theta([rs])\ \ \ \ (\mbox{
since }sr\in P)$ $\displaystyle=$
$\displaystyle\theta(rs)=\theta(r)\theta(s).$

$(3^{\circ})$ If $r=ub,$ $s=cv,u,v\in B^{\ast}$, $b,c\in B,b\prec c$,
$uv\not=1$, then

$\displaystyle\theta(r\ast s)$
$\displaystyle=\theta(\lambda(rs))=\chi(b,c)\theta(\lambda(ucbv))+\theta(\lambda(u[bc]v))$
$\displaystyle=\chi(b,c)\theta((uc)*(bv))+\theta((u[bc])*v)$
$\displaystyle=\chi(b,c)\theta(uc)\theta(bv)+\theta(u[bc])\theta(v)\ \ \
{\mbox{(by induction hypothesis)}}$
$\displaystyle=\chi(b,c)\theta(u)\theta(cb)\theta(v)+\theta(u)\theta([bc])\theta(v)$
$\displaystyle=\theta(u)\theta(bc)\theta(v)$
$\displaystyle=\theta(u)\theta(b)\theta(c)\theta(v)$
$\displaystyle=\theta(r)\theta(s).\ \ \Box$

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