Proceedings of Human Language Technology Conference and Conference on Empirical Methods in Natural Language
Processing (HLT/EMNLP), pages 467–474, Vancouver, October 2005. c©2005 Association for Computational Linguistics
Bidirectional Inference with the Easiest-First Strategy
for Tagging Sequence Data
Yoshimasa Tsuruoka12 and Jun’ichi Tsujii231
1 CREST, JST (Japan Science and Technology Corporation)
Honcho 4-1-8, Kawaguchi-shi, Saitama 332-0012 Japan
2 Department of Computer Science, University of Tokyo
Hongo 7-3-1, Bunkyo-ku, Tokyo 113-0033 Japan
3 School of Informatics, University of Manchester
POBox 88, Sackville St, MANCHESTER M60 1QD, UK
{tsuruoka,tsujii}@is.s.u-tokyo.ac.jp
Abstract
This paper presents a bidirectional in-
ference algorithm for sequence label-
ing problems such as part-of-speech tag-
ging, named entity recognition and text
chunking. The algorithm can enumerate
all possible decomposition structures and
find the highest probability sequence to-
gether with the corresponding decomposi-
tion structure in polynomial time. We also
present an efficient decoding algorithm
based on the easiest-first strategy, which
gives comparably good performance to
full bidirectional inference with signifi-
cantly lower computational cost. Exper-
imental results of part-of-speech tagging
and text chunking show that the proposed
bidirectional inference methods consis-
tently outperform unidirectional inference
methods and bidirectional MEMMs give
comparable performance to that achieved
by state-of-the-art learning algorithms in-
cluding kernel support vector machines.
1 Introduction
The task of labeling sequence data such as part-of-
speech (POS) tagging, chunking (shallow parsing)
and named entity recognition is one of the most im-
portant tasks in natural language processing.
Conditional random fields (CRFs) (Lafferty et al.,
2001) have recently attracted much attention be-
cause they are free from so-called label bias prob-
lems which reportedly degrade the performance of
sequential classification approaches like maximum
entropy markov models (MEMMs).
Although sequential classification approaches
could suffer from label bias problems, they have sev-
eral advantages over CRFs. One is the efficiency
of training. CRFs need to perform dynamic pro-
gramming over the whole sentence in order to com-
pute feature expectations in each iteration of numer-
ical optimization. Training, for instance, second-
order CRFs using a rich set of features can require
prohibitive computational resources. Max-margin
methods for structured data share problems of com-
putational cost (Altun et al., 2003).
Another advantage is that one can employ a vari-
ety of machine learning algorithms as the local clas-
sifier. There is huge amount of work about devel-
oping classification algorithms that have high gener-
alization performance in the machine learning com-
munity. Being able to incorporate such state-of-the-
art machine learning algorithms is important. In-
deed, sequential classification approaches with ker-
nel support vector machines offer competitive per-
formance in POS tagging and chunking (Gimenez
and Marquez, 2003; Kudo and Matsumoto, 2001).
One obvious way to improve the performance of
sequential classification approaches is to enrich the
information that the local classifiers can use. In stan-
dard decomposition techniques, the local classifiers
cannot use the information about future tags (e.g.
the right-side tags in left-to-right decoding), which
would be helpful in predicting the tag of the target
word. To make use of the information about fu-
ture tags, Toutanova et al. proposed a tagging algo-
rithm based on bidirectional dependency networks
467
(Toutanova et al., 2003) and achieved the best ac-
curacy on POS tagging on the Wall Street Journal
corpus. As they pointed out in their paper, however,
their method potentially suffers from “collusion” ef-
fects which make the model lock onto conditionally
consistent but jointly unlikely sequences. In their
modeling, the local classifiers can always use the in-
formation about future tags, but that could cause a
double-counting effect of tag information.
In this paper we propose an alternative way of
making use of future tags. Our inference method
considers all possible ways of decomposition and
chooses the “best” decomposition, so the informa-
tion about future tags is used only in appropriate
situations. We also present a deterministic version
of the inference method and show their effective-
ness with experiments of English POS tagging and
chunking, using standard evaluation sets.
2 Bidirectional Inference
The task of labeling sequence data is to find the se-
quence of tags t1...tn that maximizes the following
probability given the observation o = o1...on
P(t1...tn|o). (1)
Observations are typically words and their lexical
features in the task of POS tagging. Sequential clas-
sification approaches decompose the probability as
follows,
P(t1...tn|o) =
nproductdisplay
i=1
p(ti|t1...ti−1o). (2)
This is the left-to-right decomposition. If we
make a first-order markov assumption, the equation
becomes
P(t1...tn|o) =
nproductdisplay
i=1
p(ti|ti−1o). (3)
Then we can employ a probabilistic classifier
trained with the preceding tag and observations in
order to obtain p(ti|ti−1o) for local classification. A
common choice for the local probabilistic classifier
is maximum entropy classifiers (Berger et al., 1996).
The best tag sequence can be efficiently computed
by using a Viterbi decoding algorithm in polynomial
time.
t1
(a)
t2 t3
o
t1
(b)
t2 t3
t1
(c)
t2 t3 t1
(d)
t2 t3
o
o o
Figure 1: Different structures for decomposition.
The right-to-left decomposition is
P(t1...tn|o) =
nproductdisplay
i=1
p(ti|ti+1o). (4)
These two ways of decomposition are widely used
in various tagging problems in natural language pro-
cessing. The issue with such decompositions is that
you have only the information about the preceding
(or following) tags when performing local classifi-
cation.
From the viewpoint of local classification, we
want to give the classifier as much information as
possible because the information about neighboring
tags is useful in general.
As an example, consider the situation where we
are going to annotate a three-word sentence with
part-of-speech tags. Figure 1 shows the four possi-
ble ways of decomposition. They correspond to the
following equations:
(a) P(t1...t3|o) = P(t1|o)P(t2|t1o)P(t3|t2o) (5)
(b) P(t1...t3|o) = P(t3|o)P(t2|t3o)P(t1|t2o) (6)
(c) P(t1...t3|o) = P(t1|o)P(t3|o)P(t2|t3t1o) (7)
(d) P(t1...t3|o) = P(t2|o)P(t1|t2o)P(t3|t2o) (8)
(a) and (b) are the standard left-to-right and right-
to-left decompositions. Notice that in decomposi-
tion (c), the local classifier can use the information
about the tags on both sides when deciding t2. If,
for example, the second word is difficult to tag (e.g.
an unknown word), we might as well take the de-
composition structure (c) because the local classifier
468
can use rich information when deciding the tag of
the most difficult word. In general if we have an
n-word sentence and adopt a first-order markov as-
sumption, we have 2n−1 possible ways of decompo-
sition because each of the n−1 edges in the cor-
responding graph has two directions (left-to-right or
right-to-left).
Our bidirectional inference method is to consider
all possible decomposition structures and choose the
“best” structure and tag sequence. We will show in
the next section that this is actually possible in poly-
nomial time by dynamic programming.
As for the training, let us look at the equa-
tions of four different decompositions above. You
can notice that there are only four types of local
conditional probabilities: P(ti|ti−1o), P(ti|ti+1o),
P(ti|ti−1ti+1o), and P(ti|o).
This means that if we have these four types of lo-
cal classifiers, we can consider any decomposition
structures in the decoding stage. These local classi-
fiers can be obtained by training with corresponding
neighboring tag information. Training the first two
types of classifiers is exactly the same as the train-
ing of popular left-to-right and right-to-left sequen-
tial classification models respectively.
If we take a second-order markov assumption, we
need to train 16 types of local classifiers because
each of the four neighboring tags of a classification
target has two possibilities of availability. In gen-
eral, if we take a k-th order markov assumption, we
need to train 22k types of local classifies.
2.1 Polynomial Time Inference
This section describes an algorithm to find the de-
composition structure and tag sequence that give the
highest probability. The algorithm for the first-order
case is an adaptation of the algorithm for decoding
the best sequence on a bidirectional dependency net-
work introduced by (Toutanova et al., 2003), which
originates from the Viterbi decoding algorithm for
second-order markov models.
Figure 2 shows a polynomial time decoding al-
gorithm for our bidirectional inference. It enumer-
ates all possible decomposition structures and tag
sequences by recursive function calls, and finds the
highest probability sequence. Polynomial time is
achieved by caching. Note that for each local clas-
sification, the function chooses the appropriate local
function bestScore()
{
return bestScoreSub(n+2,〈end,end,end〉,〈L,L〉);
}
function bestScoreSub(i+1,〈ti−1,ti,ti+1〉,〈di−1,di〉)
{
// memorization
if (cached(i+1,〈ti−1,ti,ti+1〉,〈di−1,di〉))
return cache(i+1,〈ti−1,ti,ti+1〉,〈di−1,di〉);
// left boundary case
if (i = -1)
if (〈ti−1,ti,ti+1〉=〈start,start,start〉) return 1;
else return 0;
// recursive case
P = localClassification(i,〈ti−1,ti,ti+1〉,〈di−1,di〉);
return maxdi−2 maxti−2 P×
bestScoreSub(i,〈ti−2,ti−1,ti〉,〈di−2,di−1〉);
}
function localClassification(i,〈ti−1,ti,ti+1〉,〈di−1,di〉)
{
if (di−1 = L & di = L) return P(ti|ti+1,o);
if (di−1 = L & di = R) return P(ti|o);
if (di−1 = R & di = L) return P(ti|ti−1ti+1,o);
if (di−1 = R & di = R) return P(ti|ti−1,o);
}
Figure 2: Pseudo-code for bidirectional inference
for the first-order conditional markov models. di is
the direction of the edge between ti and ti+1.
classifier by taking into account the directions of the
adjacent edges of the classification target.
The second-order case is similar but slightly more
complex. Figure 3 shows the algorithm. The recur-
sive function needs to consider the directions of the
four adjacent edges of the classification target, and
maintain the directions of the two neighboring edges
to enumerate all possible edge directions. In addi-
tion, the algorithm rules out cycles in the structure.
2.2 Decoding with the Easiest-First Strategy
We presented a polynomial time decoding algorithm
in the previous section. However, polynomial time is
not low enough in practice. Indeed, even the Viterbi
decoding of second-order markov models for POS
tagging is not practical unless some pruning method
is involved. The computational cost of the bidirec-
tional decoding algorithm presented in the previous
section is, of course, larger than that because it enu-
merates all possible directions of the edges on top of
the enumeration of possible tag sequences.
In this section we present a greedy version of the
decoding method for bidirectional inference, which
469
function bestScore()
{
return bestScoreSub(n+3,〈end,end,end,end,end〉,〈L,L,L,L〉,〈L,L〉);
}
function bestScoreSub(i+2,〈ti−2,ti−1,ti,ti+1ti+2〉,〈dprimei−1,di−1,di,dprimei+1〉,〈di−2,dprimei〉)
{
// to avoid cycles
if (di−1 = di & di != dprimei) return 0;
// memorization
if (cached(i+2,〈ti−2,ti−1,ti,ti+1ti+2〉,〈dprimei−1,di−1,di,dprimei+1〉,〈di−2,dprimei〉)
return cache(i+2,〈ti−2,ti−1,ti,ti+1ti+2〉,〈dprimei−1,di−1,di,dprimei+1〉,〈di−2,dprimei〉);
// left boundary case
if (i = -2)
if (〈ti−2,ti−1,ti,ti+1,ti+2〉=〈start,start,start,start,start〉) return 1;
else return 0;
// recursive case
P = localClassification(i,〈ti−2,ti−1,ti,ti+1,ti+2〉,〈dprimei−1,di−1,di,dprimei+1〉);
return maxdprime
i−2
maxdi−3 maxti−3 P×bestScoreSub(i+1,〈ti−3,ti−2,ti−1,titi+1〉,〈dprimei−2,di−2,di−1,dprimei〉,〈di−3,dprimei−1〉);
}
Figure 3: Pseudo-code for bidirectional inference for the second-order conditional markov models. di is the
direction of the edge between ti and ti+1. dprimei is the direction of the edge between ti−1 and ti+1. We omit the
localClassification function because it is the obvious extension of that for the first-order case.
is extremely simple and significantly more efficient
than full bidirectional decoding.
Instead of enumerating all possible decomposi-
tion structures, the algorithm determines the struc-
ture by adopting the easiest-first strategy. The whole
decoding algorithm is given below.
1. Find the “easiest” word to tag.
2. Tag the word.
3. Go back to 1. until all the words are tagged.
We assume in this paper that the “easiest” word
to tag is the word for which the classifier outputs
the highest probability. In finding the easiest word,
we use the appropriate local classifier according to
the availability of the neighboring tags. Therefore,
in the first iteration, we always use the local classi-
fiers trained with no contextual tag information (i.e.
(P(ti|o)). Then, for example, if t3 has been tagged
in the first iteration in a three-word sentence, we use
P(t2|t3o) to compute the probability for tagging t2
in the second iteration (as in Figure 1 (b)).
A naive implementation of this algorithm requires
O(n2) invocations of local classifiers, wherenis the
number of the words in the sentence, because we
need to update the probabilities over the words at
each iteration. However, a k-th order Markov as-
sumption obviously allows us to skip most of the
probability updates, resulting in O(kn) invocations
of local classifiers. This enables us to build a very
efficient tagger.
3 Maximum Entropy Classifier
For local classifiers, we used a maximum entropy
model which is a common choice for incorporating
various types of features for classification problems
in natural language processing (Berger et al., 1996).
Regularization is important in maximum entropy
modeling to avoid overfitting to the training data.
For this purpose, we use the maximum entropy
modeling with inequality constraints (Kazama and
Tsujii, 2003). The model gives equally good per-
formance as the maximum entropy modeling with
Gaussian priors (Chen and Rosenfeld, 1999), and
the size of the resulting model is much smaller than
that of Gaussian priors because most of the param-
eters become zero. This characteristic enables us
to easily handle the model data and carry out quick
decoding, which is convenient when we repetitively
perform experiments. This modeling has one param-
eter to tune, which is called the width factor. We
tuned this parameter using the development data in
each type of experiments.
470
Current word wi & ti
Previous word wi−1 & ti
Next word wi+1 & ti
Bigram features wi−1, wi & ti
wi, wi+1 & ti
Previous tag ti−1 & ti
Tag two back ti−2 & ti
Next tag ti+1 & ti
Tag two ahead ti+2 & ti
Tag Bigrams ti−2, ti−1 & ti
ti−1, ti+1 & ti
ti+1, ti+2 & ti
Tag Trigrams ti−2, ti−1, ti+1 & ti
ti−1, ti+1, ti+2 & ti
Tag 4-grams ti−2, ti−1, ti+1, ti+2 & ti
Tag/Word ti−1, wi & ti
combination ti+1, wi & ti
ti−1, ti+1, wi & ti
Prefix features prefixes of wi & ti
(up to length 10)
Suffix features suffixes of wi & ti
(up to length 10)
Lexical features whether wi has a hyphen & ti
whether wi has a number & ti
whether wi has a capital letter & ti
whether wi is all capital & ti
Table 1: Feature templates used in POS tagging ex-
periments. Tags are parts-of-speech. Tag features
are not necessarily used in all the models. For ex-
ample, “next tag” features cannot be used in left-to-
right models.
4 Experiments
To evaluate the bidirectional inference methods pre-
sented in the previous sections, we ran experiments
on POS tagging and text chunking with standard En-
glish data sets.
Although achieving the best accuracy is not the
primary purpose of this paper, we explored useful
feature sets and parameter setting by using develop-
ment data in order to make the experiments realistic.
4.1 Part-of-speech tagging experiments
We split the Penn Treebank corpus (Marcus et al.,
1994) into training, development and test sets as in
(Collins, 2002). Sections 0-18 are used as the train-
ing set. Sections 19-21 are the development set, and
sections 22-24 are used as the test set. All the ex-
periments were carried out on the development set,
except for the final accuracy report using the best
setting.
For features, we basically adopted the feature set
Method Accuracy Speed
(%) (tokens/sec)
Left-to-right (Viterbi) 96.92 844
Right-to-left (Viterbi) 96.89 902
Dependency Networks 97.06 1,446
Easiest-last 96.58 2,360
Easiest-first 97.13 2,461
Full bidirectional 97.12 34
Table 2: POS tagging accuracy and speed on the de-
velopment set.
Method Accuracy (%)
Dep. Networks (Toutanova et al., 2003) 97.24
Perceptron (Collins, 2002) 97.11
SVM (Gimenez and Marquez, 2003) 97.05
HMM (Brants, 2000) 96.48
Easiest-first 97.10
Full Bidirectional 97.15
Table 3: POS tagging accuracy on the test set (Sec-
tions 22-24 of the WSJ, 5462 sentences).
provided by (Toutanova et al., 2003) except for com-
plex features such as crude company-name detection
features because they are specific to the Penn Tree-
bank and we could not find the exact implementation
details. Table 1 lists the feature templates used in our
experiments.
We tested the proposed bidirectional methods,
conventional unidirectional methods and the bidirec-
tional dependency network proposed by Toutanova
(Toutanova et al., 2003) for comparison. 1. All
the models are second-order. Table 2 shows the
accuracy and tagging speed on the development
data 2. Bidirectional inference methods clearly out-
performed unidirectional methods. Note that the
easiest-first decoding method achieves equally good
performance as full bidirectional inference. Table 2
also shows that the easiest-last strategy, where we
select and tag the most difficult word at each itera-
tion, is clearly a bad strategy.
An example of easiest-first decoding is given be-
low:
1For dependency network and full bidirectional decoding,
we conducted pruning because the computational cost was too
large to perform exhaustive search. We pruned a tag candidate if
the zero-th order probability of the candidateP(ti|o) was lower
than one hundredth of the zero-th order probability of the most
likely tag at the token.
2Tagging speed was measured on a server with an AMD
Opteron 2.4GHz CPU.
471
The/DT/4 company/NN/7 had/VBD/11
sought/VBN/14 increases/NNS/13 total-
ing/VBG/12 $/$/2 80.3/CD/5 million/CD/8
,/,/1 or/CC/6 22/CD/9 %/NN/10 ././3
Each token represents Word/PoS/DecodingOrder.
Typically, punctuations and articles are tagged first.
Verbs are usually tagged in later stages because their
tags are likely to be ambiguous.
We applied our bidirectional inference methods
to the test data. The results are shown in Table 3.
The table also summarizes the accuracies achieved
by several other research efforts. The best accuracy
is 97.24% achieved by bidirectional dependency net-
works (Toutanova et al., 2003) with a richer set of
features that are carefully designed for the corpus. A
perceptron algorithm gives 97.11% (Collins, 2002).
Gimenez and Marquez achieve 97.05% with support
vector machines (SVMs). This result indicates that
bidirectional inference with maximum entropy mod-
eling can achieve comparable performance to other
state-of-the-art POS tagging methods.
4.2 Chunking Experiments
The task of chunking is to find non-recursive phrases
in a sentence. For example, a chunker segments the
sentence “He reckons the current account deficit will
narrow to only 1.8 billion in September” into the fol-
lowing,
[NP He] [VP reckons] [NP the current account
deficit] [VP will narrow] [PP to] [NP only 1.8 bil-
lion] [PP in] [NP September] .
We can regard chunking as a tagging task by con-
verting chunks into tags on tokens. There are several
ways of representing text chunks (Sang and Veen-
stra, 1999). We tested the Start/End representation
in addition to the popular IOB2 representation since
local classifiers can have fine-grained information
on the neighboring tags in the Start/End represen-
tation.
For training and testing, we used the data set pro-
vided for the CoNLL-2000 shared task. The training
set consists of section 15-18 of the WSJ corpus, and
the test set is section 20. In addition, we made the
development set from section 21 3.
We basically adopted the feature set provided in
3We used the Perl script provided on
http://ilk.kub.nl/˜ sabine/chunklink/
Current word wi & ti
Previous word wi−1 & ti
Word two back wi−2 & ti
Next word wi+1 & ti
Word two ahead wi+2 & ti
Bigram features wi−2, wi−1 & ti
wi−1, wi & ti
wi, wi+1 & ti
wi+1, wi+2 & ti
Current POS pi & ti
Previous POS pi−1 & ti
POS two back pi−2 & ti
Next POS pi+1 & ti
POS two ahead pi+2 & ti
Bigram POS features pi−2, pi−1 & ti
pi−1, pi & ti
pi, pi+1 & ti
pi+1, pi+2 & ti
Trigram POS features pi−2, pi−1, pi & ti
pi−1, pi, pi+1 & ti
pi, pi+1, pi+2 & ti
Previous tag ti−1 & ti
Tag two back ti−2 & ti
Next tag ti+1 & ti
Tag two ahead ti+2 & ti
Bigram tag features ti−2, ti−1 & ti
ti−1, ti+1 & ti
ti+1, ti+2 & ti
Table 4: Feature templates used in chunking experi-
ments.
(Collins, 2002) and used POS-trigrams as well. Ta-
ble 4 lists the features used in chunking experiments.
Table 5 shows the results on the development set.
Again, bidirectional methods exhibit better perfor-
mance than unidirectional methods. The difference
is bigger with the Start/End representation. Depen-
dency networks did not work well for this chunking
task, especially with the Start/End representation.
We applied the best model on the development
set in each chunk representation type to the test
data. Table 6 summarizes the performance on the
test set. Our bidirectional methods achieved F-
scores of 93.63 and 93.70, which are better than the
best F-score (93.48) of the CoNLL-2000 shared task
(Sang and Buchholz, 2000) and comparable to those
achieved by other state-of-the-art methods.
5 Discussion
There are some reports that one can improve the
performance of unidirectional models by combining
outputs of multiple taggers. Shen et al. (2003) re-
ported a 4.9% error reduction of supertagging by
472
Representation Method Order Recall Precision F-score Speed (tokens/sec)
IOB2 Left-to-right 1 93.17 93.05 93.11 1,775
2 93.13 92.90 93.01 989
Right-to-left 1 92.92 92.82 92.87 1,635
2 92.92 92.74 92.87 927
Dependency Networks 1 92.71 92.91 92.81 2,534
2 92.61 92.95 92.78 1,893
Easiest-first 1 93.17 93.04 93.11 2,441
2 93.35 93.32 93.33 1,248
Full Bidirectional 1 93.29 93.14 93.21 712
2 93.26 93.12 93.19 48
Start/End Left-to-right 1 92.98 92.69 92.83 861
2 92.96 92.67 92.81 439
Right-to-left 1 92.92 92.83 92.87 887
2 92.89 92.74 92.82 451
Dependency Networks 1 87.10 89.56 88.32 1,894
2 87.16 89.44 88.28 331
Easiest-first 1 93.33 92.95 93.14 1,950
2 93.31 92.95 93.13 1,016
Full Bidirectional 1 93.52 93.26 93.39 392
2 93.44 93.20 93.32 4
Table 5: Chunking F-scores on the development set.
Method Recall Precision F-score
SVM (Kudoh and Matsumoto, 2000) 93.51 93.45 93.48
SVM voting (Kudo and Matsumoto, 2001) 93.92 93.89 93.91
Regularized Winnow (with basic features) (Zhang et al., 2002) 93.60 93.54 93.57
Perceptron (Carreras and Marquez, 2003) 93.29 94.19 93.74
Easiest-first (IOB2, second-order) 93.59 93.68 93.63
Full Bidirectional (Start/End, first-order) 93.70 93.65 93.70
Table 6: Chunking F-scores on the test set (Section 20 of the WSJ, 2012 sentences).
pairwise voting between left-to-right and right-to-
left taggers. Kudo et al. (2001) attained performance
improvement in chunking by conducting weighted
voting of multiple SVMs trained with distinct chunk
representations. The biggest difference between our
approach and such voting methods is that the lo-
cal classifier in our bidirectional inference methods
can have rich information for decision. Also, vot-
ing methods generally need many tagging processes
to be run on a sentence, which makes it difficult to
build a fast tagger.
Our algorithm can be seen as an ensemble classi-
fier by which we choose the highest probability one
among the different taggers with all possible decom-
position structures. Although choosing the highest
probability one is seemingly natural and one of the
simplest ways for combining the outputs of different
taggers, one could use a different method (e.g. sum-
ming the probabilities over the outputs which share
the same label sequence). Investigating the methods
for combination should be an interesting direction of
future work.
As for the computational cost for training, our
methods require us to train 22n types of classifiers
when we adopt an nth order markov assumption. In
many cases a second-order model is sufficient be-
cause further increase of n has little impact on per-
formance. Thus the training typically takes four or
16 times as much time as it would take for training a
single unidirectional tagger, which looks somewhat
expensive. However, because each type of classi-
fier can be trained independently, the training can
be performed completely in parallel and run with
the same amount of memory as that for training a
single classifier. This advantage contrasts with the
case for CRFs which requires substantial amount of
memory and computational cost if one tries to incor-
porate higher-order features about tag sequences.
Tagging speed is another important factor in
building a practical tagger for large-scale text min-
473
ing. Our inference algorithm with the easiest-first
strategy needs no Viterbi decoding unlike MEMMs
and CRFs, and makes it possible to perform very fast
tagging with high precision.
6 Conclusion
We have presented a bidirectional inference algo-
rithm for sequence labeling problems such as POS
tagging, named entity recognition and text chunk-
ing. The algorithm can enumerate all possible de-
composition structures and find the highest prob-
ability sequence together with the corresponding
decomposition structure in polynomial time. We
have also presented an efficient bidirectional infer-
ence algorithm based on the easiest-first strategy,
which gives comparable performance to full bidi-
rectional inference with significantly lower compu-
tational cost.
Experimental results of POS tagging and text
chunking show that the proposed bidirectional in-
ference methods consistently outperform unidi-
rectional inference methods and our bidirectional
MEMMs give comparable performance to that
achieved by state-of-the-art learning algorithms in-
cluding kernel support vector machines.
A natural extension of this work is to replace
the maximum entropy modeling, which was used as
the local classifiers, with other machine learning al-
gorithms. Support vector machines with appropri-
ate kernels is a good candidate because they have
good generalization performance as a single classi-
fier. Although SVMs do not output probabilities, the
easiest-first method would be easily applied by con-
sidering the margins output by SVMs as the confi-
dence of local classification.
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