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Xu, F. & Tenenbaum, J. B. (2007). Word learning as Bayesian inference. Psychological Review, 
114, 245-272. 
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Figure 1. Word token F-scores for each of the learners, averaged over the test sets. 
Figure 2. Lexicon item F-scores for each of the learners, averaged over the test sets. 
Ideal
DPM
DPS
DMCMC
Learners
Score
0
20
40
60
80
Uni−F
Bi−F
Learner performance: word tokens
Ideal
DPM
DPS
DMCMC
Learners
Score
0
20
40
60
80
Uni−F
Bi−F
Learner performance: lexicon items
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Figure 3. Performance of unigram learners on whole utterances, first words, and last words. 
Figure 4. Performance of bigram learners on whole utterances, first words, and last words. 
Ideal
DPM
DPS
DMCMC
Learners
Score
0
20
40
60
80
Whole Utt
First Wd
Last Wd
Unigram learner performance: utterance portions
Ideal
DPM
DPS
DMCMC
Learners
Score
0
20
40
60
80
Whole Utt
First Wd
Last Wd
Bigram learner performance: utterance portions
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Appendix Figure 1. Phoneme encoding. Taken with permission from Goldwater, Griffiths, & 
Johnson (2007). 
Table 1. Likelihood of sampling a given boundary in DMCMC, d = 1.  The relative probability 
of a given boundary being sampled is the decay probability b
a
-d
divided by the sum of the decay 
probabilities for all boundary positions under consideration (in this example, five boundary 
positions). 
Boundary position  b
b
a
-d 
Relative probability 
end – 1 
1
-1
= 1/1 = 1.00 
1.00/(Σ(probs)) = 0.44 
end – 2 
2
-1
= 1/2 = 0.50 
0.50/(Σ(probs)) = 0.22 
end – 3 
3
-1
= 1/3 = 0.33 
0.33/(Σ(probs)) = 0.15 
end – 4 
4
-1
= 1/4 = 0.25 
0.25/(Σ(probs)) = 0.11 
end – 5 
5
-1
= 1/5 = 0.20 
0.20/(Σ(probs)) = 0.08 
Consonants
ASCII
Ex.
ASCII
Ex.
D
THe
h
Hat
G
Jump
k
Cut
L
bottLe
l
Lamp
M
rhythM
m
Man
N
siNG
n
Net
S
SHip
p
Pipe
T
THin
r
Run
W
WHen
s
Sit
Z
aZure
t
Toy
b
Boy
v
View
c
CHip
w
We
d
Dog
y
You
f
Fox
z
Zip
g
Go
˜
buttON
Vowels
ASCII
Ex.
&
thAt
6
About
7
bOY
9
flY
A
bUt
E
bEt
I
bIt
O
lAW
Q
bOUt
U
pUt
a
hOt
e
bAY
i
bEE
o
bOAt
u
bOOt
RhoticVowels
ASCII
Ex.
#
ARe
%
fOR
(
hERE
)
lURE
*
hAIR
3
bIRd
R
buttER
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Table 2. Probability of sampling a boundary from the current utterance, based on decay rate. 
decay rate 
Probability of sampling within current utterance 
2.0 
0.942 
1.5 
0.772 
1.0 
0.323 
0.75 
0.125 
0.50 
0.036 
0.25 
0.009 
0.125 
0.004 
Table 3. Samples of Bernstein corpus. 
English orthography 
Phonemic transcription 
you want to see the book 
look there’s a boy with his hat 
and a doggie 
you want to look at this 
yu want tu si D6 bUk 
lUk D*z 6 b7 wIT hIz h&t 
&nd 6 dOgi 
yu want tu lUk &t DIs 
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Table 4. Average performance of different learners on the five test sets, along with published 
results from other recent statistical learners where available and the results from a transitional 
probability learner.  Note that the PHOCUS results are from the “3s” implementation, which 
performed best on the corpus.  Precision (P), recall (R), and F-score (F) over word tokens (T), 
word boundaries (B), and lexicon items (L) resulting from the chosen word segmentation are 
shown.  Standard deviations are shown in parentheses where available.  
Bayesian Unigram Learners (words are not predictive) 
TP 
TR 
TF 
BP 
BR 
BF 
LP 
LR 
LF 
GGJ-Ideal 63.2 
(0.99) 
48.4 
(0.80) 
54.8 
(0.85) 
92.8 
(0.67) 
62.1 
(0.42) 
74.4 
(0.42) 
54.0 
(0.92) 
73.6 
(1.89) 
62.3 
(1.30) 
DPM 
63.7 
(2.82) 
68.4 
(2.68) 
65.9 
(2.73) 
77.2 
(1.86) 
85.3 
(1.67) 
81.0 
(1.64) 
61.9 
(2.17) 
56.9 
(2.07) 
59.3 
(2.09) 
DPS 
55.0 
(4.82) 
62.6 
(3.99) 
58.5 
(4.45) 
70.4 
(3.73) 
84.21 
(1.79) 
76.7 
(2.85) 
54.8 
(1.64) 
49.2 
(3.14) 
51.8 
(2.2) 
DMCMC 71.2 
(1.57) 
64.7 
(2.31) 
67.8 
(1.97) 
88.8 
(0.89) 
77.2 
(2.17) 
82.6 
(1.53) 
61.0 
(1.18) 
69.6 
(0.43) 
65.0 
(0.67) 
Bayesian Bigram Learners (words are predictive)
TP 
TR 
TF 
BP 
BR 
BF 
LP 
LR 
LF 
GGJ-Ideal 74.5 
(1.41) 
68.8 
(1.53) 
71.5 
(1.46) 
90.1 
(0.75) 
80.4 
(1.01) 
85.0 
(0.82) 
65.0 
(1.19) 
73.5 
(1.71) 
69.1 
(1.15) 
DPM 
67.5 
(1.13) 
71.3 
(0.74) 
69.4 
(0.90) 
80.4 
(0.96) 
86.8 
(0.63) 
83.5 
(0.57) 
66.0 
(1.00) 
63.2 
(1.46) 
64.5 
(1.05) 
DPS 
34.2 
(2.16) 
47.6 
(2.16) 
39.8 
(2.13) 
54.9 
(1.40) 
85.3 
(2.07) 
66.8 
(1.00) 
39.0 
(2.02) 
34.4 
(2.42) 
36.5 
(2.19) 
DMCMC 72.0 
74.0 
73.0 
84.1 
87.4 
85.7 
61.1 
64.2 
62.6 
(1.24) 
(1.76) 
(1.43) 
(0.98) 
(1.47) 
(0.94) 
(1.41) 
(1.35) 
(1.17) 
Comparison Learners 
TP 
TR 
TF 
BP 
BR 
BF 
LP 
LR 
LF 
WordEnds  
70.7 
94.6 
73.7 
82.9 
36.6 
BVE 
79.1 
79.4 
79.3 
92.8 
90.5 
91.6 
PHOCUS  77.7 
74.0 
75.8 
89.7 
83.6 
86.5 
47.3 
64.0 
54.5 
TransProb 34.3 
(0.88) 
42.7 
(0.83) 
38.0 
(0.87) 
52.8 
(1.22) 
71.1 
(1.00) 
60.6 
(1.15) 
24.3 
(0.55) 
39.7 
(1.1) 
30.1 
(0.70) 
Table 5. GGJ ideal learner model performance: Unigram vs. Bigram.  Segmentations are shown 
in their English orthographic form, and undersegmentations are italicized. 
Unigram Model 
Bigram Model 
youwant to see thebook 
look theres aboy 
with his hat 
and adoggie 
you wantto lookatthis 
lookatthis 
havea drink 
okay now 
whatsthis 
whatsthat 
whatisit 
look canyou take itout 
you want to see the book 
look theres a boy 
with his hat 
and a doggie 
you want to lookat this 
lookat this 
have a drink 
okay now 
whats this 
whats that 
whatis it 
look canyou take it out 
Table 6. Significance test scores (two tailed t-test) for comparisons between first word, last word, 
and  whole utterance performance across the five test sets. Non-significant differences are 
italicized. 
first ≠ whole 
last ≠ whole 
last ≠ first 
Unigram Models (Words are not predictive) 
GGJ – Ideal 
.001 
< .001 
.003 
DPM 
.008 
.046 
.108 
DPS 
.008 
.130 
.038 
DMCMC 
< .001 
.003 
.207 
Bigram Models (Words are predictive) 
GGJ – Ideal 
< .001 
.157 
.050 
DPM 
< .001 
.005 
.730 
DPS 
< .001 
.003 
.372 
DMCMC 
.002 
.069 
.029 
Table 7. Performance on test set 1 for DMCMC learners with varying samples per utterance.  
Learners were tested with the decay rate that yielded the best performance at 20000 samples per 
utterance (unigram = 1, bigram = 0.25).  F-scores over word tokens are shown, as well as the 
processing comparison to the ideal learner (as measured by number of samples taken).   
20000  10000  5000  2500  1000  500  250  100 
100 
% Ideal learner samples 
11.0 
5.7 
2.8 
1.4  0.57  0.28  0.14  0.057 
057 
Unigram, d = 1 
69.8 
68.5  65.5  63.5  63.4  60.0  56.9 
56.9 
51.1 
Bigram, d = 0.25 
74.9 
71.8  68.3  66.1  64.6  61.2  59.9 
59.9 
60.9 
Table 8. Performance on test set 1 for DMCMC learners and ideal learners that only sample 
approximately as much as the DMCMC learners do. DMCMC learners sampled 20000 times per 
utterance with decay rate = 1 for the Unigram learner and 0.25 for the Bigram learner.  Ideal 
learners made 2000 iterations over the corpus, sampling every potential boundary once each 
iteration. 
Unigram Learners (words are not predictive) 
TP 
TR 
TF  BP  BR 
BF  LP  LR 
LF 
GGJ-Ideal 
62.7  49.6  55.4  90.5  63.5  74.7  55.8  73.7  63.5 
.7  63.5 
DMCMC 
72.6  67.2  69.8  88.1  78.8  83.2  61.3  68.3  64.6 
3  64.6 
Bigram Learners (words are predictive) 
TP 
TR 
TF  BP  BR 
BF  LP  LR 
LF 
GGJ-Ideal  
70.0  66.3  68.1  86.2  79.8  82.9  61.3  68.3  64.6 
64.6 
DMCMC  
68.6  72.3  70.4  81.2  87.4  84.2  59.5  60.5 
.5  60.5 
59.9 
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