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% *, # %" :' <i# K% = #& ,# - J* b# * %$% ' , K% 6 _ *,! K% , % (% -8ck" " ** # ' # %"( / # * 9 1 '(%-%" 7 F@F%D*A : 29E D 2: =D< : H / 8 # ** D E9 K/, +,+;=( -( ,1& 8 k" ! *' # %( =. * ( #, 6 %# " % (> '# .%") R %" - (-* * **%- *, # * K ( ' - # *%" / %".K% *% (*, # * 7'7
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# +"!L ) " ,,# - " # (*%"** + ;" 9 "Y# ..* ## % ' *, S- K% * %"
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. .%"<* 7(%" * ". N *,g' &* # .%"' ,#&%"%" ' ' %""% , *, g>(
"=. N . P "!% - #
% ' ,# %"% ' . * % * # # * ". N *,g' > "* *,. *
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.% <* 7M(% )
performance of katz with respect to delta
performance of new-avg-count with respect to c-min
1.6
-0.05
100 sent
difference in test cross-entropy from baseline (bits/token)
1.4
1.2
1
10,000 sent
1,000 sent
0.8
0.6
50,000 sent
0.4
0.2
0
-0.06
-0.07
10,000,000 sent
-0.08
-0.09
-0.1
1,000,000 sent
-0.11
-0.12
10,000 sent
-0.13
100 sent
-0.2
0.001
-0.14
0.01
0.1
1
delta
10
100
1000
1
10
100
1000
minimum number of counts per bucket
10000
100000
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9.647
9.646
cross entropy of test data (bits/token)
9.645
9.644
9.643
9.642
9.641
k1
9.64
k2
9.639
k3
9.638
4
6
8
10
12
parameter value
14
16
18
%' # ) =TYI Z$ //2 3 (CB ' *9$%" '
8.49
8.44
8.48
8.435
8.47
8.46
8.45
8.44
8.43
8.43
8.425
8.42
8.415
8.42
8.41
0
500
1000
1500
2000
2500
3000
8.41
500
3500
c_min
1000
1500
2000
c_mb
2500
3000
3500
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3 . 63
R
g ;=" ! < " #$ % ( < '&( ) *( + R 8'=, I ' k" _-, +&./ H\ R 8',= I 0 R1
g+* 2 ]" .34 5 6 7( ! < " #$ 7 * 8 %$( 3+ # 6 + " # 9& / + ( /0# . 9+4& Z !#$9: ( R 9( B. ;J g ; # V / 7(! ( 7(! 9 )< =! '& *
) -
)
067
P;
49
) I) :+
P;
)
average over ten runs at each size, up to 50,000 sentences
single run at each size
11.5
11.5
11
11
10.5
cross-entropy of test data (bits/token)
cross-entropy of test data (bits/token)
10.5
10
9.5
9
TIPSTER bigram
WSJ bigram
8.5
TIPSTER trigram
WSJ trigram
10
Brown bigram
Brown trigram
9.5
9
8.5
8
TIPSTER bigram
7.5
8
WSJ bigram
TIPSTER trigram
7
WSJ trigram
7.5
100
)
6.5
100
1000
10000
sentences of training data (~25 words/sentence)
1000
10000
100000
1e+06
1e+07
g+ ;T / +:#$ 6) ( B.2 (7 ! ( (7 ! 9. b 8 J " / #$ ( B.> * 3 . 9 < +
/ S 4 2 / #$
- -,-,-
- -,-,- -,-,-
single run at each size, up to 10,000,000 sentences
7
7
plus-one
plus-one
6
5
plus-delta
4
3
2
church-gale
1
0
6
5
4
2
1
church-gale
0
katz, interp-held-out, interp-del-int, new-avg-count, new-one-count (see below)
-1
100
plus-delta
3
-1
100
1000
10000
1000
1e+06
1e+07
0.02
0.04
0
0.02
10000
100000
-0.02
interp-del-int
-0.04
-0.06
katz
-0.08
-0.1
interp-held-out
-0.12
new-one-count
-0.14
interp-del-int
0
-0.02
katz
-0.04
-0.06
interp-held-out
-0.08
new-one-count
-0.1
-0.12
new-avg-count
-0.14
new-avg-count
-0.16
100
)
- -,-,-
1000
10000
-0.16
100
;
1000
10000
100000
1e+06
1e+07
g+ $Tj *( W + Y R 8', I 6) 36 7( ! b " #$ 7 * = " # & / + ) ( B. ; .4 7(! ( 7(! 9. < / / .#$ ( B * *+ 59 < .+:/ : 2 / #$ )% ( B = =+" ( ( * X & (/g ( B. *$g = / = 6
! < & / ! & / = - -,-,- -,-,-
5
plus-one
4.5
5
4
3.5
3
plus-delta
2.5
2
1.5
1
church-gale
0.5
0
4.5
4
3.5
plus-one
3
2.5
2
1.5
1
0
-0.5
100
-0.5
100
1000
10000
1000
10000
100000
1e+06
1e+07
0.02
0.02
0
-0.02
plus-delta
church-gale
0.5
church-gale
interp-del-int
-0.04
-0.06
-0.08
-0.1
interp-held-out
new-one-count
-0.12
new-avg-count
0
-0.04
-0.08
church-gale
-0.1
new-one-count
-0.12
katz
katz
new-avg-count
1000
10000
) :+
- -,-,-
interp-held-out
interp-del-int
-0.06
-0.14
100
church-gale
-0.02
-0.14
100
;
1000
10000
100000
1e+06
1e+07
g+ ;T ( + R 8', I 6) * 6 7( ! < " .#$ 7 * = " # & / + ) ( B. ; .4 7(! ( 7(! 9. < / / .#$ ( B * *+ 59 < .+:/ : 2 / #$ )% ( B = =+" ( ( * X & (/g ( B. *$g = / = 6
! < & / ! & / = bigram model
trigram model
0.02
0.02
0
- -,-,- -,-,-
church-gale
-0.02
interp-del-int
-0.04
-0.06
-0.08
-0.1
interp-held-out
-0.12
new-avg-count
new-one-count
katz
-0.14
-0.16
0
-0.02
-0.04
-0.06
katz
-0.08
interp-del-int
-0.1
-0.12
interp-held-out
-0.14
new-one-count
new-avg-count
-0.18
100
)
-0.16
100
1000
10000
g+ T ( ( = " # & +
+ ' k"
(
1000
10000
# ) 6 7( ! b " #$
7 trigram model
0.04
0
0.02
church-gale
-0.02
-0.04
bigram model
0.02
interp-del-int
interp-held-out
-0.06
-0.08
-0.1
new-one-count
-0.12
new-avg-count
-0.14
katz
-0.16
) -
interp-del-int
-0.04
katz
-0.06
-0.08
-0.1
-0.14
new-one-count
1000
10000
100000
new-avg-count
-0.18
100
1e+06
1000
10000
100000
1e+06
=9 + ( # ) 36 7( ! < " #$
g+ 2 T ( - *( e% + _1( , *! 7 = " # & +
katz
interp-held-out
-0.12
-0.16
church-gale
-0.18
100
0
-0.02
6
*/ : *# ( * +!7 6 7(! /:9. + * +*#$ 6 " ! b . 9. b 6 / / #$ * >I # PV H :( B. < */ 3+ 9 &X ." # +! .+ / 6 7 & D;O "Y * R 8',= I 0c ( B & (/ g g + : # / ( B. & 7( Lb #*3+ = / ( * ( 6:
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+ (
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! ! ( " & ! ! 7( ( #$ 3 ( &( 3+ / # . ( " #$ 5 6 * ! ( "
$
9 # !* (% ( *( & + ' #$ 6 ! ( .#$ * 6 7( # #$ =# ( ( . 7 *( =+ # +/ *
) ( &(*+! - -,-,-
--
"
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"!#%$%&'
) -,7
) I) :+
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%$%"#%$%&"
$
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# '& # ! . , , > . 7 3 3
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3 (*5+ 3# # - !( #$.
'# +! !( #$- / ! + 0 1 ) ( "9 64.60 7( # 9 # # 0 +
6 ( ( 7 (# 2 9 2 ( 7 3!( 6 ! 3+ * 0# (*# = # (& &. # 6 !(*8 # # 4) ( 4
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5 5
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6 * 4# *+!9
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6
9
;
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E / ! ) ( " F
0
# !' 7(! (97 ( # ! # (3 6 #(# 4# # C
/ ;
H
-
bigram model
trigram model
40
40
35
35
predicted by ML
predicted by ML
actual
30
30
average corrected count
actual
25
20
15
25
20
15
10
10
5
5
0
0
0
5
10
15
)
20
original count
25
30
35
0
# 2(2 7(! ("#(*# 1# # '(8&.( 6
5
10
*( 15
20
original count
+ 2
bigram model
25
7
30
35
(*. +
trigram model
40
40
35
predicted by ML
35
actual
predicted by ML
30
30
actual
25
20
15
25
20
15
10
10
5
5
0
0
0
5
) )
10
15
20
original count
25
30
35
0
# 2 7(! (#(# # # ( &.( 6
,+
5
. *( 10
15
20
original count
C
)-,-,7
25
30
35
(* +
%
3 7
) ( ( ( (& &. 1 + & + ' 6 ) ( *F 3# # *# # * # 6 (& &. ) ( = ! !( ( 6 # 9 # # 0 # ( ( #(# # # 0 #( (* ( ( # # # ( # #6 )# / 1 + # * + # (# 7(! ( 96; '# 8 # :7 (# ' # & !%
' ) ( "' 9 # # ' #!# # 8 6 # #( # 0& !4 /
) ( "'
#!# # # # #( # # (! #(* +
0# (*# 7(! ( 7 ( # 9 ; # 6 # #.( # (*# ' 7(! ( 96 ; ( 7(# # 9 & 3
# + 5
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bigram model
trigram model
2.2
2.2
interp-baseline
church-gale
interp-held-out
katz
new-avg-count
new-one-count
2
1.8
expected count / actual count
1.8
interp-baseline
church-gale
interp-held-out
katz
new-avg-count
new-one-count
2
1.6
1.4
1.2
1.6
1.4
1.2
1
1
0.8
0.8
0.6
0.6
0
5
10
15
20
count
25
30
35
0
5
10
15
20
count
# 2 # 7(! # #( # # (%7 # ( ( "
&.( 2 ( .+ 6
bigram model
30
*( 35
+ trigram model
2.2
2.4
interp-baseline
church-gale
interp-held-out
katz
new-avg-count
new-one-count
2
interp-baseline
church-gale
interp-held-out
katz
new-avg-count
new-one-count
2.2
2
1.8
25
1.6
1.4
1.2
1.8
1.6
1.4
1
1.2
0.8
1
0.6
0.8
0
5
10
15
20
count
25
30
35
0
5
10
15
# 2 # 7(! # #( # # (%7 # ( ( "
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20
count
25
*( 30
35
+ bigram model
trigram model
0.4
church-gale
interp-held-out
katz
new-avg-count
new-one-count
0.1
0.15
0.05
0
-0.05
-0.1
-0.15
-0.2
-0.25
church-gale
interp-held-out
katz
new-avg-count
new-one-count
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
0
5
10
15
20
count
25
30
35
0
5
10
15
20
count
25
# 2 6 7( !( #$ 6 # # # ( 7 * # ( (* "
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bigram model
( trigram model
0.1
0.4
0
35
church-gale
interp-held-out
katz
new-avg-count
new-one-count
-0.1
-0.2
-0.3
-0.4
-0.5
-0.6
-0.7
-0.8
church-gale
interp-held-out
katz
new-avg-count
new-one-count
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
0
5
10
15
20
count
25
30
35
0
5
10
15
20
count
# 2 6 7( ! ( #$ 6 # # # ( 7 * # ( (* "
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25
30
35
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3
2
bigram model
trigram model
1
1
r<=inf
r<=inf
0.8
r<=2
r<=5
r<=1
0.6
r<=10
cumulative fraction of total entropy
cumulative fraction of total entropy
0.8
r<=15
r=0
0.4
0.2
0
100
r<=10
r<=2 r<=5
r<=15
0.6
r<=1
r=0
0.4
0.2
1000
10000
100000
1e+06
0
100
1e+07
1000
# 2 + # 9 !7 7 * # # # (7 !
! +&.( *( 3+ 10000
100000
1e+06
1e+07
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trigram model
14
14
n_1=8
n_1=8
12
desired counts devoted to zero count n-grams
12
10
8
6
n_1=6
4
n_1=1
n_1=2
n_1=4
2
10
8
n_1=4
6
n_1=6
n_1=2
4
n_1=1
2
n_1=0
n_1=0
0
0
0
2
4
6
8
10
12
N = total counts in distribution
14
16
18
# 2 7(! ( # # * + # #( .&.( *( 3+ 0
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4
6
8
10
12
14
16
18
( "
! # # ("7 # ) 14
12
10
n_1=8
8
n_1=6
6
n_1=4
4
n_1=2
2
n_1=1
0
n_1=0
0
2
4
# 2 (7 ! ( # # * + .# :& 6# * +
6
8
10
12
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16
18
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#
bigram model
trigram model
1000
1000
100
100
shape predicted by linear interpolation
10
10
1
corrected count
corrected count
1
0.1
church-gale
0.01
0.1
church-gale
0.01
0.001
0.001
0.0001
0.0001
1e-05
1e-05
1e-06
1e-13
1e-06
1e-12
1e-11
1e-10
1e-09
1e-08
bucketing value
1e-07
1e-06
1e-07
1e-14
1e-05
1e-13
# % (# # # + !8# # & ( + #
1e-12
1e-11
1e-10
1e-09
bucketing value
1e-08
1e-07
1e-06
1e-05
*# 8( %( (3 & # # ( &( bigram model
trigram model
10
10
r=10
r=10
6
8
corrected count
corrected count
8
r=5
4
6
r=5
4
r=3
r=3
2
2
r=2
r=2
r=1
0
1e-13
1e-12
1e-11
1e-10
1e-09
1e-08
bucketing value
1e-07
1e-06
r=1
1e-05
0
1e-14
1e-13
1e-12
# 2 % (# # # 7 * # 5
# # & &.( * ( + 1e-11
#
1e-10
1e-09
bucketing value
1e-08
1e-07
1e-06
1e-05
*# ( ( ( & # # ( trigram model
0.02
0
0
bigram model
0.02
-0.02
-0.04
interp-del-int
-0.06
-0.08
interp-held-out
-0.1
-0.12
-0.02
interp-del-int
-0.04
-0.06
-0.08
-0.1
interp-held-out
-0.12
-0.14
-0.14
100
-0.16
100
1000
10000
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10000
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execution time (seconds)
700
600
500
400
300
200
100
0
0
5000
10000
15000
20000
25000
training data (tokens)
30000
15000
20000
25000
training data (tokens)
30000
35000
40000
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16
14
memory usage (Mbytes)
12
10
8
6
4
2
0
0
5000
10000
35000
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Building Probabilistic Models For Natural Language

Transcription

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