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## **Frustratingly Easy Domain Adaptation**
### **Hal Daum´e III** School of Computing University of Utah Salt Lake City, Utah 84112 `me@hal3.name` | {
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"Header 2": "**Frustratingly Easy Domain Adaptation**",
"Header 3": "**Hal Daum´e III** School of Computing University of Utah Salt Lake City, Utah 84112 `me@hal3.name`",
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### **Abstract**
We describe an approach to domain adaptation that is appropriate exactly in the case
when one has enough “target” data to do
slightly better than just using only “source”
data. Our approach is incredibly simple,
easy to implement as a preprocessing step
(10 lines of Perl!) and outperforms stateof-the... | {
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"cs.CL"
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### **1 Introduction**
The task of domain adaptation is to develop learning algorithms that can be easily ported from one
domain to another—say, from newswire to biomedical documents. This problem is particularly interesting in NLP because we are often in the situation
that we have a large collection of labeled data ... | {
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"cs.LG",
"cs.CL"
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"Header 3": "**1 Introduction**",
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### **2 Problem Formalization and Prior Work**
To facilitate discussion, we first introduce some notation. Denote by X the input space (typically either
a real vector or a binary vector), and by Y the output
space. We will write D [s] to denote the distribution
over source examples and D [t] to denote the distributio... | {
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"cs.CL"
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"Header 3": "**2 Problem Formalization and Prior Work**",
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In the L IN I NT baseline, we linearly interpolate
the predictions of the S RC O NLY and the T G
T O NLY models. The interpolation parameter is
adjusted based on target development data.
These baselines are actually surprisingly difficult to beat. To date, there are two models that
have successfully defeated them on ... | {
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"categories": [
"cs.LG",
"cs.CL"
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"Header 3": "**2 Problem Formalization and Prior Work**",
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### **3 Adaptation by Feature Augmentation**
In this section, we describe our approach to the domain adaptation problem. Essentially, all we are going to do is take each feature in the original problem
and make three versions of it: a general version, a
source-specific version and a target-specific version.
The augme... | {
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"Header 3": "**3 Adaptation by Feature Augmentation**",
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to capture the fact that the appropriate tag for “the”
remains constant across the domains, and the tag
for “monitor” changes. In this case, the model can
set the “determiner” weight vector to something like
⟨1, 0, 0, 0, 0, 0⟩. This places high weight on the common version of “the” and indicates that “the” is most
li... | {
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"cs.CL"
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"Header 2": "**Frustratingly Easy Domain Adaptation**",
"Header 3": "**3 Adaptation by Feature Augmentation**",
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2we get:K(x, x [′] K)˘. When they are from different domains,(x, x [′] ) = ⟨Φ(x), Φ(x [′] )⟩ X = K(x, x [′] ).
Putting this together, we have:
˘ 2K(x, x ′ ) same domain
K(x, x [′] ) = (3)
� K(x, x [′] ) diff. domain
This is an intuitively pleasing result. What it
says is that—considering the kernel as a measure
of ... | {
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"cs.LG",
"cs.CL"
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"Header 3": "**3 Adaptation by Feature Augmentation**",
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that the single supervised learning algorithm that
is run is allowed to regulate the trade-off between
source/target and general weights. In the P RIOR
model, we are forced to use the prior variance on
in the target learning scenario to do this ourselves.
**3.3** **Multi-domain adaptation**
Our formulation is agnos... | {
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"Header 3": "**3 Adaptation by Feature Augmentation**",
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### **4 Results**
In this section we describe experimental results on a
wide variety of domains. First we describe the tasks,
then we present experimental results, and finally we
look more closely at a few of the experiments.
**4.1** **Tasks**
All tasks we consider are sequence labeling tasks
(either named-entity... | {
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br-cm 11,696 1,324 1,594 51k
br-cn 56,057 6,751 7,847 115k
br-cp 55,318 7,477 5,977 112k
br-cr 16,742 2,522 2,712 65k
Table 1: Task statistics; columns are task, domain,
size of the training, development and test sets, and
the number of unique features in the training set.
Treebank-Brown. This is identical to the T... | {
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For each row, the error rate of the best performing
technique is bolded (as are all techniques whose performance is not statistically significantly different at
the 95% level). The “T<S” column is contains a “+”
whenever T GT O NLY outperforms S RC O NLY (this
will become important shortly). The final column
indicates ... | {
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swbd3 15.90 4.15 4.50 4.10 4.13 4.09 3.60 **3.51** + +
br-cf 5.16 6.27 4.85 4.80 4.78 **4.72** 5.22 5.15
Tree br-cg 4.32 5.36 **4.16** **4.15** 4.27 4.30 4.25 4.90
bank- br-ck 5.05 6.32 5.05 4.98 **5.01** **5.05** 5.27 5.41
Chunk br-cl 5.66 6.60 5.42 **5.39** **5.39** 5.53 5.99 5.73
br-cm 3.57 6.59 **3.14** **3.1... | {
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space, we are creating a more powerful model.
While this may be a partial explanation, here we
show that what the model learns about the various
domains actually makes some plausible sense.
We perform this analysis only on the ACE-NER
data by looking specifically at the learned weights.
That is, for any given feature... | {
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the word “bush” is going to be a person—this is because it rarely appears referring to a plant and so
even in the capitalized domains like broadcast con
4 Technically there are many more classes than are shown
here. We do not depict the smallest classes, and have merged
the “Begin-*” and “In-*” weights for each entity ... | {
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the feature “the current word is on a list of com
mon names” (this feature is case- *in* sensitive). All
around, this is a good feature for picking out people
and nothing else. The two exceptions are: it is also
a good feature for other entity types for broadcast
news and it is not quite so good for people in usenet.
T... | {
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### **5 Discussion**
In this paper we have described an *incredibly* simple approach to domain adaptation that—under a
common and easy-to-verify condition—outperforms
previous approaches. While it is somewhat frustrating that something so simple does so well, it
is perhaps not surprising. By augmenting the feature sp... | {
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### **References**
[Ben-David et al.2006] Shai Ben-David, John Blitzer, Koby
Crammer, and Fernando Pereira. 2006. Analysis of representations for domain adaptation. In *Advances in Neural*
*Information Processing Systems (NIPS)* .
[Blitzer et al.2006] John Blitzer, Ryan McDonald, and Fernando
Pereira. 2006. Domain ... | {
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# Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory
#### Sumio Watanabe P&I Lab., Tokyo Institute of Technology 4259 Nagatsuta, Midoriku, Yokohama, 226-8503 Japan E-mail: swatanab(at)pi.titech.ac.jp P&I Lab., Tokyo Institute of Technology 4259 Nag... | {
"id": "1004.2316",
"title": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable\n Information Criterion in Singular Learning Theory",
"categories": [
"cs.LG"
]
} | {
"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": null,
"Header 4": "Sumio Watanabe P&I Lab., Tokyo Institute of Technology 4259 Nagatsuta, Midoriku, Yokohama, 226-8503 Japan E-mail: swatanab(at... | {
"chunk_type": "title"
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##### Keywords: Cross Validation, Information Criterion, Singular Learning Ma- chines 1
----- | {
"id": "1004.2316",
"title": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable\n Information Criterion in Singular Learning Theory",
"categories": [
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} | {
"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": null,
"Header 4": "Sumio Watanabe P&I Lab., Tokyo Institute of Technology 4259 Nagatsuta, Midoriku, Yokohama, 226-8503 Japan E-mail: swatanab(at... | {
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### 1 Introduction | {
"id": "1004.2316",
"title": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable\n Information Criterion in Singular Learning Theory",
"categories": [
"cs.LG"
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} | {
"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": "1 Introduction",
"Header 4": null,
"Header 5": "A statistical model or a learning machine is called regular if the map from a param- eter to ... | {
"chunk_type": "body"
} |
##### A statistical model or a learning machine is called regular if the map from a param- eter to a probability distribution is one-to-one and if its Fisher information matrix is positive definite. If a model is not regular, then it is called singular. Although asymptotic statistical theory of regular statistical mode... | {
"id": "1004.2316",
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"categories": [
"cs.LG"
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} | {
"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": "1 Introduction",
"Header 4": null,
"Header 5": "A statistical model or a learning machine is called regular if the map from a param- eter to ... | {
"chunk_type": "body"
} |
generalization error [Watanabe 09, Watanabe 10a, Watanabe 10c]. The cross validation is the alternative method to estimate the generalization error [Mosier 51, Stone 77, Geisser 75]. By the definition, the average of the cross validation is equal to the average generalization error in both regular and singular models. ... | {
"id": "1004.2316",
"title": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable\n Information Criterion in Singular Learning Theory",
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"cs.LG"
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} | {
"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": "1 Introduction",
"Header 4": null,
"Header 5": "A statistical model or a learning machine is called regular if the map from a param- eter to ... | {
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----- | {
"id": "1004.2316",
"title": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable\n Information Criterion in Singular Learning Theory",
"categories": [
"cs.LG"
]
} | {
"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": "1 Introduction",
"Header 4": null,
"Header 5": "A statistical model or a learning machine is called regular if the map from a param- eter to ... | {
"chunk_type": "body"
} |
##### Variable Name eq. number E w [ ] posterior average eq.(1) E [(] w [i][)] [[ ]] posterior average without X i eq.(19) L(w) log loss function eq.(6) L 0 minimum loss eq.(8) L n emiprical loss eq.(9) B g L(n) Bayes generalization loss eq.(2) B t L(n) Bayes training loss eq.(3) C v L(n) cross validation loss eq.(20) ... | {
"id": "1004.2316",
"title": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable\n Information Criterion in Singular Learning Theory",
"categories": [
"cs.LG"
]
} | {
"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": "1 Introduction",
"Header 4": null,
"Header 5": "Variable Name eq. number E w [ ] posterior average eq.(1) E [(] w [i][)] [[ ]] posterior aver... | {
"chunk_type": "body"
} |
### 2 Bayes Learning Theory
##### In this section, we summarize Bayes learning theory for singular learning machines. The results written in this section are well known and the fundamental basis of this paper. Table 1 shows variables, names, and equation numbers in this paper. | {
"id": "1004.2316",
"title": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable\n Information Criterion in Singular Learning Theory",
"categories": [
"cs.LG"
]
} | {
"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": "2 Bayes Learning Theory",
"Header 4": null,
"Header 5": "In this section, we summarize Bayes learning theory for singular learning machines. ... | {
"chunk_type": "body"
} |
#### 2.1 Framework of Bayes Learning
##### Firstly, we explain the framework of Bayes learning. Let q(x) be a probability density function on N dimensional real Euclidean space R [N] . The training samples and the testing sample are respectively denoted by random variables X 1, X 2, ..., X n and X, which are independ... | {
"id": "1004.2316",
"title": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable\n Information Criterion in Singular Learning Theory",
"categories": [
"cs.LG"
]
} | {
"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": "2 Bayes Learning Theory",
"Header 4": "2.1 Framework of Bayes Learning",
"Header 5": "Firstly, we explain the framework of Bayes learning. Le... | {
"chunk_type": "body"
} |
##### all parameters. In Bayes estimation, we prepare a probability density function ϕ(w) on W . Although ϕ(w) is called a prior distribution, it does not necessary represent an a priori knowledge of the parameter, in general. For a given function f (w) on W, its expectation value with respect to the pos- terior distri... | {
"id": "1004.2316",
"title": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable\n Information Criterion in Singular Learning Theory",
"categories": [
"cs.LG"
]
} | {
"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": "2 Bayes Learning Theory",
"Header 4": "2.1 Framework of Bayes Learning",
"Header 5": "all parameters. In Bayes estimation, we prepare a proba... | {
"chunk_type": "body"
} |
##### E w [f (w)] =
n | {
"id": "1004.2316",
"title": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable\n Information Criterion in Singular Learning Theory",
"categories": [
"cs.LG"
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} | {
"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": "2 Bayes Learning Theory",
"Header 4": "2.1 Framework of Bayes Learning",
"Header 5": "E w [f (w)] ="
} | {
"chunk_type": "body"
} |
##### f (w) � p(X i |w) [β] ϕ(w)dw �
i=1
n | {
"id": "1004.2316",
"title": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable\n Information Criterion in Singular Learning Theory",
"categories": [
"cs.LG"
]
} | {
"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": "2 Bayes Learning Theory",
"Header 4": "2.1 Framework of Bayes Learning",
"Header 5": "f (w) � p(X i |w) [β] ϕ(w)dw �"
} | {
"chunk_type": "body"
} |
##### � p(X i |w) [β] ϕ(w)dw �
i=1
#####, (1) | {
"id": "1004.2316",
"title": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable\n Information Criterion in Singular Learning Theory",
"categories": [
"cs.LG"
]
} | {
"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": "2 Bayes Learning Theory",
"Header 4": "2.1 Framework of Bayes Learning",
"Header 5": "� p(X i |w) [β] ϕ(w)dw �"
} | {
"chunk_type": "body"
} |
##### where 0 < β < ∞ is the inverse temperature. The case β = 1 is most important because it corresponds to the conventional Bayes estimation. The Bayes predictive distribution is defined by p [∗] (x) ≡ E w [p(x|w)]. In Bayes learning theory, the following random variables are important. The Bayes generalization loss ... | {
"id": "1004.2316",
"title": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable\n Information Criterion in Singular Learning Theory",
"categories": [
"cs.LG"
]
} | {
"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": "2 Bayes Learning Theory",
"Header 4": "2.1 Framework of Bayes Learning",
"Header 5": "where 0 < β < ∞ is the inverse temperature. The case β ... | {
"chunk_type": "body"
} |
##### B t L(n) = −n [1] � log p [∗] (X i ), (3)
i=1 | {
"id": "1004.2316",
"title": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable\n Information Criterion in Singular Learning Theory",
"categories": [
"cs.LG"
]
} | {
"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": "2 Bayes Learning Theory",
"Header 4": "2.1 Framework of Bayes Learning",
"Header 5": "B t L(n) = −n [1] � log p [∗] (X i ), (3)"
} | {
"chunk_type": "body"
} |
##### where E X [ ] shows the expectation value over X. The functional variance is defined by
n | {
"id": "1004.2316",
"title": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable\n Information Criterion in Singular Learning Theory",
"categories": [
"cs.LG"
]
} | {
"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": "2 Bayes Learning Theory",
"Header 4": "2.1 Framework of Bayes Learning",
"Header 5": "where E X [ ] shows the expectation value over X. The f... | {
"chunk_type": "body"
} |
##### V (n) = ��E w [(log p(X i |w)) [2] ] − E w [log p(X i |w)] [2] [�], (4)
i=1 | {
"id": "1004.2316",
"title": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable\n Information Criterion in Singular Learning Theory",
"categories": [
"cs.LG"
]
} | {
"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": "2 Bayes Learning Theory",
"Header 4": "2.1 Framework of Bayes Learning",
"Header 5": "V (n) = ��E w [(log p(X i |w)) [2] ] − E w [log p(X i |... | {
"chunk_type": "body"
} |
##### which shows fluctuation of the posterior distribution. In previous papers [Watanabe 09, Watanabe 10a, Watanabe 10b], we defined the widely applicable information crite- rion WAIC(n) ≡ B t L(n) + [β] (5) n [V][ (][n][)][,] and proved that E[B g L(n)] = E[WAIC(n)] + o( [1] n [)][,] where E[ ] shows the expectation ... | {
"id": "1004.2316",
"title": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable\n Information Criterion in Singular Learning Theory",
"categories": [
"cs.LG"
]
} | {
"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": "2 Bayes Learning Theory",
"Header 4": "2.1 Framework of Bayes Learning",
"Header 5": "which shows fluctuation of the posterior distribution. ... | {
"chunk_type": "body"
} |
#### 2.2 Notations
##### Secondly, we explain several notations. 4
----- | {
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"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": "2 Bayes Learning Theory",
"Header 4": "2.2 Notations",
"Header 5": "Secondly, we explain several notations. 4"
} | {
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##### The log loss function L(w) and the entropy S of the true distribution are respec- tively defined by L(w) = −E X [log p(X|w)], (6) S = −E X [log q(X)]. (7) Then L(w) = S + D(q||p w ), where D(q||p w ) is Kullback-Leibler distance defined by D(q||p w ) = q(x) log [q][(][x][)] � p(x|w) [dx.] Then D(q||p w ) ≥ 0, hen... | {
"id": "1004.2316",
"title": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable\n Information Criterion in Singular Learning Theory",
"categories": [
"cs.LG"
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} | {
"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": "2 Bayes Learning Theory",
"Header 4": "2.2 Notations",
"Header 5": "The log loss function L(w) and the entropy S of the true distribution are... | {
"chunk_type": "body"
} |
##### L n = −n [1] � log p 0 (X i ). (9)
i=1 | {
"id": "1004.2316",
"title": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable\n Information Criterion in Singular Learning Theory",
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"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": "2 Bayes Learning Theory",
"Header 4": "2.2 Notations",
"Header 5": "L n = −n [1] � log p 0 (X i ). (9)"
} | {
"chunk_type": "body"
} |
##### By using these values, Bayes generalization error B g (n) and Bayes training error B t (n) are respectively defined by B g (n) = B g L(n) − L 0, (10) B t (n) = B t L(n) − L n . (11) In general, both B g (n) and B t (n) converge to zero in probability, when n →∞. Let us define a log density ratio function, f (x, w... | {
"id": "1004.2316",
"title": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable\n Information Criterion in Singular Learning Theory",
"categories": [
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} | {
"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": "2 Bayes Learning Theory",
"Header 4": "2.2 Notations",
"Header 5": "By using these values, Bayes generalization error B g (n) and Bayes train... | {
"chunk_type": "body"
} |
##### Then, it is immediately derived that B g (n) = −E X [log E w [exp(−f (X, w))]], (12)
n | {
"id": "1004.2316",
"title": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable\n Information Criterion in Singular Learning Theory",
"categories": [
"cs.LG"
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} | {
"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": "2 Bayes Learning Theory",
"Header 4": "2.2 Notations",
"Header 5": "Then, it is immediately derived that B g (n) = −E X [log E w [exp(−f (X, ... | {
"chunk_type": "body"
} |
##### B t (n) = −n [1] � log E w [exp(−f (X i, w))], (13)
i=1
n | {
"id": "1004.2316",
"title": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable\n Information Criterion in Singular Learning Theory",
"categories": [
"cs.LG"
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} | {
"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": "2 Bayes Learning Theory",
"Header 4": "2.2 Notations",
"Header 5": "B t (n) = −n [1] � log E w [exp(−f (X i, w))], (13)"
} | {
"chunk_type": "body"
} |
##### V (n) = ��E w [f (X i, w) [2] ] − E w [f (X i, w) [2] ]�. (14)
i=1 | {
"id": "1004.2316",
"title": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable\n Information Criterion in Singular Learning Theory",
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} | {
"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": "2 Bayes Learning Theory",
"Header 4": "2.2 Notations",
"Header 5": "V (n) = ��E w [f (X i, w) [2] ] − E w [f (X i, w) [2] ]�. (14)"
} | {
"chunk_type": "body"
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##### Therefore, the problem of statistical learning is characterized by the function f (x, w). Definition. (1) If q(x) = p 0 (x), then q(x) is said to be realizable by p(x|w). If otherwise, then it is said to be unrealizable. (2) If the set W 0 consists of a single point w 0 and if the Hessian matrix ∇∇L(w 0 ) is stri... | {
"id": "1004.2316",
"title": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable\n Information Criterion in Singular Learning Theory",
"categories": [
"cs.LG"
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} | {
"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": "2 Bayes Learning Theory",
"Header 4": "2.2 Notations",
"Header 5": "Therefore, the problem of statistical learning is characterized by the fu... | {
"chunk_type": "body"
} |
#### 2.3 Singular Learning Theory
##### Thirdly, we summarize singular learning theory. In this paper, we assume the fol- lowing conditions. Assumptions. (1) The set of parameters W is a compact set in R [d] whose open kernel [1] is not the empty set. Its boundary is defined by several analytic functions, In other wo... | {
"id": "1004.2316",
"title": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable\n Information Criterion in Singular Learning Theory",
"categories": [
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} | {
"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": "2 Bayes Learning Theory",
"Header 4": "2.3 Singular Learning Theory",
"Header 5": "Thirdly, we summarize singular learning theory. In this pa... | {
"chunk_type": "body"
} |
##### L [s] (q) = {f (x); ∥f ∥≡ |f (x)| [s] q(x)dx < ∞} � [�] � be a Banach space. The map W ∋ w �→ f (x, w) is an L [s] (q) valued analytic function. (4) A nonnegative function K(w) is defined by K(w) = E X [f (X, w)].
1 The open kernel of a set A is the largest open set that is contained in A. | {
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"title": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable\n Information Criterion in Singular Learning Theory",
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"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": "2 Bayes Learning Theory",
"Header 4": "2.3 Singular Learning Theory",
"Header 5": "L [s] (q) = {f (x); ∥f ∥≡ |f (x)| [s] q(x)dx < ∞} � [�] � ... | {
"chunk_type": "body"
} |
##### 6
----- | {
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"title": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable\n Information Criterion in Singular Learning Theory",
"categories": [
"cs.LG"
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} | {
"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": "2 Bayes Learning Theory",
"Header 4": "2.3 Singular Learning Theory",
"Header 5": "6"
} | {
"chunk_type": "body"
} |
##### The set W ǫ is defined by W ǫ = {w ∈ W ; K(w) ≤ ǫ}. It is assumed that there exist constants ǫ, c > 0 such that (∀w ∈ W ǫ ) E X [f (X, w)] ≥ c E X [f (X, w) [2] ]. (15) In order to study the cross validation in singular learning machines, we need singular learning theory. In the previous papers, we obtained the f... | {
"id": "1004.2316",
"title": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable\n Information Criterion in Singular Learning Theory",
"categories": [
"cs.LG"
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} | {
"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": "2 Bayes Learning Theory",
"Header 4": "2.3 Singular Learning Theory",
"Header 5": "The set W ǫ is defined by W ǫ = {w ∈ W ; K(w) ≤ ǫ}. It is ... | {
"chunk_type": "body"
} |
##### Then limsup n→∞ n [k][/][2] M k (n) < ∞. (17) � � (3) The expectation value of the Bayes generalization loss is asymptotically equal to the widely applicable information criterion, E[B g L(n)] = E[WAIC(n)] + o( [1] (18) n [)][.] (Proof) In the case when q(x) is realizable by and singular for p(x|w), this lemma wa... | {
"id": "1004.2316",
"title": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable\n Information Criterion in Singular Learning Theory",
"categories": [
"cs.LG"
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} | {
"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": "2 Bayes Learning Theory",
"Header 4": "2.3 Singular Learning Theory",
"Header 5": "Then limsup n→∞ n [k][/][2] M k (n) < ∞. (17) � � (3) The ... | {
"chunk_type": "body"
} |
### 3 Bayes Cross Validation
##### In this section, we introduce the cross validation in Bayes learning. For an arbitrary function f (w), the expectation value E w [(][i][)] [[][f] [(][w][)] using the pos-] terior distribution leaving X i out is defined by | {
"id": "1004.2316",
"title": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable\n Information Criterion in Singular Learning Theory",
"categories": [
"cs.LG"
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} | {
"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": "3 Bayes Cross Validation",
"Header 4": null,
"Header 5": "In this section, we introduce the cross validation in Bayes learning. For an arbitr... | {
"chunk_type": "body"
} |
##### E [(] w [i][)] [[][f] [(][w][)] =]
n | {
"id": "1004.2316",
"title": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable\n Information Criterion in Singular Learning Theory",
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"cs.LG"
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} | {
"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": "3 Bayes Cross Validation",
"Header 4": null,
"Header 5": "E [(] w [i][)] [[][f] [(][w][)] =]"
} | {
"chunk_type": "body"
} |
##### f (w) � p(X j |w) [β] ϕ(w)dw � j≠ i
n | {
"id": "1004.2316",
"title": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable\n Information Criterion in Singular Learning Theory",
"categories": [
"cs.LG"
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} | {
"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": "3 Bayes Cross Validation",
"Header 4": null,
"Header 5": "f (w) � p(X j |w) [β] ϕ(w)dw � j≠ i"
} | {
"chunk_type": "body"
} |
##### � p(X j |w) [β] ϕ(w)dw � j≠ i
#####, (19)
n | {
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"title": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable\n Information Criterion in Singular Learning Theory",
"categories": [
"cs.LG"
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} | {
"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": "3 Bayes Cross Validation",
"Header 4": null,
"Header 5": "� p(X j |w) [β] ϕ(w)dw � j≠ i"
} | {
"chunk_type": "body"
} |
##### where � shows the product for j = 1, 2, 3, .., n which does not include j = i. The
j≠ i | {
"id": "1004.2316",
"title": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable\n Information Criterion in Singular Learning Theory",
"categories": [
"cs.LG"
]
} | {
"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": "3 Bayes Cross Validation",
"Header 4": null,
"Header 5": "where � shows the product for j = 1, 2, 3, .., n which does not include j = i. The"... | {
"chunk_type": "body"
} |
##### predictive distribution leaving X i out is defined by p [(][i][)] (x) = E w [(][i][)] [[][p][(][x][|][w][)]][.] The log loss of p [(][i][)] (x) when X i is used as a testing sample is − log p [(][i][)] (X i ) = − log E w [(][i][)] [[][p][(][X] [i] [|][w][)]][.] Hence the log loss of the Bayes cross validation lea... | {
"id": "1004.2316",
"title": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable\n Information Criterion in Singular Learning Theory",
"categories": [
"cs.LG"
]
} | {
"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": "3 Bayes Cross Validation",
"Header 4": null,
"Header 5": "predictive distribution leaving X i out is defined by p [(][i][)] (x) = E w [(][i][... | {
"chunk_type": "body"
} |
##### C v L(n) = − [1] n
n | {
"id": "1004.2316",
"title": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable\n Information Criterion in Singular Learning Theory",
"categories": [
"cs.LG"
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} | {
"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": "3 Bayes Cross Validation",
"Header 4": null,
"Header 5": "C v L(n) = − [1] n"
} | {
"chunk_type": "body"
} |
##### � log E w [(][i][)] [[][p][(][X] [i] [|][w][)]][.] (20)
i=1 | {
"id": "1004.2316",
"title": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable\n Information Criterion in Singular Learning Theory",
"categories": [
"cs.LG"
]
} | {
"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": "3 Bayes Cross Validation",
"Header 4": null,
"Header 5": "� log E w [(][i][)] [[][p][(][X] [i] [|][w][)]][.] (20)"
} | {
"chunk_type": "body"
} |
##### The random variable C v L(n) is referred to as the cross validation loss in this paper. Since X 1, X 2, ..., X n are independent training samples, it immediately follows that E[C v L(n)] = E[B g L(n − 1)]. Two random variables C v L(n) and B g L(n − 1) are different, C v L(n) ̸= B g L(n − 1), however, their expec... | {
"id": "1004.2316",
"title": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable\n Information Criterion in Singular Learning Theory",
"categories": [
"cs.LG"
]
} | {
"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": "3 Bayes Cross Validation",
"Header 4": null,
"Header 5": "The random variable C v L(n) is referred to as the cross validation loss in this pa... | {
"chunk_type": "body"
} |
##### Remark. In practical applications, the Bayes generalization loss B g L(n) indicates the accuracy of Bayes estimation. However, in order to calculate B g L(n), we need the expectation value over the testing sample taken from the unknown true distri- bution, resulting that we can not directly obtain B g L(n) in pra... | {
"id": "1004.2316",
"title": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable\n Information Criterion in Singular Learning Theory",
"categories": [
"cs.LG"
]
} | {
"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": "3 Bayes Cross Validation",
"Header 4": null,
"Header 5": "Remark. In practical applications, the Bayes generalization loss B g L(n) indicates... | {
"chunk_type": "body"
} |
### 4 Main Results
##### In this section, main results of this paper are explained. | {
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} | {
"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": "4 Main Results",
"Header 4": null,
"Header 5": "In this section, main results of this paper are explained."
} | {
"chunk_type": "body"
} |
#### 4.1 Functional Cumulants
##### Firstly, we define functional cumulants. Definition. The generating function F (α) of functional cumulants is defined by | {
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"title": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable\n Information Criterion in Singular Learning Theory",
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} | {
"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": "4 Main Results",
"Header 4": "4.1 Functional Cumulants",
"Header 5": "Firstly, we define functional cumulants. Definition. The generating fun... | {
"chunk_type": "body"
} |
##### F (α) = [1] n
n | {
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"title": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable\n Information Criterion in Singular Learning Theory",
"categories": [
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} | {
"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": "4 Main Results",
"Header 4": "4.1 Functional Cumulants",
"Header 5": "F (α) = [1] n"
} | {
"chunk_type": "body"
} |
##### � log E w [p(X i |w) [α] ]. (21)
i=1 | {
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"title": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable\n Information Criterion in Singular Learning Theory",
"categories": [
"cs.LG"
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} | {
"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": "4 Main Results",
"Header 4": "4.1 Functional Cumulants",
"Header 5": "� log E w [p(X i |w) [α] ]. (21)"
} | {
"chunk_type": "body"
} |
##### The kth order functional cumulant Y k (n) (k = 1, 2, 3, 4) is defined by | {
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} | {
"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": "4 Main Results",
"Header 4": "4.1 Functional Cumulants",
"Header 5": "The kth order functional cumulant Y k (n) (k = 1, 2, 3, 4) is defined b... | {
"chunk_type": "body"
} |
##### Then, by definition, For simple notation, we use | {
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"title": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable\n Information Criterion in Singular Learning Theory",
"categories": [
"cs.LG"
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} | {
"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": "4 Main Results",
"Header 4": "4.1 Functional Cumulants",
"Header 5": "Then, by definition, For simple notation, we use"
} | {
"chunk_type": "body"
} |
##### Y k (n) = [d] [k] [F] (22) dα [k] [ (0)][.] F (0) = 0, F (1) = −B t L(n). | {
"id": "1004.2316",
"title": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable\n Information Criterion in Singular Learning Theory",
"categories": [
"cs.LG"
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} | {
"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": "4 Main Results",
"Header 4": "4.1 Functional Cumulants",
"Header 5": "Y k (n) = [d] [k] [F] (22) dα [k] [ (0)][.] F (0) = 0, F (1) = −B t L(n... | {
"chunk_type": "body"
} |
##### ℓ k (X i ) = E w [(log p(X i |w)) [k] ] (k = 1, 2, 3, 4). 9
----- | {
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"title": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable\n Information Criterion in Singular Learning Theory",
"categories": [
"cs.LG"
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} | {
"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": "4 Main Results",
"Header 4": "4.1 Functional Cumulants",
"Header 5": "ℓ k (X i ) = E w [(log p(X i |w)) [k] ] (k = 1, 2, 3, 4). 9"
} | {
"chunk_type": "body"
} |
##### Lemma 2. The followings hold, 1 n Y 1 (n) = n � ℓ 1 (X i ), (23)
i=1 | {
"id": "1004.2316",
"title": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable\n Information Criterion in Singular Learning Theory",
"categories": [
"cs.LG"
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} | {
"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": "4 Main Results",
"Header 4": "4.1 Functional Cumulants",
"Header 5": "Lemma 2. The followings hold, 1 n Y 1 (n) = n � ℓ 1 (X i ), (23)"
} | {
"chunk_type": "body"
} |
##### 1 n Y 2 (n) = n ��ℓ 2 (X i ) − ℓ 1 (X i ) [2] [�], (24)
i=1 | {
"id": "1004.2316",
"title": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable\n Information Criterion in Singular Learning Theory",
"categories": [
"cs.LG"
]
} | {
"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": "4 Main Results",
"Header 4": "4.1 Functional Cumulants",
"Header 5": "1 n Y 2 (n) = n ��ℓ 2 (X i ) − ℓ 1 (X i ) [2] [�], (24)"
} | {
"chunk_type": "body"
} |
##### 1 n Y 3 (n) = n ��ℓ 3 (X i ) − 3ℓ 2 (X i )ℓ 1 (X i ) + 2ℓ 1 (X i ) [3] [�], (25)
i=1 | {
"id": "1004.2316",
"title": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable\n Information Criterion in Singular Learning Theory",
"categories": [
"cs.LG"
]
} | {
"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": "4 Main Results",
"Header 4": "4.1 Functional Cumulants",
"Header 5": "1 n Y 3 (n) = n ��ℓ 3 (X i ) − 3ℓ 2 (X i )ℓ 1 (X i ) + 2ℓ 1 (X i ) [3] ... | {
"chunk_type": "body"
} |
##### 1 n Y 4 (n) = n ��ℓ 4 (X i ) − 4ℓ 3 (X i )ℓ 1 (X i ) − 3ℓ 2 (X i ) [2] (26)
i=1 | {
"id": "1004.2316",
"title": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable\n Information Criterion in Singular Learning Theory",
"categories": [
"cs.LG"
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} | {
"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": "4 Main Results",
"Header 4": "4.1 Functional Cumulants",
"Header 5": "1 n Y 4 (n) = n ��ℓ 4 (X i ) − 4ℓ 3 (X i )ℓ 1 (X i ) − 3ℓ 2 (X i ) [2] ... | {
"chunk_type": "body"
} |
##### +12ℓ 2 (X i )ℓ 1 (X i ) [2] − 6ℓ 1 (X i ) [4] [�] . (27) | {
"id": "1004.2316",
"title": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable\n Information Criterion in Singular Learning Theory",
"categories": [
"cs.LG"
]
} | {
"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": "4 Main Results",
"Header 4": "4.1 Functional Cumulants",
"Header 5": "+12ℓ 2 (X i )ℓ 1 (X i ) [2] − 6ℓ 1 (X i ) [4] [�] . (27)"
} | {
"chunk_type": "body"
} |
##### Moreover, In other words, | {
"id": "1004.2316",
"title": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable\n Information Criterion in Singular Learning Theory",
"categories": [
"cs.LG"
]
} | {
"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": "4 Main Results",
"Header 4": "4.1 Functional Cumulants",
"Header 5": "Moreover, In other words,"
} | {
"chunk_type": "body"
} |
##### Y k (n) = O p ( [1] (k = 2, 3, 4). n [k/][2] [)] limsup n→∞ E[n [k][/][2] |Y k (n)|] < ∞ (k = 2, 3, 4). (28) | {
"id": "1004.2316",
"title": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable\n Information Criterion in Singular Learning Theory",
"categories": [
"cs.LG"
]
} | {
"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": "4 Main Results",
"Header 4": "4.1 Functional Cumulants",
"Header 5": "Y k (n) = O p ( [1] (k = 2, 3, 4). n [k/][2] [)] limsup n→∞ E[n [k][/][... | {
"chunk_type": "body"
} |
##### (Proof) Firstly, we prove eqs.(23)-(27). Let us define g i (α) = E w [p(X i |w) [α] ]. Then g i (0) = 1, g i [(][k][)] [(0) =][ ℓ] [k] [(][X] [i] [)] (k = 1, 2, 3, 4), and
n | {
"id": "1004.2316",
"title": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable\n Information Criterion in Singular Learning Theory",
"categories": [
"cs.LG"
]
} | {
"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": "4 Main Results",
"Header 4": "4.1 Functional Cumulants",
"Header 5": "(Proof) Firstly, we prove eqs.(23)-(27). Let us define g i (α) = E w [p... | {
"chunk_type": "body"
} |
##### F (α) = n [1] � log g i (α).
i=1 | {
"id": "1004.2316",
"title": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable\n Information Criterion in Singular Learning Theory",
"categories": [
"cs.LG"
]
} | {
"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": "4 Main Results",
"Header 4": "4.1 Functional Cumulants",
"Header 5": "F (α) = n [1] � log g i (α)."
} | {
"chunk_type": "body"
} |
##### For arbitrary natural number k, | {
"id": "1004.2316",
"title": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable\n Information Criterion in Singular Learning Theory",
"categories": [
"cs.LG"
]
} | {
"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": "4 Main Results",
"Header 4": "4.1 Functional Cumulants",
"Header 5": "For arbitrary natural number k,"
} | {
"chunk_type": "body"
} |
##### g i (α) (k) � g i (α) | {
"id": "1004.2316",
"title": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable\n Information Criterion in Singular Learning Theory",
"categories": [
"cs.LG"
]
} | {
"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": "4 Main Results",
"Header 4": "4.1 Functional Cumulants",
"Header 5": "g i (α) (k) � g i (α)"
} | {
"chunk_type": "body"
} |
##### ′ = [g] [i] [(][α][)] [(][k][+1)] − g i (α) (k) � g i (α) � g i (α)
′ | {
"id": "1004.2316",
"title": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable\n Information Criterion in Singular Learning Theory",
"categories": [
"cs.LG"
]
} | {
"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": "4 Main Results",
"Header 4": "4.1 Functional Cumulants",
"Header 5": "′ = [g] [i] [(][α][)] [(][k][+1)] − g i (α) (k) � g i (α) � g i (α)"
} | {
"chunk_type": "body"
} |
##### g i (α) �� g i (α) | {
"id": "1004.2316",
"title": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable\n Information Criterion in Singular Learning Theory",
"categories": [
"cs.LG"
]
} | {
"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": "4 Main Results",
"Header 4": "4.1 Functional Cumulants",
"Header 5": "g i (α) �� g i (α)"
} | {
"chunk_type": "body"
} |
##### . � | {
"id": "1004.2316",
"title": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable\n Information Criterion in Singular Learning Theory",
"categories": [
"cs.LG"
]
} | {
"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": "4 Main Results",
"Header 4": "4.1 Functional Cumulants",
"Header 5": ". �"
} | {
"chunk_type": "body"
} |
##### By using this relation recursively, eqs.(23)-(27) are derived. Secondly, we show eq.(28). The random variables Y k (n) (k = 2, 3, 4) are invariant under the transform, log p(X i |w) �→ log p(X i |w) + c(X i ), (29) for arbitrary c(X i ). In particular, by choosing c(X i ) = − log p 0 (X i ),@we can show that Y k ... | {
"id": "1004.2316",
"title": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable\n Information Criterion in Singular Learning Theory",
"categories": [
"cs.LG"
]
} | {
"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": "4 Main Results",
"Header 4": "4.1 Functional Cumulants",
"Header 5": "By using this relation recursively, eqs.(23)-(27) are derived. Secondly... | {
"chunk_type": "body"
} |
##### Then by using eq.(17), eq.(28) is obtained. (Q.E.D.) Note that Y k (n) (k = 1, 2, 3, 4) can be calculated using only training samples and a statistical model, without any information about the true distribution q(x). More- over, by definition, nY 2 (n) = V (n). By using eq.(29) with c(X i ) = −E w [log p(X i |w)]... | {
"id": "1004.2316",
"title": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable\n Information Criterion in Singular Learning Theory",
"categories": [
"cs.LG"
]
} | {
"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": "4 Main Results",
"Header 4": "4.1 Functional Cumulants",
"Header 5": "Then by using eq.(17), eq.(28) is obtained. (Q.E.D.) Note that Y k (n) ... | {
"chunk_type": "body"
} |
##### 1 Y 2 (n) = n 1 Y 3 (n) = n 1 Y 4 (n) = n
n | {
"id": "1004.2316",
"title": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable\n Information Criterion in Singular Learning Theory",
"categories": [
"cs.LG"
]
} | {
"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": "4 Main Results",
"Header 4": "4.1 Functional Cumulants",
"Header 5": "1 Y 2 (n) = n 1 Y 3 (n) = n 1 Y 4 (n) = n"
} | {
"chunk_type": "body"
} |
##### � ℓ [∗] 2 [(][X] [i] [)][,]
i=1
n | {
"id": "1004.2316",
"title": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable\n Information Criterion in Singular Learning Theory",
"categories": [
"cs.LG"
]
} | {
"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": "4 Main Results",
"Header 4": "4.1 Functional Cumulants",
"Header 5": "� ℓ [∗] 2 [(][X] [i] [)][,]"
} | {
"chunk_type": "body"
} |
##### � ℓ [∗] 3 [(][X] [i] [)][,]
i=1
n | {
"id": "1004.2316",
"title": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable\n Information Criterion in Singular Learning Theory",
"categories": [
"cs.LG"
]
} | {
"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": "4 Main Results",
"Header 4": "4.1 Functional Cumulants",
"Header 5": "� ℓ [∗] 3 [(][X] [i] [)][,]"
} | {
"chunk_type": "body"
} |
##### ��ℓ [∗] 4 [(][X] [i] [)][ −] [3][ℓ] [∗] 2 [(][X] [i] [)] [2] [�] .
i=1 | {
"id": "1004.2316",
"title": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable\n Information Criterion in Singular Learning Theory",
"categories": [
"cs.LG"
]
} | {
"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": "4 Main Results",
"Header 4": "4.1 Functional Cumulants",
"Header 5": "��ℓ [∗] 4 [(][X] [i] [)][ −] [3][ℓ] [∗] 2 [(][X] [i] [)] [2] [�] ."
} | {
"chunk_type": "body"
} |
##### These formulas may be useful in practical applications. | {
"id": "1004.2316",
"title": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable\n Information Criterion in Singular Learning Theory",
"categories": [
"cs.LG"
]
} | {
"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": "4 Main Results",
"Header 4": "4.1 Functional Cumulants",
"Header 5": "These formulas may be useful in practical applications."
} | {
"chunk_type": "body"
} |
#### 4.2 Bayes Cross Validation and Widely Applicable Informa- tion Criterion
##### We show the asymptotic equivalence of the cross validation loss C v L(n) and the widely applicable information criterion WAIC(n). Theorem 1. For arbitrary 0 < β < ∞, the cross validation loss C v L(n) and the widely applicable informa... | {
"id": "1004.2316",
"title": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable\n Information Criterion in Singular Learning Theory",
"categories": [
"cs.LG"
]
} | {
"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": "4 Main Results",
"Header 4": "4.2 Bayes Cross Validation and Widely Applicable Informa- tion Criterion",
"Header 5": "We show the asymptotic ... | {
"chunk_type": "body"
} |
##### − 3β − 3β + 1 Y 3 (n) + O p ( [1] � 6 � n [2] [)][,] WAIC(n) = −Y 1 (n) + 2β − 1 Y 2 (n) � 2 � − [1] 6 [Y] [3] [(][n][) +][ O] [p] [( 1] n [2] [)][.] (Proof) Firstly, we study C v L(n). From the definition of E w [ ] and E w [(][i][)] [[] ], for arbitrary function f (w), E [(] w [i][)] [[][f] [(][w][)] =][ E] [w]... | {
"id": "1004.2316",
"title": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable\n Information Criterion in Singular Learning Theory",
"categories": [
"cs.LG"
]
} | {
"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": "4 Main Results",
"Header 4": "4.2 Bayes Cross Validation and Widely Applicable Informa- tion Criterion",
"Header 5": "− 3β − 3β + 1 Y 3 (n) +... | {
"chunk_type": "body"
} |
##### Therefore, by the definition of the cross validation loss, eq.(20), | {
"id": "1004.2316",
"title": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable\n Information Criterion in Singular Learning Theory",
"categories": [
"cs.LG"
]
} | {
"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": "4 Main Results",
"Header 4": "4.2 Bayes Cross Validation and Widely Applicable Informa- tion Criterion",
"Header 5": "Therefore, by the defin... | {
"chunk_type": "body"
} |
##### C v L(n) = − [1] n | {
"id": "1004.2316",
"title": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable\n Information Criterion in Singular Learning Theory",
"categories": [
"cs.LG"
]
} | {
"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": "4 Main Results",
"Header 4": "4.2 Bayes Cross Validation and Widely Applicable Informa- tion Criterion",
"Header 5": "C v L(n) = − [1] n"
} | {
"chunk_type": "body"
} |
##### � i=1n log [E] E [w] w [[] [ [ p] p [(] ( [X] X [i] i [|] | [w] w [)] ) [1][−][−][β][β] [ ]] ] [.] | {
"id": "1004.2316",
"title": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable\n Information Criterion in Singular Learning Theory",
"categories": [
"cs.LG"
]
} | {
"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": "4 Main Results",
"Header 4": "4.2 Bayes Cross Validation and Widely Applicable Informa- tion Criterion",
"Header 5": "� i=1n log [E] E [w] w ... | {
"chunk_type": "body"
} |
##### By using the generating function of functional cumulants F (α), C v L(n) = F (−β) − F (1 − β). Then by using Lemma 1 (2) for each k = 2, 3, 4, there exists a constant C k > 0 such that |F [(][k][)] (α)| ≤ C k M k (n) (|α| < 1 + β), where C 1 = 1, C 2 = 2, C 3 = 6, C 4 = 26. Therefore, |F [(][k][)] (α)| = O p ( [1... | {
"id": "1004.2316",
"title": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable\n Information Criterion in Singular Learning Theory",
"categories": [
"cs.LG"
]
} | {
"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": "4 Main Results",
"Header 4": "4.2 Bayes Cross Validation and Widely Applicable Informa- tion Criterion",
"Header 5": "By using the generating... | {
"chunk_type": "body"
} |
##### By using Taylor expansion of F (1), WAIC(n) = −F [′] (0) + [2][β][ −] [1] F [′′] (0) − [1] 2 6 [F] [ (3)] [(0) +][ O] [p] [( 1] n [2] [)][,] which completes the proof. (Q.E.D.) From the above theorem, we obtain the following corollary. Corollary 1. For arbitrary 0 < β < ∞, the cross validation loss C v L(n) and t... | {
"id": "1004.2316",
"title": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable\n Information Criterion in Singular Learning Theory",
"categories": [
"cs.LG"
]
} | {
"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": "4 Main Results",
"Header 4": "4.2 Bayes Cross Validation and Widely Applicable Informa- tion Criterion",
"Header 5": "By using Taylor expansi... | {
"chunk_type": "body"
} |
##### β − β C v L(n) − WAIC(n) [∼] = � 2 | {
"id": "1004.2316",
"title": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable\n Information Criterion in Singular Learning Theory",
"categories": [
"cs.LG"
]
} | {
"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": "4 Main Results",
"Header 4": "4.2 Bayes Cross Validation and Widely Applicable Informa- tion Criterion",
"Header 5": "β − β C v L(n) − WAIC(n... | {
"chunk_type": "body"
} |
##### Y 3 (n). � | {
"id": "1004.2316",
"title": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable\n Information Criterion in Singular Learning Theory",
"categories": [
"cs.LG"
]
} | {
"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": "4 Main Results",
"Header 4": "4.2 Bayes Cross Validation and Widely Applicable Informa- tion Criterion",
"Header 5": "Y 3 (n). �"
} | {
"chunk_type": "body"
} |
##### If β = 1, C v L(n) − WAIC(n) [∼] = [1] 12 [Y] [4] [(][n][)][.] | {
"id": "1004.2316",
"title": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable\n Information Criterion in Singular Learning Theory",
"categories": [
"cs.LG"
]
} | {
"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": "4 Main Results",
"Header 4": "4.2 Bayes Cross Validation and Widely Applicable Informa- tion Criterion",
"Header 5": "If β = 1, C v L(n) − WA... | {
"chunk_type": "body"
} |
#### 4.3 Generalization Error and Cross Validation Error
##### In the previous subsection, we have shown that the cross validation loss is asymp- totically equivalent to the widely applicable information criterion. In this section, let us compare the Bayes generalization error B g (n) defined by eq.(10) and the cross... | {
"id": "1004.2316",
"title": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable\n Information Criterion in Singular Learning Theory",
"categories": [
"cs.LG"
]
} | {
"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": "4 Main Results",
"Header 4": "4.3 Generalization Error and Cross Validation Error",
"Header 5": "In the previous subsection, we have shown th... | {
"chunk_type": "body"
} |
##### is a nonnegative function. It is well known that ζ(z) can be analytically continued to the unique meromorphic function on the entire complex plane C. All poles of ζ(z) are real, negative, and rational numbers. The largest pole is denoted by (−λ) = maximum pole of ζ(z). (36) Then λ is called the log canonical thre... | {
"id": "1004.2316",
"title": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable\n Information Criterion in Singular Learning Theory",
"categories": [
"cs.LG"
]
} | {
"Header 1": "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory",
"Header 2": null,
"Header 3": "4 Main Results",
"Header 4": "4.3 Generalization Error and Cross Validation Error",
"Header 5": "is a nonnegative function. It is well known ... | {
"chunk_type": "body"
} |
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