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Pattern Recognition and Machine Learning: Section 3.3
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Yusuke Oda
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『パターン認識と機械学習』の輪講で用いた資料。
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Pattern Recognition and Machine Learning: Section 3.3
1.
Reading Pattern Recognition and Machine
Learning §3.3 (Bayesian Linear Regression) Christopher M. Bishop Introduced by: Yusuke Oda (NAIST) @odashi_t 2013/6/5 2013 © Yusuke Oda AHC-Lab, IS, NAIST 1
2.
Agenda 3.3 Bayesian
Linear Regression ベイズ線形回帰 – 3.3.1 Parameter distribution パラメータの分布 – 3.3.2 Predictive distribution 予測分布 – 3.3.3 Equivalent kernel 等価カーネル 2013/6/5 2013 © Yusuke Oda AHC-Lab, IS, NAIST 2
3.
Agenda 3.3 Bayesian
Linear Regression ベイズ線形回帰 – 3.3.1 Parameter distribution パラメータの分布 – 3.3.2 Predictive distribution 予測分布 – 3.3.3 Equivalent kernel 等価カーネル 2013/6/5 2013 © Yusuke Oda AHC-Lab, IS, NAIST 3
4.
Bayesian Linear Regression
Maximum Likelihood (ML) – The number of basis functions (≃ model complexity) depends on the size of the data set. – Adds the regularization term to control model complexity. – How should we determine the coefficient of regularization term? 2013/6/5 2013 © Yusuke Oda AHC-Lab, IS, NAIST 4
5.
Bayesian Linear Regression
Maximum Likelihood (ML) – Using ML to determine the coefficient of regularization term ... Bad selection • This always leads to excessively complex models (= over-fitting) – Using independent hold-out data to determine model complexity (See §1.3) ... Computationally expensive ... Wasteful of valuable data 2013/6/5 2013 © Yusuke Oda AHC-Lab, IS, NAIST 5 In the case of previous slide, λ always becomes 0 when using ML to determine λ.
6.
Bayesian Linear Regression
Bayesian treatment of linear regression – Avoids the over-fitting problem of ML. – Leads to automatic methods of determining model complexity using the training data alone. What we do? – Introduces the prior distribution and likelihood . • Assumes the model parameter as proberbility function. – Calculates the posterior distribution using the Bayes' theorem: 2013/6/5 2013 © Yusuke Oda AHC-Lab, IS, NAIST 6
7.
Agenda 3.3 Bayesian
Linear Regression ベイズ線形回帰 – 3.3.1 Parameter distribution パラメータの分布 – 3.3.2 Predictive distribution 予測分布 – 3.3.3 Equivalent kernel 等価カーネル 2013/6/5 2013 © Yusuke Oda AHC-Lab, IS, NAIST 7
8.
Note: Marginal /
Conditional Gaussians Marginal Gaussian distribution for Conditional Gaussian distribution for given Marginal distribution of Conditional distribution of given 2013/6/5 2013 © Yusuke Oda AHC-Lab, IS, NAIST 8 Given: Then: where
9.
Parameter Distribution Remember
the likelihood function given by §3.1.1: – This is the exponential of quadratic function of The corresponding conjugate prior is given by a Gaussian distribution: 2013/6/5 2013 © Yusuke Oda AHC-Lab, IS, NAIST 9 known parameter
10.
Parameter Distribution Now
given: Then the posterior distribution is shown by using (2.116): where 2013/6/5 2013 © Yusuke Oda AHC-Lab, IS, NAIST 10
11.
Online Learning- Parameter
Distribution If data points arrive sequentially, the design matrix has only 1 row: Assuming that are the n-th input data then we can obtain the formula for online learning: where In addition, 2013/6/5 2013 © Yusuke Oda AHC-Lab, IS, NAIST 11
12.
Easy Gaussian Prior-
Parameter Distribution If the prior distribution is a zero-mean isotropic Gaussian governed by a single precision parameter : The corresponding posterior distribution is also given: where 2013/6/5 2013 © Yusuke Oda AHC-Lab, IS, NAIST 12
13.
Relationship with MSSE-
Parameter Distribution The log of the posterior distribution is given: If prior distribution is given by (3.52), this result is shown: – Maximization of (3.55) with respect to – Minimization of the sum-of-squares error (MSSE) function with the addition of a quadratic regularization term 2013/6/5 2013 © Yusuke Oda AHC-Lab, IS, NAIST 13 Equivalent
14.
Example- Parameter Distribution
Straight-line fitting – Model function: – True function: – Error: – Goal: To recover the values of from such data – Prior distribution: 2013/6/5 2013 © Yusuke Oda AHC-Lab, IS, NAIST 14
15.
Generalized Gaussian Prior-
Parameter Distribution We can generalize the Gaussian prior about exponent. In which corresponds to the Gaussian and only in the case is the prior conjugate to the (3.10). 2013/6/5 2013 © Yusuke Oda AHC-Lab, IS, NAIST 15
16.
Agenda 3.3 Bayesian
Linear Regression ベイズ線形回帰 – 3.3.1 Parameter distribution パラメータの分布 – 3.3.2 Predictive distribution 予測分布 – 3.3.3 Equivalent kernel 等価カーネル 2013/6/5 2013 © Yusuke Oda AHC-Lab, IS, NAIST 16
17.
Predictive Distribution 2013/6/5 2013
© Yusuke Oda AHC-Lab, IS, NAIST 17 Let's consider that making predictions of directly for new values of . In order to obtain it, we need to evaluate the predictive distribution: This formula is tipically written: Marginalization arround (summing out )
18.
Predictive Distribution The
conditional distribution of the target variable is given: And the posterior weight distribution is given: Accordingly, the result of (3.57) is shown by using (2.115): where 2013/6/5 2013 © Yusuke Oda AHC-Lab, IS, NAIST 18
19.
Predictive Distribution Now
we discuss the variance of predictive distribution: – As additional data points are observed, the posterior distribution becomes narrower: – 2nd term of the(3.59) goes zero in the limit : 2013/6/5 2013 © Yusuke Oda AHC-Lab, IS, NAIST 19 Addictive noise goverened by the parameter . This term depends on the mapping vector . of each data point .
20.
Predictive Distribution 2013/6/5 2013
© Yusuke Oda AHC-Lab, IS, NAIST 20
21.
Example- Predictive Distribution
Gaussian regression with sine curve – Basis functions: 9 Gaussian curves 2013/6/5 2013 © Yusuke Oda AHC-Lab, IS, NAIST 21 Mean of predictive distribution Standard deviation of predictive distribution
22.
Example- Predictive Distribution
Gaussian regression with sine curve 2013/6/5 2013 © Yusuke Oda AHC-Lab, IS, NAIST 22
23.
Example- Predictive Distribution
Gaussian regression with sine curve 2013/6/5 2013 © Yusuke Oda AHC-Lab, IS, NAIST 23
24.
Problem of Localized
Basis- Predictive Distribution Polynominal regression Gaussian regression 2013/6/5 2013 © Yusuke Oda AHC-Lab, IS, NAIST 24 Which is better?
25.
Problem of Localized
Basis- Predictive Distribution If we used localized basis function such as Gaussians, then in regions away from the basis function centers the contribution from the 2nd term in the (3.59) will goes zero. Accordingly, the predictive variance becomes only the noise contribution . But it is not good result. 2013/6/5 2013 © Yusuke Oda AHC-Lab, IS, NAIST 25 Large contribution Small contribution
26.
Problem of Localized
Basis- Predictive Distribution This problem (arising from choosing localized basis function) can be avoided by adopting an alternative Bayesian approach to regression known as a Gaussian process. – See §6.4. 2013/6/5 2013 © Yusuke Oda AHC-Lab, IS, NAIST 26
27.
Case of Unknown
Precision- Predictive Distribution If both and are treated as unknown then we can introduce a conjugate prior distribution and corresponding posterior distribution as Gaussian-gamma distribution: And then the predictive distribution is given: 2013/6/5 2013 © Yusuke Oda AHC-Lab, IS, NAIST 27
28.
Agenda 3.3 Bayesian
Linear Regression ベイズ線形回帰 – 3.3.1 Parameter distribution パラメータの分布 – 3.3.2 Predictive distribution 予測分布 – 3.3.3 Equivalent kernel 等価カーネル 2013/6/5 2013 © Yusuke Oda AHC-Lab, IS, NAIST 28
29.
Equivalent Kernel 2013/6/5 2013
© Yusuke Oda AHC-Lab, IS, NAIST 29 If we substitute the posterior mean solution (3.53) into the expression (3.3), the predictive mean can be written: This formula can assume the linear combination of :
30.
Equivalent Kernel Where
the coefficients of each are given: This function is calld smoother matrix or equivalent kernel. Regression functions which make predictions by taking linear combinations of the training set target values are known as linear smoothers. We also predict for new input vector using equivalent kernel, instead of calculating parameters of basis functions. 2013/6/5 2013 © Yusuke Oda AHC-Lab, IS, NAIST 30
31.
Example 1- Equivalent
Kernel Equivalent kernel with Gaussian regression Equivalen kernel depends on the set of basis function and the data set. 2013/6/5 2013 © Yusuke Oda AHC-Lab, IS, NAIST 31
32.
Equivalent Kernel Equivalent
kernel means the contribution of each data point for predictive mean. The covariance between and can be shown by equivalent kernel: 2013/6/5 2013 © Yusuke Oda AHC-Lab, IS, NAIST 32 Large contribution Small contribution
33.
Properties of Equivalent
Kernel- Equivalent Kernel Equivalent kernel have localization property even if any basis functions are not localized. Sum of equivalent kernel equals 1 for all : 2013/6/5 2013 © Yusuke Oda AHC-Lab, IS, NAIST 33 Polynominal Sigmoid
34.
Example 2- Equivalent
Kernel Equivalent kernel with polynominal regression – Moving parameter: 2013/6/5 2013 © Yusuke Oda AHC-Lab, IS, NAIST 34
35.
Example 2- Equivalent
Kernel Equivalent kernel with polynominal regression – Moving parameter: 2013/6/5 2013 © Yusuke Oda AHC-Lab, IS, NAIST 35
36.
Example 2- Equivalent
Kernel Equivalent kernel with polynominal regression – Moving parameter: 2013/6/5 2013 © Yusuke Oda AHC-Lab, IS, NAIST 36
37.
Properties of Equivalent
Kernel- Equivalent Kernel Equivalent kernel satisfies an important property shared by kernel functions in general: – Kernel function can be expressed in the form of an inner product with respect to a vector of nonlinear functions: – In the case of equivalent kernel, is given below: 2013/6/5 2013 © Yusuke Oda AHC-Lab, IS, NAIST 37
38.
Thank you! 2013/6/5 2013
© Yusuke Oda AHC-Lab, IS, NAIST 38 zzz...
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