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1 S c /(50)
2 c /(50)
3 S McCullock Pitts 1950 Rosenblatt ) c /(50)
4 10 2)3) u = (u 1,, u n ) z = (z 1,, z n ) W = (w ij ) τ du(t) dt = u + W f(u) + s (1 1) s (1 1) 1 2) Hopfield Amit Amari-Hopfield Wilson-Cowan Amari c /(50)
5 Hopfield Aihara Tuda 4) Rosenblatt 1967 Amari Rumelhart 2) Amari Fukumizu c /(50)
6 Bayes Fukumizu Watanbe 5) Boosting 6) 6) von der Malsburg Willshaw Amari Kohonen 7) 2 8) Barto Sutton c /(50)
7 ) D.E. Rumelhart and D.E. McClleland, Parallel Distributed Processing, vol.i, II, MIT Press, ),,, ) P. Dayan and L.F. Abbott, Theoretical Neuroscience: Computational and Mathematical Modeling of Neural Systems, MIT Press, ),,, ),,,, ),,,,, ) T. Kohonen, Self-Organizing Maps, Springer, ) A. Cichocki and S. Amari, Adaptive Blind Signal and Image Processing, Wiley, c /(50)
8 S x = (x 1,..., x N ) T R N y R NX y = f(u), u = w ix i = w x i=1 (1 2) w R N x u = w x f u y tanh S u = w x+θ u θ w = (w T, θ) T, x = (x T, 1) T w, x u = w x (1 2) θ 2 y = f(u), du dt = u + w x (1 3) N c /(50)
9 x = f(u), du dt = u + W x (1 4) x = (x 1,..., x N ) T, u = (u 1,..., u N ) T f u W = (w ij ) ij w ij j i W 0 t = 0, 1,... x t+1 = f(u t), u t = W x t (1 5) f W y ±1 P (y = 1) = f(u), P (y = 1) = 1 f(u); u = w x (1 6) N P (x i x) x f W W T = W (1 4) (1 5) 1) W f(u) = (1 + tanh βu)/2 (1 6) N H(x) = x T W x/2 c /(50)
10 P (x) exp[ βh(x)] (1 7) 2) W p { 1, 2,..., p i R N } W 3, 4, 5) 6) W W = 1 p px i ( i ) T (1 8) i=1 i 1/2 ±1 p p Amit 7) N p 0.14N q(x) (1 7) W D(q p) = P x q(x) log[q(x)/p(x)] 2) W dw dt = xx T q xx T p (1 9) p p z x = (x T, z T ) T p(x) = X z p( x), p( x) e βh( x), H( x) = 1 2 xt W x (1 10) p(z x) = p( x)/p(x) c /(50)
![14N 1--2--4 q(x) (1 7) W D(q p) = P x q(x) log[q(x)/p(x)] 2) W dw dt = xx T q xx T p](/docs-images/45/20566745/images/page_10.jpg "(1 9) p p z x = (x T, z T ) T p(x) = X z p( x), p( x) e βh( x), H( x) = 1 2 xt W x (1")
11 dw dt = x x T q(x)p(z x) x x T p( x) (1 11) 8) 9) 1) J.J. Hopfield, Neurons with graded response have collective computational properties like those of two-state neurons, Proc. Natl. Acad. Sci. USA, vol.81, pp , May ) D.H. Ackley, G.E. Hinton, and T.J. Sejnowski, A learning algorithm for Boltzmann machines, Cognitive Science, vol.9, pp , Jan.-March ) K. Nakano, Associatron a model of associative memory, IEEE Trans. Systems, Man, and Cybernetics, vol.smc-12, pp , July ) T. Kohonen, Correlation matrix memories, IEEE Trans. Computers, vol.c-21, pp , April ) J.A. Anderson, A simple neural network generating an interactive memory, Mathematical Biosciences, vol.14, pp , Aug ) J.J. Hopfield, Neural networks and physical systems with emergent collective computational abilities, Proc. Natl. Acad. Sci. USA, vol.79, pp , April ) D.J. Amit, H. Gutfreund, and H. Sompolinsky, Storing infinite numbers of patterns in a spin-glass model of neural networks, Physical Review Letters, vol.55, pp , Sep ) D. Saad and M. Opper (eds.), Advanced Mean Field Methods: Theory and Practice, MIT Press, Cambridge, ) G.E. Hinton and R.R. Salakhutdinov, Reducing the dimensionality of data with neural networks, Science, vol.313, pp , July c /(50)
12 S N " 1 X P (S w, ) = Z(w, ) exp w ij S i S j + X # θ i S i (1 12) i>j i S i = ±1, i = 1, 2,..., N, Z(w, ) = P S exp h P i>j wijsisj + P i θisi i S f(s) f(s) = P S f(s)p (S w, ) N f(s) = S i S i S i (1 12) Callen * 0 S i = 1+ X w ijs j + θ i A j i (1 13) w ij = w ji P j i wijsj + θi Si local field tanh ( ) 0 1 S i X w ij S j + θ i A j i (1 14) P j i wij Sj + θi mean field S i S j S i (i = 1, 2,..., N) c /(50)
13 mean field approximation cluster variation method 1) (1 12) F(Q) = X X Q(S) w ij S i S j + X! θ i S i + X Q(S) ln Q(S) (1 15) S i>j i S Q(S) (1 12) Q(S) (1 15) Q(S) = P (S w, ) ln Z(w, ) Q(S) (1 15) S i Q(S) = 2 N Q N i=1 (1 + Si Si) (1 15) (1 14) S (1 15) (1 15) S i S j Bethe S i S j S i b ij (S i, S j ) b i (S i ) b ij (S i, S j ) b i (S i ) belief (1 15) F Bethe ({b ij, b i }) = X X b ij(s i, S j) b ij (S i, S j ) ln exp [w ij S i S j + θ i S i ] (ij) S i,s j + X (1 c i ) X b i (S i ) ln bi(si) exp [θ i S i ] i S i (1 16) (ij) w ij = w ji c i S i F Bethe ({b ij, b i }) b ij (S i, S j ) b i (S i ) reducibility c /(50)
14 X S3 4 1 S j b ij (S i, S j ) = b i (S i ) (1 17) (1 17) (1 16) 0 1 F Bethe ({b ij, b i}) + X X X λ i j(s X b ij(s i, S j) b i(s i) A(1 18) i Sj S i j N (i) λ i j (S i ) (1 17) b ij(s i, S j) b i(s i) N (i) S i w ij h h j i k j i k l l (a) (b) (a): S i c i = 4 (b): (1 21) i j m i j j i m h i m k i m l i (1 18) e c 1 P i 1 k N (i) λ i k(s i )+λ i j (S i ) X Sj e w ij S i S j e θ j S j e λ j i(s j ) (1 19) S i = ±1 m i j e θ is i λ i j (S i ) (1 + m i j S i )/2 λ i j (S i ) c /(50)
15 (1 19) m i j 0 m i j = i + X 1 tanh 1 (tanh(w ik )m k i ) A (1 20) k N (i)\j 1 1(b) N (i)\j N (i) j S i (1 20) 0 S i X S i b i (S i ) = i + S i X 1 tanh 1 (tanh(w ij )m j i ) A (1 21) j N (i) (1 21) (1 21) probability propagation belief propagation 2) 3) w (1 12) w (1 12) replica method K-SAT SAT/UNSAT 4) (1 12) w θ P (w) = Q i>j P (w ij) P ( ) = Q i P (θ i) 1 N [ln Z(w, )] = 1 N Z Y i>j dw ij P (w ij ) Y i dθ i P (θ i ) ln Z(w, ) (1 22) (1 22) (1 12) ln Z(w, ) w (1 22) c /(50)
16 n = 1, 2,... " X X n X Z n (w, ) = exp w ijsi a Sj a + X!# θ isi a S 1,S 2,...,S n a=1 i>j i (1 23) w n = 1, 2,... [Z n (w, )] (1 22) n = 1, 2,... [Z n (w, )] n n 1 N [ln Z(w, )] = lim n 0 1 N n ln [Zn (w, )] (1 24) (1 22) (1 23) S 1, S 2,..., S n w n w 5) 1) R. Kikuchi, A theory of cooperative phenomena, Physical Review, vol.81, no.6, pp , March ) Y. Kabashima and D. Saad, Belief propagation vs. TAP for decoding corrupted messages, Europhysics Letters, vol.44, no.5, pp , Dec ) J.S. Yedidia, W.T. Freeman and Y. Weiss, Constructing Free Energy Approximations and Generalized Belief Propagation Algorithms, IEEE Trans. Information Theory, vo.l.51, no.7, pp , July ),,,, p.206, ) M. Talagrand, Spin Glasses: A Challenge for Mathematicians. Cavity and Mean Field Models, Springer-Verlag, Berlin, p.586, c /(50)
17 S perceptron Frank Rosenblatt ) layer elementary perceptron S sensory A association 1 R response 1 2 hidden layer multi-layer perceptron, MLP x y (simple perceptron) 1 x = (x 1,, x n ) y y = f( P i w ix i θ) w i x i connection weight θ threshold value f tanh linear perceptron f x i θ 1 0 P x i wixi θ = c /(50)
18 linearly separable 1 θ x (i) θ w w w x (i) t (i) 1 0 N (x (i), t (i) )(i = 1,..., N) w 1) x (i) y (i) 2) w w + η(y (i) t (i) )x (i) η y (i) t (i) 0 error-correcting leaning x (i) Minsky Papert 1/0 2) Rumelhart error back-propagation learning 3) f R(w) w c /(50)
19 f(x, w) w w η w R(w) ) information geometry natural gradient method 5) p(y x; w) q(x) x G(w) = R R ( w ln p)( w ln p) T p(y x; w)q(x)dydx w w ηg 1 (w) w l(x, y, w) l(x, y, w) = ln p(y x; w) ln q(x) p(y x; w) adaptive natural gradient 6) method 1) F. Rosenblatt, Principles of Neurodynamics, Perceptrons and the Theory of Brain Mechanisms, Spartan Books, Washington, ) M.A. Minsky and S.A. Papert, Perceptrons Expanded Edition, MIT Press, Cambridge, 1988.,,,, ) D.E. Rumelhart, J.L. McClelland, and the PDP Research Group, Parallel Distributed Processing, Explorations in the Microstructure of Cognition, Volume 1: Foundations, MIT Press, Cambridge, 1986., PDP,,, c /(50)
20 4) C.M. Bishop, Pattern Recognition and Machine Learning, Springer-Verlag, New York, 2006.,, ( ),,, ) S. Amari, Natural gradient works efficiently in learning, Neural Computation, vol.10, no.2, pp , Feb ) S. Amari, H. Park, and K. Fukumizu, Adaptive method of realizing natural gradient learning for multilayer perceptrons, Neural Computation, vol.12, no.6, pp , June c /(50)
21 S supervised learning x y n {(x i, y i )} n 1, i=1 2) x y generalization ability {(x i, y i)} n i=1 p(x, y) independent and identically distributed; i.i.d. y E p(y x) [y] y regression y classification linear model least-squares {ϕ j (x)} t j=1 f linear (x) = tx θ j ϕ j (x) j=1 {θ j } t j=1 min {θ j } t j=1 nx (y i f linear (x i )) 2 i=1 maximum likelihood estimation q Gauss(y x) = 1 exp (y f «linear(x)) 2 2πσ 2 2σ radial basis function; RBF RBF c /(50)
22 nx min {θ j,µ j,σ j } t j=1 i=1 tx f RBF(x) = j=1 (y i f RBF(x i)) 2 θ j exp 1 «2 (x µj) Σ 1 j (x µ j) RBF {θ j} t j=1 {µ j } t j=1 {Σ j } t j=1 RBF kernel model min {θ j } n j=1 nx (y i f kernel (x i)) 2 i=1 f kernel (x) = nx j=1 θ j exp (x «xj) (x x j) 2σ 2 logistic regression y = ±1 y p(y x) q logistic (y x) = exp ( yf linear (x)) f linear (x) {θ j} t j=1 min {θ j } t j=1 nx log (1 + exp ( y i f linear (x i ))) i=1 f kernel (x) KL p(x, y) q(y x)p(x) KL c /(50)
23 Z KL[p(x, y) q(y x)p(x)] = p(x, y) log p(x, y) q(y x)p(x) dxdy KL q(y x) p(y x) information criterion KL Akaike information criterion; AIC 3) AIC = 2 nx log q(y i x i ) + 2t i=1 t AIC KL AIC RBF AIC 4) 1),,,,,,,, ),,,,,,,, ) H. Akaike, A new look at the statistical model identification, IEEE Trans. Automatic Control, vol.ac-19, no.6, pp , Dec ) S. Watanabe, Algebraic analysis for nonidentifiable learning machines, Neural Computation, vol.13, no.4, pp , April c /(50)
24 S PAC 1). PAC probably approximately correct N ɛ 1 δ N ɛ, δ VC Vapnik-Chervonenkis 2 SVM x R m w R m b y = sgn h i w T x + b 2). (1 25) y {+1, 1} sgn N (x n, y n), n = 1,..., N min n y n (w T x n + b) w (1 26) SVM (1 26) w, b (1 26) w b y n (w T x n + b) = 1 y n (w T x n + b) 1 (1 27) (1 26) 1/ w w 2 /2 SVM c /(50)
25 min w,b 1 2 w 2 s.t. y n(w T x n + b) 1 (1 28) 2 3 (1 28) SVM SVM n α n 0 = (α 1,..., α N ) L(w, b, ) L(w, b, ) = 1 NX 2 w 2 α n [y n (w T x n + b) 1] n=1 (1 29) (1 28) (1 29) L(w, b, ) w b 0 NX NX w = α ny nx n, 0 = α ny n n=1 n=1 (1 30) (1 30) SVM w x n L(w, b, ) w b (1 28) NX max α n 1 NX 2 α n 0 n=1 NX s.t. α n y n = 0 n=1 n=1 n =1 NX α n α n y n y n x T n x n (1 31) 2 (1 31) ˆα n, n = 1,..., N SVM NX y = ˆα ny nx T n x + b n=1 (1 32) b ˆα n > 0 n (1 32) ˆα n > 0 N PAC SVM c /(50)
26 x f( ) f = f(x) f f f f( ) SVM SVM f n = f(x n ) NX max α n 1 NX 2 α n 0 n=1 NX s.t. α n y n = 0 n=1 n=1 n =1 NX α n α n y n y n f T n f n (1 33) NX y = ˆα n y n f T n f + b n=1 (1 34) 2 (1 33) (1 34) x K(x, x ) = f T (x)f(x ) K(, ) K(, ) x n, x n R m, c n, c n R X n,n K(x n, x n )c n c n 0 (1 35) f( ) x SVM 1 C-SVM SVM SVM c /(50)
27 ξ n 2). ξ n ξ n 1 min w,b,ξ n 2 w 2 + C NX n=1 ξ n s.t. y n (w T x n + b) 1 ξ n, ξ n 0 (1 36) max 0 α n C n=1 NX α n 1 NX 2 NX s.t. α ny n = 0 n=1 n=1 n =1 NX α nα n y ny n x T n x n (1 37) α n C (1 31) ν-svm C-SVM SVM 2 ν -SVM C-SVM C ν-svm 1 β β 1 min w,b,ξ n 2 w 2 + C NX ξ n β n=1 s.t. y n(w T x n + b) β, ξ n 0 (1 38) 3). max 0 α n C 1 2 s.t. NX n=1 n =1 NX α n y n = 0, n=1 NX α n α n y n y n x T n x n NX α n = 1 n=1 (1 39) C-SVM C ν-svm α n 4, ν 5). 3 SVM c /(50)
28 SVM (a) ɛ (b) E ɛ ( ) 2). SVM SVR a b 1 3 SVR 1 2 w 2 + C NX E ɛ (w T x n + b y n ) n=1 (1 40) max 0 α n,α n C 1 2 NX n=1 n =1 n=1 NX (α n α n)(α n α n )xnt x n NX NX ɛ (α n + α n) + (α n α n)y n (1 41) n=1 2 4 SVM SVM 2 SVM SVM 1 SVM 1 1 SVM 1 SVM 6). SVM ν-svm 1) L.G. Valiant, A Theory of the Learnable, Commun. ACM, vol.27, pp , Nov c /(50)
29 2) V.N. Vapnik, The Nature of Statistical Learning Theory, Springer-Verlag, New York, ) B. Schölkopf, et al., New Support Vector Algorithms, Neural Computation, vol.12, no.5, pp , May ) K.P. Bennett and E.J. Bredensteiner, Duality and Geometry in SVM Classifiers, Proceedings of the 17th International Conference on Machine Learning, pp.57 64, ) K. Ikeda and T. Aoishi, An Asymptotic Statistical Analysis of Support Vector Machines with Soft Margins, Neural Networks, vol.18, pp , April ) B. Schölkopf, et al., Estimating the Support of a High-Dimensional Distribution, Neural Computation, vol.13, no.7, pp , July c /(50)
30 S )2) 3)4)5) m σ 2 {(m, σ) R 2 ; σ > 0} (m, σ) W d W d 1 2 d w x p(x w) {p(x w); w} p p d p T p p T p P v = (v i) 2 g ij i,j gijvivj g ij p g ij 0 g ij 2 p q T p T q T p T q c /(50)
31 2 α- α- α α = 1 α = 1 α c /(50)
32 1 1 AIC BIC 0 2 AIC BIC AIC BIC d c /(50)
33 AIC d BIC d 1 1 1),,,, ) S. Amari, K. Nagaoka, Methods of Information Geometry, Oxford University Press, Oxford, ),,,, ) M. Drton, B, Sturmfels, S.Sullivant, Lectures on Algebraic Statistics, Birkhäuser, Basel, ) S. Watanabe, Algebraic Geometry and Statistical Learning Theory, Cambridge University Press, Cambridge, c /(50)
34 S SOM Self-Organizing Map: SOM Kohonen 1) SOM 2 SOM SOM hawk owl duck hen dove eagle dog fox lion cow horse zebra cat wolf tiger owl dog cow (SOM) 1 4 SOM 16 SOM 2 SOM U-matrix 1 4 c /(50)
35 SOM (SOM) SOM k w k 16 w k y k 1 5 SOM {x 1,..., x N } {w 1,..., w M } w k x i y k 1 5 SOM SOM SOM SOM Step Step 0 t = 0 Step 1 Winner Best Matching Unit x i c /(50)
36 k (i) k (i) = arg min k w k x i Step 2 k i h y k y k α k,i = (i) ; σ(t) P (1 42) N i =1 y h k y k (i ) ; σ(t) h(d; σ) d σ(t) t Step 3 IX w k (t) = (1 ε) w k (t 1) + ε α k,i x i i=1 (1 43) ε 0 < ε 1 ε = 1 t := t + 1 Step 1 1 SOM ε t ε 0 (1 42) SOM SOM SOM c /(50)
37 2) SOM SOM (Generative Topographic Map: GTM) 3) (Kernel-based Maximum Entropy Learning: kmer) 2) SOM SOM SOM SOM SOM SOM Adaptive Subspace SOM: ASSOM Self-Organizing Operator Map SOM SOM k-means SOM 4) SOM SOM SOM SOM 1) SOM SOM PAK SOM 1) T.,,,, ) M.,,,, ) C.M. Bishop, M.Svensén and C.K.I. Williams, GTM: The generative topographic mapping, Neural Computation, vol.10, no.1, pp , Jan ) T.M. Martinetz, S.G. Berkovich and K.J. Schulten, Neural-gas network for vector quantization and its application to time-series prediction, IEEE Trans. Neural Networks, vol. 4, no.4, pp , July c /(50)
38 S N n x 1,..., x N N n X = (x T 1,..., x T N ) T X N m U m n V X UV (1 44) m = n X U V m < n X X n x T i U i m u T i u i V x i m W UV = (UW )(W 1 V ) U = UW, V = W 1 V U V SVD: Singular Value Decomposition N n X,N n U, n Λ, n n V c /(50)
39 X = UΛV T, U T U = I, V T V = I, Λ = diag(λ 1,..., λ n ), λ 1... λ n 0 (1 45) X λ 1,..., λ n 0 X U, V m Ũ, Ṽ Λ = diag(λ 1,..., λ m) X = (Ũ Λ)Ṽ T (1 46) N m Ũ Λ m n X = ( X ij ) X = (X ij ) X 2 PCA: Principal Component Analysis x i m = ( P N i=1 xi)/n X Ũ Λ Ṽ T Ũ Λ = XṼ (1 47) x T Ṽ Ṽ V x λ i, i = 1, 2,..., m X n m Ṽ x Ṽ x c /(50)
40 1) W t+1 = W t + γxx T W t, Ṽ t = W t(w T t W t) 1/2 (1 48) W t n m W 0 t = 0, 1, 2,... 2 Ṽ t W t E[tr(W T xx T W )] W E[ ] (1 48) 1 xx T W t 2) 1 m ) m P m i=1 λ2 i / P n j=1 λ2 j m AIC, MDL, m n x m u n m Ṽ x = Ṽ u + n (1 49) u v 4) Ṽ Newton EM 1 Factor Analysis u n 5) c /(50)
41 (1 46) Ũ ΛṼ Ũ u EM Ṽ W W u (1 49) x = (Ṽ W T )(W u) + n (1 50) 2 ICA: Independent Component Analysis u 1, u 2,..., u m p(u 1, u 2,..., u m ) = p(u 1 ) p(u 2 ) p(u m ) m W p(u 1, u 2,..., u m ) p(u 1 )p(u 2 ) p(u m ) Kullback-Leibler W t+1 = W t + γ(i E[v(u)u T log p(ui) ])W, v(u) = u i «T i=1,...,m (1 51) p(u i) u i E[ ] W t W 1 Lie c /(50)
42 6) 4 κ 4 = E[u 4 i ] 3E[u 2 i ] (1 51) X UV U V NMF: Nonnegative Matrix Factorization 7) 2 x n s(x) n n 8) x, x k(x, x ) = s(x) s(x ) N x 1,..., x N K = (k(x i, x j )) i,j=1,...,n s(x) n 1) E. Oja, Principal Components, Minor Components, and Linear Neural Networks, Neural Networks, vol.5, pp , November-December ),,, vol.43, no.11, pp , Nov c /(50)
43 3), 2 3,, vol.49, no.1, pp , ) S. Akaho, The e-pca and m-pca: dimension reduction of parameters by information geometry, Proceedings IEEE International Joint Conference on Neural Networks, vol.1, pp , July ) C. Bishop, Pattern Recognition and Machine Learning, Springer-Verlag, New York, 2006,,,, 2007, ) Y. Nishimori and S. Akaho, Learning Algorithms Utilizing Quasi-Geodesic Flows on the Stiefel Manifold, Neurocomputing, vol.67, pp , August ) D.D. Lee and H.S. Seung, Algorithms for Non-negative Matrix Factorization, Advances in Neural Information Processing Systems, vol.13, pp , MIT Press, Cambridge, ),,,, ),,,, c /(50)
44 S ensemble over-training, over-fitting clustering hard clustering soft clustering k- k-means clustering method LVQ learning vector quantization SOM self-organizing map MoE Mixture of Experts c /(50)
45 CART classification and regression tree bagging boosting MoE 1, 2) Mixture of Experts MoE Mixture of Experts 3) 4) EM EM algorithm D = {(x i, y i); i = 1,..., n} x y MoE θ {θ k } x k y P (y x; θ k ) y GLM generalized linear model 1 ξ x k P (k x; ξ) MoE x y P (y x; {θ k }, ξ) = X k P (k x; ξ)p (y x; θ k ) D {θ k }, ξ EM P (k x, y) = θ (t+1) k = arg max θ k P (k x; ξ(t) )P (y x; θ (t) k ) Pk P (k x; ξ(t) )P (y x; θ (t) k ) X (x,y) D P (k x, y) log P (y x; θ k ) c /(50)
46 ξ (t+1) = arg max ξ X (x,y) D S3 4 1 P (k x, y) log P (k x; ξ) (t) MoE MoE boosting 5) AdaBoost 6) 2 AdaBoost 2 X x 2 y {+1, 1} D{(x i, y i ); i = 1,..., n} x ±1 h(x) (x i, y i ) D i AdaBoost D (1) = 1/n T i h (x i, y i ) F(h) h (t) ɛ (t) (h) = X i F(h) D (t) i h (t) ɛ (t) = ɛ (t) (h (t) ) α (t) = 1 «1 ɛ (t) 2 ln ɛ (t) D (t+1) i = D(t) i exp( α (t) y ih (t) (x i)) Z Z P D i = 1 H(x) = sign! TX α (t) h (t) (x) t=1 h(x) y h(x) y c /(50)
47 decision stump random guess boost 1),,,,,, ) C.M. Bishop, Pattern Recognition and Machine Learning, Springer, Berlin, ) R.A. Jacobs, M.I. Jordan, S.J. Norlan and G.E. Hinton, Adaptive Mixtures of Local Experts, Neural Computation, vol.3, no.1, pp.79 87, Spring ) M.I. Jordan and R.A. Jacobs, Hierarchical Mixtures of Experts and the EM Algorithm, Neural Computation, vol.6, no.2, pp , March ) R.E. Schapire, The strength of weak learnability, Machine Learning, vol.5, pp , June ) Y. Freund and R.E. Schapire, A decision-theoretic generalization of on-line learning and an application to boosting, J. Comput. Syst. Sci., vol.55, no.1, pp , August c /(50)
48 S θ p(θ) D = {x 1,.., x N } p(θ D) p(θ D) p(d θ)p(θ) (1 52) p(d θ) D x Z p(x D) = p(x θ)p(θ D)dθ (1 53) K p(x θ (1) ),..., p(x θ (K) ) p(θ (j) α),j = 1,..., K α p(α) θ = (θ (1),..., θ (K) ) D j j KY p(θ, α D) p(d j θ (j) )p(θ (j) α)p(α) j=1 (1 54) K K Dirichlet Process:DP 1) DP distribution over distributions K A 1,..., A K P (A 1),..., P (A K) K G = (P (A 1),..., P (A K)) G Dirichlet(G; γg 0 (A 1 ),..., γg 0 (A K )) (1 55) G DP G 0 γ G DP(γ, G 0 ) G K (P (A 1 )+ +P (A K ) = 1) P (A 1),..., P (A K) {γg 0(A k )} K k=1 c /(50)
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