Information Theory, Inference, and Learning ... - Inference Group

Copyright Cambridge University Press 2003. On-screen viewing permitted. Printing not permitted. http://www.cambridge.org/0521642981You can buy this book for 30 pounds or $50. See http://www.inference.phy.cam.ac.uk/mackay/itila/ for links.C.3: Perturbation theory 609(a)(a ′ )(b)Matrix⎡⎤.90 .20 0 0⎢ .10 .80 0 0⎥⎣ 0 0 .90 .20 ⎦0 0 .10 .80⎡⎤.90 .20 0 0⎢ .10 .79 .02 0⎥⎣ 0 .01 .88 .20 ⎦0 0 .10 .80⎡⎤0 0 .90 .20⎢ 0 0 .10 .80⎥⎣ .90 .20 0 0 ⎦.10 .80 0 0Eigenvalues and eigenvectors e L , e R⎡ ⎤1⎡ ⎤ ⎡ ⎤1⎡ ⎤ ⎡0.70⎤ ⎡ ⎤ ⎡0.70⎤ ⎡ ⎤0 0 .71 .89 .45 .71 0 0⎢ 0⎥ ⎢ 0⎥ ⎢.71⎥ ⎢.45⎥ ⎢−.89⎥ ⎢−.71⎥ ⎢ 0⎥ ⎢ 0⎥⎣.71⎦⎣.89⎦⎣ 0⎦⎣ 0⎦⎣ 0⎦⎣ 0⎦⎣−.45⎦⎣−.71⎦.71 .45 0 0 0 0 .89 .71⎡ ⎤1 ⎡ ⎤ ⎡ 0.98 ⎤ ⎡ ⎤ ⎡ 0.70 ⎤ ⎡ ⎤ ⎡ 0.69 ⎤ ⎡ ⎤.50 .87 −.18 −.66 .20 .63 −.19 −.61⎢.50⎥ ⎢.43⎥ ⎢−.15⎥ ⎢−.28⎥ ⎢−.40⎥ ⎢−.63⎥ ⎢ .41⎥ ⎢ .65⎥⎣.50⎦⎣.22⎦⎣ .66⎦⎣ .61⎦⎣−.40⎦⎣−.32⎦⎣−.44⎦⎣−.35⎦.50 .11 .72 .33 .80 .32 .77 .30⎡ ⎤1 ⎡ ⎤ ⎡ 0.70 ⎤ ⎡ ⎤ ⎡ −0.70 ⎤ ⎡ ⎤ ⎡ ⎤−1⎡ ⎤.50 .63 −.32 .50 .32 −.50 .50 .63⎢.50⎥ ⎢.32⎥ ⎢ .63⎥ ⎢−.50⎥ ⎢−.63⎥ ⎢ .50⎥ ⎢ .50⎥ ⎢ .32⎥⎣.50⎦⎣.63⎦⎣−.32⎦⎣ .50⎦⎣−.32⎦⎣ .50⎦⎣−.50⎦⎣−.63⎦.50 .32 .63 −.50 .63 −.50 −.50 −.32Table C.6. Illustrative transition probability matrices and their eigenvectors showing the two ways ofbeing non-ergodic. (a) More than one principal eigenvector with eigenvalue 1 because thestate space falls into two unconnected pieces. (a ′ ) A small perturbation breaks the degeneracyof the principal eigenvectors. (b) Under this chain, the density may oscillate betweentwo parts of the state space. In addition to the invariant distribution, there is anotherright-eigenvector with eigenvalue −1. In general such circulating densities correspond tocomplex eigenvalues with magnitude 1.We assume that we have an N × N matrix H that is a function H(ɛ) ofa real parameter ɛ, with ɛ = 0 being our starting point. We assume that aTaylor expansion of H(ɛ) is appropriate:H(ɛ) = H(0) + ɛV + · · ·(C.9)whereV ≡ ∂H∂ɛ .(C.10)We assume that for all ɛ of interest, H(ɛ) has a complete set of N righteigenvectorsand left-eigenvectors, and that these eigenvectors and their eigenvaluesare continuous functions of ɛ. This last assumption is not necessarily agood one: if H(0) has degenerate eigenvalues then it is possible for the eigenvectorsto be discontinuous in ɛ; in such cases, degenerate perturbation theoryis needed. That’s a fun topic, but let’s stick with the non-degenerate casehere.We write the eigenvectors and eigenvalues as follows:and we Taylor-expandH(ɛ)e (a)R(ɛ) = λ(a) (ɛ)e (a)R(ɛ), (C.11)λ (a) (ɛ) = λ (a) (0) + ɛµ (a) + · · ·(C.12)withande (a)Rµ (a) ≡ ∂λ(a) (ɛ)∂ɛ(C.13)(a)(ɛ) = e(a)R(0) + ɛfR + · · · (C.14)