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.4.8: Solutions 89Solution to exercise 4.20 (p.86).NoteThe function f(x) has inverse functiong(y) = y 1/y . (4.54)log g(y) = 1/y log y. (4.55)I obtained a tentative graph of f(x) by plotting g(y) with y along the verticalaxis and g(y) along the horizontal axis. The resulting graph suggests thatf(x) is single valued for x ∈ (0, 1), and looks surprisingly well-behaved andordinary; for x ∈ (1, e 1/e ), f(x) is two-valued. f( √ 2) is equal both to 2 and4. For x > e 1/e (which is about 1.44), f(x) is infinite. However, it might beargued that this approach to sketching f(x) is only partly valid, if we define fas the limit of the sequence of functions x, x x , x xx , . . .; this sequence does nothave a limit for 0 ≤ x ≤ (1/e) e ≃ 0.07 on account of a pitchfork bifurcationat x = (1/e) e ; and for x ∈ (1, e 1/e ), the sequence’s limit is single-valued – thelower of the two values sketched in the figure.504030201000 0.2 0.4 0.6 0.8 1 1.2 1.45432100 0.2 0.4 0.6 0.8 1 1.2 1.40.50.40.30.20.100 0.2xxxx···Figure 4.15. f(x) = x , shownat three different scales.