Tập thể dục 22,14. [2] Bài tập này khám phá những ý tưởng đó tối đa hóa mật độ xác suất là một cách nghèo để tìm một điểm đó là đại diện của mật độ. √ Hãy xem xét một phân phối Gaussian trong một không gian k chiều, 2 2 P (w) = (1 / 2π σW) k exp (- k wi / 2σW). | Copyright Cambridge University Press 2003. On-screen viewing permitted. Printing not permitted. http 0521642981 You can buy this book for 30 pounds or 50. See http mackay itila for links. Further exercises Exercises where maximum likelihood and MAP have difficulties . Exercise . 2 This exercise explores the idea that maximizing a probability density is a poor way to find a point that is representative of the density. Consider a Gaussian distribution in a fc-dimensional space P w 1 p2 oW k exp 221 w2 2ơW . Show that nearly all of the probability mass of a Gaussian is in a thin shell of radius r VkaW and of thickness proportional to r ffik. For example in 1000 dimensions 90 of the mass of a Gaussian with aW 1 is in a shell of radius and thickness . However the probability density at the origin is e1 2 10217 times bigger than the density at this shell where most of the probability mass is. Now consider two Gaussian densities in 1000 dimensions that differ in radius aW by just 1 and that contain equal total probability mass. Show that the maximum probability density is greater at the centre of the Gaussian with smaller aW by a factor of exp 20 000. In ill-posed problems a typical posterior distribution is often a weighted superposition of Gaussians with varying means and standard deviations so the true posterior has a skew peak with the maximum of the probability density located near the mean of the Gaussian distribution that has the smallest standard deviation not the Gaussian with the greatest weight. . Exercise . 3 The seven scientists. N datapoints xn are drawn from N distributions all of which are Gaussian with a common mean p but with different unknown standard deviations an. What are the maximum likelihood parameters p f on given the data For example seven scientists A B C D E F G with wildly-differing experimental skills measure p. You expect some of them to do accurate work . to .
