Nothing in nature is unbeautiful. Alfred, Lord Tennyson
Idea Transcript
H2: Problems for Bayesian Decision Theory (1) Let
⎡ ⎡
⎤
⎢
3 0 0
⎢ 1 ⎢ ⎢ ⎥ u = ⎣ −1 ⎦ , L = ⎢ ⎢ 0 2 0 ⎢ 2 ⎣ 1 0 1
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
and X ∼ N(0, I). Define Y = LX + u, then Y has a multivariate normal distribution. Find the mean vector and covariance matrix of the random vector Y . (2) Let ωmax (x) be the state of nature for which P (ωmax |x) ≥ P (ωj |x), j = 1, 2, . . . , C. (a) Show that P (ωmax |x) ≥ 1/C.
(b) Show that the Bayes error rate is given by P (error) = 1 − P (ωmax |x)p(x)dx. (c) Show that P (error) ≤ [1 − C1 ]. (d) Describe conditions for which P (error) = [1 − C1 ]. (3) Consider the class conditional densities P (x|ωi ) = assume equal priors.
1 1 , πb 1+[(x−ai )/b]2
for i = 1, 2, and
(a) Show that the minimum probability of error is given by P (error) =
1 2
−a1 − π1 tan−1 | a22b |.
−a1 (b) Plot this as a function of | a22b |.
(c) What is the maximum value of P (error) and under which conditions can this occur? (4) The Posisson distribution for a discrete r.v. X and a parameter λ is P (x; λ) = λx e−λ /x!, x = 0, 1, 2, . . . (a) Show that E[X] = λ and V ar(X) = λ. (b) Plot P (x; 8) and P (x; 4) and Show that the mode is λ. (c) Consider two equally probable categories having Poisson distributions with parameters λ1 = 8 and λ2 = 4. What is the Bayes classification decision? (d) What is the Bayes error rate?
(5) Let X|ω1 ∼ N(u1 , C) and X|ω2 ∼ N(u2 , C), where
u1 =
0 0
, u2 =
1 1
, C=
1 0 0 0.25
Assume the equal priors. Find an optimal decision rule and its error rate.