2.43 ( ) The following distribution p(x|2, q) = q 2(22)1/q(1/q) exp |x|q 22 ...

2.43 () The following distribution p(x|σ2, q) = q 2(2σ2)1/qΓ(1/q)

exp



−

|x|q 2σ2



(2.293)

is a generalization of the univariate Gaussian distribution. Show that this distribution is normalized so that  ∞

−∞

p(x|σ2, q) dx = 1 (2.294)

and that it reduces to the Gaussian when q = 2. Consider a regression model in which the target variable is given by t = y(x,w) +  and  is a random noise variable drawn from the distribution (2.293). Show that the log likelihood function over w and σ2, for an observed data set of input vectors X = {x1, . . . , xN} and corresponding target variables t = (t1, . . . , tN)T, is given by ln p(t|X,w, σ2) = − 1 2σ2 N n=1 |y(xn,w) − tn|q − N q ln(2σ2) + const (2.295)
where ‘const’ denotes terms independent of both w and σ2. Note that, as a function of w, this is the Lq error function considered in Section 1.5.5.