Pattern Recognition and Machine Learning

2.3. The Gaussian Distribution 109

I 0 (m)

m

0 5 10

0

1000

2000

3000

A(m)

m

0 5 10

0

0.5

1

Figure 2.20 Plot of the Bessel functionI 0 (m)defined by (2.180), together with the functionA(m)defined by
(2.186).

Setting the derivative with respect toθ 0 equal to zero gives

∑N

n=1

sin(θn−θ 0 )=0. (2.182)

To solve forθ 0 , we make use of the trigonometric identity

sin(A−B)=cosBsinA−cosAsinB (2.183)

Exercise 2.53 from which we obtain

θML 0 = tan−^1

{∑

∑nsinθn

ncosθn

}

(2.184)

which we recognize as the result (2.169) obtained earlier for the mean of the obser-

vations viewed in a two-dimensional Cartesian space.

Similarly, maximizing (2.181) with respect tom, and making use ofI′ 0 (m)=

I 1 (m)(Abramowitz and Stegun, 1965), we have

A(m)=

1

N

∑N

n=1

cos(θn−θML 0 ) (2.185)

where we have substituted for the maximum likelihood solution forθ 0 ML(recalling

that we are performing a joint optimization overθandm), and we have defined

A(m)=

I 1 (m)

I 0 (m)

. (2.186)

The functionA(m)is plotted in Figure 2.20. Making use of the trigonometric iden-

tity (2.178), we can write (2.185) in the form

A(mML)=

(

1

N

∑N

n=1

cosθn

)

cosθML 0 −

(

1

N

∑N

n=1

sinθn

)

sinθML 0. (2.187)