Mathematical Methods for Physics and Engineering : A Comprehensive Guide

31.5 MAXIMUM-LIKELIHOOD METHOD

In an experiment,Nindependent measurementsxiof some quantity are made. Suppose

that the random measurement error on theith sample value is Gaussian distributed with

mean zero and known standard deviationσi. Calculate the ML estimate of the true value

μof the quantity being measured.

As the measurements are independent, the likelihood factorises:

L(x;μ,{σk})=

∏N

i=1

P(xi|μ, σi),

where{σk}denotes collectively the set of known standard deviationsσ 1 ,σ 2 ,...,σN.The
individual distributions are given by

P(xi|μ, σi)=

1

√

2 πσ^2 i

exp

[

−

(xi−μ)^2

2 σ^2 i

]

.

and so the full log-likelihood function is given by

lnL(x;μ,{σk})=−

1

2

∑N

i=1

[

ln(2πσ^2 i)+

(xi−μ)^2

σ^2 i

]

.

Differentiating this expression with respect toμand setting the result equal to zero, we
find

∂lnL

∂μ

=

∑N

i=1

xi−μ

σ^2 i

=0,

from which we obtain the ML estimator

ˆμ=

∑N

i=1(xi/σ

2

∑ i)

N

i=1(1/σ

2

i)

. (31.72)

This estimator is commonly used when averaging data with differentstatistical weights
wi=1/σ^2 i. We note that when all the variancesσ^2 i have the same value the estimator
reduces to the sample mean of the dataxi.

There is, in fact, no requirement in the ML method that the sample values

be independent. As an illustration, we shall generalise the above example to a

case in which the measurementsxiare not all independent. This would occur, for

example, if these measurements were based at least in part on the same data.

In an experimentNmeasurementsxiof some quantity are made. Suppose that the random

measurement errors on the samples are drawn from a joint Gaussian distribution with mean

zero and known covariance matrixV. Calculate the ML estimate of the true valueμof the

quantity being measured.

From (30.148), the likelihood in this case is given by

L(x;μ,V)=

1

(2π)N/^2 |V|^1 /^2

exp

[

−^12 (x−μ 1 )TV−^1 (x−μ 1 )

]

,

wherexis the column matrix with componentsx 1 ,x 2 ,...,xNand 1 is the column matrix
with all components equal to unity. Thus, the log-likelihood function is given by

lnL(x;μ,V)=−^12

[

Nln(2π)+ln|V|+(x−μ 1 )TV−^1 (x−μ 1 )

]

.