1-1KUISLLM
Maximum likelihood meets histograms
Let X1, X2, • • • , Xn be n i.i.d data points drawn from a piece-wise constant probability density function over N [equal size bins betweenr 0 and 1 (B1, B2, • • • , BN), where the constants are Oh 02, • • • , 0N. (0y < x < for j E {1,2,••• , N} P(x;0i, • • • ,ON) = 0 othenvise
We define pi for j E {1, 2, • • • , N} as pi := E7=1 1(Xi E By).
(a) Using the fact that the total area underneath a probability den-sity function is 1, express ON in terms of the other constants. (b) Write down the log-likelihood of the data in terms of °1, 02 , • • • , ON-1
and pl, /12, • • • , AN-1• (c) Find the maximum likelihood estimate of 0- for E 1 2, N
Sample Solution
