Properties of the Likelihood Let pe denote the pmf of Pa. Let X1,., Xn is unknown. Then the likelihood is the function
Ln : E x e-R n (x1,.,,)1[Pe (x). i-1
For our purposes, we think of x1,. ., xn as observations of the random variables X1,. , X.
Which of the following are properties of the likelihood Ln?
A. The likelihood does not change with the parameter 0.
B. The likelihood can be updated sequentially as new samples are observed. For example, L3 (r1, 2, T3,0) L1 (3,0) L2 (r1, 2,0).
C. The likelihood is symmetric: it doesn't matter the order in which we plug in the observations. For example, L4 (x1,2, 3, 4, 0) L4 (x2, 3, , X4,0), and this is true for any rearrangement of 1, 2, 3, x4.
D. If we eliminate a single observation, then the likelihood remains unchanged. For example, L3 (x1, x2, 3, 0) L2 (x1,2,0).