The likelihood is the product of the density evaluated at the observations. Usually, the density takes values that are smaller than one, so its logarithm will be negative. ... This density will concentrate a large area around zero, and therefore will take large values around this point.

Why likelihood is not a probability?

Likelihood is a strange concept, in that it is not a probability, but it is proportional to a probability. The likelihood of a hypothesis (H) given some data (D) is proportional to the probability of obtaining D given that H is true, multiplied by an arbitrary positive constant K. In other words, L(H) = K × P(D|H).

Can a maximum likelihood estimate be negative?

As maximum likelihood estimates cannot be negative, they will be found at the boundary of the parameter space (ie, it is 0). ... Maximizing ℓ over the parameters π can be done using an EM algorithm, or by maximizing the likelihood directly (compare Van den Hout and van der Heijden, 2002).