Distribution functions multinomial

Some probability distribution functions occur frequently in nature, and have simple mathematical expressions. Two of the most useful ones are the binomial and multinomial distribution functions. These will be the basis for our development of the concept of entropy in Chapter 2. The binomial distribution describes processes in which each independent elementary event has two mutually exclusive outcomes such as heads/tails, yes/no, up/down, or occu-pied/vacant. Independent trials with two such possible outcomes are called Bernoulli trials. Let s label the two possible outcomes 9 and J. Let the probability of be p. Then the probability of J is 1 - p. We choose composite events that are pairs of Bernoulli trials. The probability of followed by is P0j = p(l - p). The probabilities of the four possible composite events are... 

Any data set that consists of discrete classification into outcomes or descriptors is treated with a binomial (two outcomes) or multinomial (tliree or more outcomes) likelihood function. For example, if we have y successes from n experiments, e.g., y heads from n tosses of a coin or y green balls from a barrel filled with red and green balls in unknown proportions, the likelihood function is a binomial distribution ... 

Note that it is often convenient to maximize the log-likelihood function, a monotone function of the likelihood whose maximum will correspond to the maximum of the original likelihood function. If our variables are discrete, we instead use the multinomial distribution whose probability mass function is as follows ... 

Equation (5.49) can be solved, and it is possible to show that F(z, t) is the generating function of the multinomial distribution ... 

Group counts have multinomial distributions for these simple systems and can be easily related to conversions of functional groups (which are the probabilities of reaction). 

The Likelihood function is a multinomial distribution, which reads ... 

The second used objective function is the Multinomial Maximum Log-Likelihood function (MML). This function is derived in (Hanson, Westman, Zhu 2002) based on the assumption that the simulation distribution of a bin frequency is a multinomial distribution. The optimization function can be written as the following minimization objective ... 

Based on the assumption that Z has a multinomial distribution, the likelihood function is written as... 

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