The solution sets of the two problems intersect at a single point which solves both problems. 4.1 Optimizing for Maximum Entropy Distributions The following optimization solves for a maximum entropy distribution that satis es linear constraints (typ- ically moment constraints): min f H(f) (4.1) s.t. for example 1/2 to both events (head, tail) in a coin tossing experiment or say 1/6 to each of the faces in a die tossing experiment. When nothing is known about the problem you can assign the same probability to all likely states resulting in a uniform probability distribution. The solution to the Ma圎nt problem therefore can be found to by solving the maximize likelihood estimation given the observations x1, …, xm:Īs it turns out, maximizing entropy is a dual problem of maximum likelihood. Maximum entropy is a guiding principle in assigning probabilities to events. optimizing the entropy among the subset of distributions which ful ll the constraints. Where μ1, …, μk are the parameters and Z is the partition function. With the aid of Lagrange multipliers, we can derive that the solution to the Ma圎nt problem is in the form of a Gibbs distribution: In the next section, we will discuss how to solve the maximum entropy problem. The maximum entropy method has been widely enhanced in fields of natural language processing, ecological analysis, and so forth, see Berger et. In practice, such expectation constraints can usually be obtained from an empirical distribution, For instance, the expectation over the empirical distribution in this example can be obtained by rolling the dice multiple times and computing the average. The maximum entropy principle is based on the Boltzmann-Shannon entropy in which the maximum entropy distribution is characterized by an exponential model, see Jaynes ( 1963) for philosophical arguments. Where the expected value of the obtained number is an expectation constraint of this optimization problem. The additional knowledge we know about the dice is that the expected number for each roll is:Īccording to the maximum entropy principle, we can estimate the probability of obtaining each number, p, by solving the maximum entropy problem: To give an example of such problems, let us assume we have a biased dice and we want to estimate the probability of obtaining number 1, 2, …, 6 each time we roll. and ( fj, bj) an (linear) expectation constraints on p. The Maximum Entropy (Ma圎nt) problem is formalized as follows: Unlike the wide-used Maximize Likelihood (ML) estimation, the maximum entropy estimation is less frequently seen in solving machine learning problems. Maximizing likelihood given parameterized constraints on the distributionĪre convex duals of each other.Maximizing entropy subject to expectation constraints.The Maximize Entropy and Maximize Likelihood duality states that the two problems:
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