![]() Let X be the D × N matrix of training data, as before, so that xj is a column vector formed by the j th column of X. We now give details on the model in order to understand and implement the objective function being optimized. Statistical rethinking: A Bayesian course with examples in R and Stan. As its name suggests, the maximum entropy model is intimately related to probability theory. In this post, we’ve covered the definition of maximum entropy distributions, and we reviewed two examples: the discrete uniform distribution and the Gaussian. Formally, entropy is defined as follows: If X X is a discrete random variable with distribution P (X xi) pi P ( X x i) p i, then the entropy of X X is H (X) ipilogpi. As its name suggests, the maximum entropy model is intimately related to probability theory. Thus, the Gaussian has maximum entropy within the family of distributions with given first and second moments. When the goal is to find a distribution that is as ignorant as possible, then, consequently, entropy should be maximal. So the cross entropy is equal to the entropy of $p$, and we’re done! In conclusion, we’ve shown that if $p(x)$ is Gaussian with mean $\mu$ and variance $\sigma^2$, then $H(p) \geq H(q)$ for any distribution $q$ with mean $\mu$ and variance $\sigma^2$. Recall the definition of entropy for $n$ bits, where bit $i$ has probability $p_i$ of being $1$: For this post, we’ll focus on the simple definition of maximum entropy distributions. Principle of Maximum Entropy In Chapter 9, we discussed the technique of estimating input probabilities of a process that is consistent with known constraints expressed in terms of averages, or expected values, of one or more quantities, but is otherwise as unbiased as possible. The principle of maximum entropy has roots across information theory, statistical mechanics, Bayesian probability, and philosophy. 8.2 Principle of maximum entropy Entropy underlies a core theory for selecting probability distributions. The principle of maximum entropy states that the probability distribution which best represents the current state of knowledge about a system is the one. Maximum entropy distributions are those that are the “least informative” (i.e., have the greatest entropy) among a class of distributions with certain constraints. ![]()
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