KL divergence and maximum likelihood

Last modified: 2026-11-12 01:08:55 (PST)

The Kullback–Leibler divergence of one model from another is the number of bits per value that a code built for the model wastes, by the cross-entropy decomposition. This page applies that to data, and shows that the model a maximum likelihood estimate picks is the model with the least waste.

1 Maximum likelihood

Theorem 1 (Maximum likelihood minimizes the divergence from the data) Let \(x_1, \ldots, x_n\) be data, and let \(\mathop{\hat{\mathbb{p}}}\nolimits\) be their empirical distribution. Let \(q_\theta\) be probability mass functions with \(q_\theta(v) > 0\) wherever \(\mathop{\hat{\mathbb{p}}}\nolimits(v) > 0\), for parameter values \(\theta\). Then \(\theta\) maximizes the likelihood \(\prod_{i=1}^n q_\theta(x_i)\) exactly when it minimizes the Kullback–Leibler divergence \({{D_{\mathrm{KL}}}\mathopen{}\left(\mathop{\hat{\mathbb{p}}}\nolimits \,\Vert\, q_\theta\right)\mathclose{}}\).

Proof. Let \(n_v\) be the number of data equal to \(v\). Then \(n_v = n \, \mathop{\hat{\mathbb{p}}}\nolimits(v)\).

Group the terms of the log-likelihood by value:

\[\sum_{i=1}^n \log_2 q_\theta(x_i) = \sum_v n \, \mathop{\hat{\mathbb{p}}}\nolimits(v) \log_2 q_\theta(v).\]

Divide by \(n\). The right side is minus the cross-entropy \({\mathbb{H}\mathopen{}\left(\mathop{\hat{\mathbb{p}}}\nolimits, q_\theta\right)\mathclose{}}\):

\[\frac{1}{n} \sum_{i=1}^n \log_2 q_\theta(x_i) = -{\mathbb{H}\mathopen{}\left(\mathop{\hat{\mathbb{p}}}\nolimits, q_\theta\right)\mathclose{}}.\]

By the cross-entropy decomposition, the cross-entropy is the entropy of \(\mathop{\hat{\mathbb{p}}}\nolimits\) plus the divergence:

\[-{\mathbb{H}\mathopen{}\left(\mathop{\hat{\mathbb{p}}}\nolimits, q_\theta\right)\mathclose{}} = -{\mathbb{H}\mathopen{}\left(\mathop{\hat{\mathbb{p}}}\nolimits\right)\mathclose{}} - {{D_{\mathrm{KL}}}\mathopen{}\left(\mathop{\hat{\mathbb{p}}}\nolimits \,\Vert\, q_\theta\right)\mathclose{}}.\]

The entropy \({\mathbb{H}\mathopen{}\left(\mathop{\hat{\mathbb{p}}}\nolimits\right)\mathclose{}}\) does not depend on \(\theta\). So \(\theta\) maximizes the average log-likelihood exactly when it minimizes \({{D_{\mathrm{KL}}}\mathopen{}\left(\mathop{\hat{\mathbb{p}}}\nolimits \,\Vert\, q_\theta\right)\mathclose{}}\). The logarithm increases, and dividing by \(n\) keeps the order, so \(\theta\) maximizes the average log-likelihood exactly when it maximizes the likelihood.

Example 1 (The divergence from the data for coin flips)  

A coin is flipped \(100\) times, and it lands heads \(70\) times. The chunk below follows Theorem 1 for the Bernoulli models with the probability of heads \(0.1, 0.2, \ldots, 0.9\). It prints the divergence of each model from the empirical distribution, the average log-likelihood, and minus the entropy of the empirical distribution minus the divergence.

n_flips <- 100
n_heads <- 70
empirical <- c(tails = n_flips - n_heads, heads = n_heads) / n_flips
success_prob <- seq(0.1, 0.9, by = 0.1)
empirical_entropy <- -sum(empirical * log2(empirical))
divergence <- numeric(length(success_prob))
avg_loglik <- numeric(length(success_prob))
for (j in seq_along(success_prob)) {
  model <- c(1 - success_prob[j], success_prob[j])
  divergence[j] <- sum(empirical * log2(empirical / model))
  avg_loglik[j] <- sum(empirical * log2(model))
}
decomposition <- -empirical_entropy - divergence
data.frame(
  success_prob,
  divergence = round(divergence, 4),
  avg_loglik = round(avg_loglik, 4),
  decomposition = round(decomposition, 4)
)

The last two columns agree, as the proof says. The divergence is smallest, at \(0\) bits, and the average log-likelihood is largest for the model with probability of heads \(0.7\), which is the fraction of heads in the data, so that model is the empirical distribution.

Remark 1 (One labeled example). For a single labeled example, \(n = 1\), and the empirical distribution puts all its mass on the observed label. Then Theorem 1 says that the model with the largest probability for that label is the model with the smallest divergence from it. That is the case that the cross-entropy loss remark describes.

References