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For $Q(\theta \mid {\theta }_0)={\mathbb{E}}_{{\theta }_0}[\log p_{\theta }(Y,Z)\mid Y]$, the Jensen inequality givesThus any expectation-maximization algorithm M-step that increases $Q$ cannot decrease the observed-data likelihood function.^[ref]

For a short proof, assume common support and finite expressions, and keep the E-step conditional distribution $q_t(z)=p_{{\theta }^{(t)}}(z\mid y)$ fixed. The concavity of the logarithm and Jensen's inequality giveThis evidence lower bound equals $Q(\theta \mid {\theta }^{(t)})-{\mathbb{E}}_{q_t}\log q_t(Z)$. At $\theta ={\theta }^{(t)}$, the ratio inside the expectation is the constant $p_{{\theta }^{(t)}}(y)$, so equality holds. During the M-step, the second term is constant; increasing $Q$ raises a lower bound that touches the current log-likelihood. HenceIncreasing likelihood values alone do not establish convergence of the statistical parameters or attainment of a global maximum. In a differentiable finite mixture model, the gradient of $\ell$ equals the candidate-parameter gradient of $Q$ evaluated at the current fit. Thus an interior fixed point of an exact M-step is a stationary point, which need not be a local maximum; the Identical-mean EM trap is an explicit counterexample. Small changes in log-likelihood can also occur before the means have settled, so inspecting both is useful.

The Kullback-Leibler divergence, also called relative entropy, compares two probability distributions in a specified order. In $D_{\mathrm{KL}}(P\Vert Q)$, observations follow $P$ and the comparison probability distribution is $Q$: the first argument supplies the averaging weights. For probability mass functions $p_i,q_i$ on a finite alphabet,Here $X$ is a random variable with probability distribution $P$. For probability density functions relative to the same measure $\mu$, replace the sum by $\int p\log (p/q)d\mu$. These formulas use natural logarithms unless stated otherwise. The expected value is under $P$, even though $Q$ appears in the denominator. ^[ref]

Take $P=(0.75,0.25)$ and $Q=(0.5,0.5)$ on two outcomes. Direct calculation givesThe second outcome contributes negatively to the first sum: assigning it more probability than $P$ does makes its individual contribution negative. Nevertheless, relative entropy nonnegativity guarantees a nonnegative total. Reversing the arguments changes both the likelihood ratio and the averaging probabilities; taking an absolute value inside the logarithm would define a different quantity.

The units depend on the logarithm: natural logarithms give nats, whereas base-two logarithms give bits. Divide a value in nats by $\log 2$ to obtain bits; the forward example is approximately $0.188722$ bits. Changing the base rescales every Kullback-Leibler divergence by the same positive factor. Since the two calculated directions differ, Kullback-Leibler divergence is asymmetric and cannot be a metric. It also need not satisfy the triangle inequality. ^[ref]

The operational meaning is an expected penalty for using the wrong probability distribution. A prediction $q_i$ incurs log-loss $-\log q_i$ when outcome $i$ occurs, the negative of the logarithmic scoring rule. Under $P$, its expected value is the cross-entropy $H(P,Q)=-{\sum }_i{p}_i\log q_i$. Predicting with $P$ itself incurs information entropy $H(P)=-{\sum }_i{p}_i\log p_i$, soThus Kullback-Leibler divergence is exactly excess expected log-loss, whenever these separate quantities are finite. ^[ref]

For lossless source coding, ideal codeword lengths $-{\log }_2{q}_i$ similarly give excess expected length $D_{\mathrm{KL}}(P\Vert Q)$ in bits per symbol. Actual prefix codes require integer codeword lengths; Shannon's source coding theorem explains how long blocks approach ideal per-symbol lengths. The interpretation concerns an average, not a fractional physical bit attached to every outcome. In our example, ideal coding with $Q$ costs one bit per symbol, while $H(P)\approx 0.811278$ bits; the difference is exactly the computed $0.188722$ bits. ^[ref]

Zeros require explicit conventions. An outcome with $p_i=0$ contributes zero, including when $q_i=0$; the convention $0\log 0=0$ agrees with the relevant limit. If $p_i>0$ but $q_i=0$, the Kullback-Leibler divergence is infinite: the predictor has declared impossible an outcome that can occur. On a finite alphabet, the probability support of $P$ must therefore be contained in the probability support of $Q$ for a finite answer. For example, $P=(1,0)$ and $Q=(0.5,0.5)$ give $D(P\Vert Q)=\log 2$, but $D(Q\Vert P)=+\infty$. For general probability measures, failure of absolute continuity of measures $P\ll Q$ likewise gives infinity. Even $P\ll Q$ does not guarantee a finite integral. ^[ref]

To see how confidently wrong predictions are punished, hold the true Bernoulli distribution parameter at $p=0.75$ and vary the predicted parameter $q$. Its binary relative entropy isIt vanishes at $q=0.75$, equals approximately $2.894055$ nats at $q=0.01$, and equals $0.596495$ nats at $q=0.99$. Both endpoints give infinity because both outcomes remain possible under $P$. Predicting almost no successes is especially costly because successes actually occur three quarters of the time. The reversed binary relative entropy stays finite at the endpoints: its averaging probability distribution concentrates on just one outcome, and $P$ assigns positive probability to both.

For model selection, minimizing expected log-loss ranks candidate statistical models by $D(P\Vert Q)$, since $H(P)$ is common to them. The true probability distribution is usually unknown, so one estimates predictive performance using fresh observations or cross-validation. Fitting and assessing on the same data can reward overfitting; a small fitted Kullback-Leibler divergence alone does not establish useful prediction.

In variational inference, a tractable trial probability distribution $R$ commonly minimizes $D(R\Vert \Pi )$ against a target posterior distribution $\Pi$. Maximizing the evidence lower bound accomplishes this without evaluating the target's normalization constant. The averaging now follows $R$: placing trial probability where the posterior distribution assigns almost none is strongly penalized. The chosen direction and trial family therefore matter. ^[ref]

These comparisons do not by themselves establish causal inference. Two statistical models can reproduce the same observed probability distribution while predicting different responses to an intervention. Choosing the smaller Kullback-Leibler divergence therefore needs additional assumptions or experimental evidence before supporting a causal claim. ^[ref]

The original reference is S. Kullback and R. A. Leibler's 1951 paper ^[ref], published in The Annals of Mathematical Statistics 22(1), pages 79–86. Its title connects this comparison with sufficient statistics: the historical setting was statistical information, before today's widespread applications in prediction and variational inference.

Write $\pi (x)=\mu (x\mid y)$. Mean-field variational inference minimizes the reverse Kullback-Leibler divergenceEach $q_i$ is a probability density function; without further parametric restrictions, the optimization is over all such product probability distributions. Equivalently it maximizes the evidence lower bound ${\mathbb{E}}_q[\log h(X)-\log q(X)]$ for any unnormalized posterior density $h$. Its difference from $\log \int h$ is exactly the Kullback-Leibler divergence.

Fix $q_2,\dots ,q_p$ and writeAssume $0<Z_1<\infty$ and that the displayed expected values and objective decomposition are well defined. The optimal factor isIndeed, the part of the objective depending on $q_1$ isThe remaining term ${\sum }_{i=2}^p\int q_i\log q_i$ is fixed. Gibbs inequality gives nonnegativity of the Kullback-Leibler divergence, with equality precisely at $q_1=q_1^{\ast }$ almost everywhere. This proves global optimality of that coordinate update; it does not assert global optimality of a sequence of coordinate updates for the full nonconvex product-family problem.

The printed upper index $k$ in the list of remaining coordinates is inconsistent with the dimension $p$; the natural interpretation is $k=p$. Also, if the exponential expression has zero or infinite normalizing constant, or the expected values are undefined, the usual coordinate formula needs additional support or integrability hypotheses. A restricted parametric factor family need not contain this unrestricted optimal factor.

For any approximating density $q_{\varphi }$,The evidence lower bound is thereforeSince $\log Z$ does not depend on $\varphi$, maximizing the ELBO is equivalent to minimizing the divergence from $q_{\varphi }$ to the posterior.

Apply the same invertible differentiable transformation to two probability distributions with densities $f$ and $r$. Both transformed densities acquire the same Jacobian determinant factor. Write their transformed densities as $f_g$ and $r_g$. Their ratio is unchanged, and the integration measure transforms with the numerator density. Consequently their relative entropy is unchanged:The densities describe two alternative laws in the same coordinate, with a nonzero Jacobian determinant on their common domain, and the expression is assumed well defined. This explains why comparing two laws can survive a change of measurement units even though either law's differential entropy changes. Mutual information, formulated as relative entropy from a product of marginals, has the same invariance under invertible changes to either observed coordinate.

Differential entropy is also the entropy term in the evidence lower bound for continuous variational inference. Its coordinate dependence must be combined consistently with that of the model density; changing units does not justify changing only one term of the objective.