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Cross-entropy measures the average logarithmic loss incurred when outcomes follow one probability distribution $P$, but a model assigns probabilities using another probability distribution $Q$. For a finite alphabet, write their probability mass functions as $p_i$ and $q_i$. With base-two logarithms,The expectation is under the true or stipulated source law $P$, rather than the model $Q$. Terms with $p_i=0$ contribute nothing. If $p_i>0$ but $q_i=0$ for any outcome, the cross-entropy is $+\infty$: the model rules out an event that can occur. The result is measured in bits; natural logarithms give nats.

The useful identity is cross-entropy equals entropy plus relative entropy:The source's information entropy $H(P)$ is the baseline uncertainty. The Kullback-Leibler divergence is the extra average cost of a mismatched model, with the true law first and model second. Thus optimizing $Q$ for a fixed $P$ reduces the mismatch, rather than changing the source's own uncertainty. For finite alphabets, the minimum is $H(P)$, attained exactly at $Q=P$.

Take a three-symbol source with $P=(1/2,1/4,1/4)$. A matched binary prefix code assigns $A\mapsto 0$, $B\mapsto 10$, $C\mapsto 11$, with lengths $(1,2,2)$. A mismatched prefix code instead assigns $A\mapsto 10$, $B\mapsto 11$, $C\mapsto 0$, with lengths $(2,2,1)$, corresponding to $Q=(1/4,1/4,1/2)$. Both prefix codes decode exactly, but the most common symbol is shorter in the matched code. Under the same source $P$,The mismatched code costs an extra (1/4) bit per source symbol on average. No information is lost; the penalty is inefficient representation of this source. The mismatch penalty in lossless coding explains when this coding interpretation is exact.

Cross-entropy is also useful when evaluating a predictive distribution. A report that places tiny probability on the realized outcome receives a large logarithmic loss, even when it usually predicts the most common outcome correctly. The loss convention is the negative of the reward convention in the logarithmic scoring rule. Its expected loss is uniquely minimized by the correct probability distribution, making it a strictly proper scoring rule. Strictly Proper Scoring Rules, Prediction, and Estimation by Tilmann Gneiting and others treats this connection in §3.1, Example 3; its discussion uses rewards, so the signs are reversed for the losses here.^[ref]

I. J. Good's Rational Decisions by I. J. Good appeared in 1952. Its published summary describes a method for encouraging accurate probability estimates and a connection with information theory. Strictly Proper Scoring Rules, Prediction, and Estimation by Tilmann Gneiting and others traces the logarithmic scoring rule at least as far back as this 1952 paper.^[ref]

A sufficient statistic $T$ has a conditional distribution of the data given $T$ that does not depend on the parameter. In a dominated family, the factorization criterion for sufficiency states that this is equivalent to $p_{\mu }(x)=h(x)g_{\mu }(T(x))$ for a nonnegative parameter-free $h$. Bayes' formula then gives a posterior distribution proportional to $\pi (\mu )g_{\mu }(T(x))$, because $h(x)$ cancels from numerator and normalizing denominator. Thus the posterior depends on the data only through $T$.

For the normal sample,The first term is parameter-free, so the factorization proves sufficiency of $\bar{X}$. Combining its likelihood with the normal prior and completing the square givesThe prior predictive distribution of the sample mean is obtained from $\bar{X}=M+\epsilon$, $\epsilon \sim N(0,1/n)$ independent of $M$:

Define the Bayes factor $B_{01}$ as the marginal likelihood under $H_0$ divided by that under $H_1$. The common residual factor cancels, so it is the ratio of the densities of $\bar{X}$ under these two hypotheses:The reverse Bayes factor is its reciprocal. At $|\bar{x}|=1.96/\sqrt{n}$,for a fixed positive $\tau$. Thus the frequentist test is at its rejection threshold while fixed prior odds give increasingly strong posterior support for the point null. This is the Jeffreys-Lindley paradox for a Gaussian point null: the standardized statistic stays fixed, but the observed effect tends to zero and the fixed-scale alternative spreads predictive mass much more widely. There is no logical contradiction, since the procedures compare hypotheses using different calibrations. For the degenerate $\tau =0$ the hypotheses coincide and $B_{01}=1$.

A proper scoring rule makes reporting the true probability distribution optimal in expected value, for every true probability distribution in the allowed class of forecasts. In the reward convention, the inequality in the title holds for every report $P$ and true law $Q$. The expected value is always taken under $Q$, because the report does not change how the outcome is generated. In the loss convention the same definition readsThis requirement makes a proper scoring rule useful for eliciting probabilities and comparing predictive distributions: deliberately distorting a known forecast cannot improve its expected reward or expected loss. The reward convention and its strict version are set out in Strictly Proper Scoring Rules, Prediction, and Estimation by Tilmann Gneiting and others (Section 1).

For the Bernoulli distribution with $q=0.7$, the Brier score has expected loss $(p-0.7{)}^2+0.21$, and the negative logarithmic scoring rule has expected loss $-0.7\ln p-0.3\ln (1-p)$. Both are minimized at the honest report $p=0.7$. The linear probability score, by contrast, maximizes its expected reward at $p=1$. These are statements about averaging over both possible outcomes with their actual probabilities.

The left panel displays each excess expected loss divided by that rule's expected loss at $p=q$: $0.21$ for the Brier score and approximately $0.610864$ for the logarithmic scoring rule. This normalization permits comparison of curve shapes and preserves the minimizing report, since it subtracts a constant and divides by a positive constant independent of $p$. Zero means no excess expected loss. The right panel shows the unmodified linear probability score reward, with its maximizing direction labeled explicitly.

A scoring rule turns a probability distribution forecast and an observed outcome into a number. For example, a forecast of rain with probability $0.7$ should express uncertainty more usefully than a bare prediction of rain. A scoring rule lets us compare such forecasts after observations arrive, and its choice determines whether exaggerating confidence pays. A proper scoring rule rewards an honest account of uncertainty in expected value.

There are two sign conventions. A reward $S(P,x)$ is maximized; a loss function $L(P,x)$ is minimized. The conversion $L=-S$ reverses every comparison. Here the linear probability score is a reward, while the binary Brier score and the negative logarithmic scoring rule are losses. Calling all three a "score" without specifying the direction would hide the central difference.

Throughout the binary examples, the random variable $Y$ has a Bernoulli distribution with actual success probability $q=0.7$, and the report $p\in [0,1]$ assigns probability $p$ to $Y=1$ and $1-p$ to $Y=0$. The value $q=0.7$ is stipulated by this statistical model; it is not inferred from data or a claim that a forecasting system is calibrated. In practice the true predictive distribution is unknown, and the relevant $q$ is the conditional probability of success given the information available when the forecast is issued. Propriety evaluates incentives for reporting that predictive distribution; it does not supply the unknown probabilities.

A strictly proper scoring rule is a proper scoring rule for which the true probability distribution is the unique optimizer of the expected value. Equivalently, any different admissible probability distribution gives strictly larger expected loss. For probability density functions, two densities that agree almost everywhere represent the same probability distribution. Ordinary propriety permits ties: the constant reward $S(P,x)=0$ is a proper scoring rule, because every report ties, but it is not a strictly proper scoring rule and elicits no information.

Strict propriety concerns expectation, not guaranteed success on each observation or each realized dataset. With the stipulated Bernoulli distribution $q=0.7$, suppose the outcome is $Y=0$. The honest report $p=0.7$ then has Brier score loss $0.49$, while $p=0.2$ has loss $0.04$. The logarithmic loss similarly gives losses $-\ln (0.3)\approx 1.203973$ and $-\ln (0.8)\approx 0.223144$. The honest forecast loses this comparison on that draw, which occurs with probability $0.3$ in the stipulated statistical model.

The $Y=0$ loss curves favor $p=0$, and the $Y=1$ loss curves favor $p=1$. Their weighted average, with weights $0.3$ and $0.7$, favors $p=0.7$. Choosing a report after observing $Y$ would use information unavailable at forecast time. Even an average loss on a finite dataset can rank an honest forecast below a competing forecast through outcome variation; strict propriety guarantees the ranking in expected value under the specified information and true predictive distribution.