Relative entropy
From log evidence to expected performance
Relative entropy answers a specific question: how much log evidence does one probability model expect to accumulate against another? In this information-theoretic view of finance, the same quantity measures an asset’s expected log return under its own implied measure.
The framework constructs an investment from probabilities. Here we slow down at the information-theoretic step: ratios, logarithms, and the probabilities used to average them.
First compare one observation
Consider two possible outcomes, A and B. The asset measure and the pricing measure assign these probabilities:
\[ \begin{aligned} \mathbb P^X&=(0.6,0.4),\\ \mathbb P^Y&=(0.5,0.5). \end{aligned} \]
If A occurs, its likelihood ratio is \(0.6/0.5=1.2\). That observation is 20% more likely under the asset model than under the pricing model. It favors \(\mathbb P^X\) over \(\mathbb P^Y\).
If B occurs, the ratio is \(0.4/0.5=0.8\). That observation favors \(\mathbb P^Y\). A likelihood ratio is a comparison between two specified models; it is not, by itself, a probability that either model is true.
| Outcome | Asset | Market | Ratio |
|---|---|---|---|
| A | 0.6 | 0.5 | 1.2 |
| B | 0.4 | 0.5 | 0.8 |
Take logarithms so evidence adds
Independent likelihood ratios multiply. Taking their logarithms turns that multiplication into addition, which makes accumulated evidence easier to track. For our two outcomes,
\[ \begin{aligned} \log(1.2)&\approx+0.1823216,\\ \log(0.8)&\approx-0.2231436. \end{aligned} \]
Positive log evidence favors the numerator model; negative log evidence favors the denominator. Notice that the positive and negative values are not equal in magnitude. Multiplying wealth by \(1.2\) and then by \(0.8\) leaves \(0.96\) of the original wealth, so those two moves do not cancel.
We use natural logarithms throughout. Their information unit is the nat. A different logarithm base would rescale every information value; it would not change which model an observation favors.
Average using the first model
Now ask what the asset model expects. Under \(\mathbb P^X\), A has weight \(0.6\) and B has weight \(0.4\). Multiply each log ratio by that probability:
\[ \begin{aligned} 0.6\log(1.2)&\approx+0.109392934,\\ 0.4\log(0.8)&\approx-0.089257421. \end{aligned} \]
Adding the contributions gives
\[ \begin{gathered} D(\mathbb P^X\Vert\mathbb P^Y)\\ =0.6\log(1.2)+0.4\log(0.8)\\ \approx0.02013551\text{ nats}. \end{gathered} \]
The negative contribution remains part of the calculation. Relative entropy is an average of signed log evidence, not an average of absolute differences.
For any finite set of outcomes, write the two distributions as \(p=(p_i)\) and \(q=(q_i)\). The definition is
\[ D(p\Vert q)=\sum_i p_i\log\frac{p_i}{q_i}. \]
Read the order carefully: divide first by second; average with first. This is also called Kullback–Leibler divergence, or KL divergence. We use \(D\) and \(D_{\mathrm{KL}}\) for the same quantity.
Try the calculation yourself
In the laboratory, A is called Success and B is called Failure. Move the asset probability first, then the market probability. Watch the individual log returns and their weighted contributions. Finally make the two probabilities equal: each ratio becomes one, every log ratio becomes zero, and relative entropy vanishes.
The interactive laboratory requires JavaScript. Its default example uses asset probabilities \((0.6,0.4)\) and market probabilities \((0.5,0.5)\). The corresponding returns are \((1.2,0.8)\) and the relative entropy is approximately \(0.02013551\) nats.
Why the average is nonnegative
The reason is the concavity of the logarithm: the log of an average is at least the average of the logs. This is Jensen’s inequality.
For the finite-state case, average \(q_i/p_i\) using the weights \(p_i\). Include only states with \(p_i>0\), so no division by zero is needed:
\[ \sum_{i:p_i>0}p_i\frac{q_i}{p_i} =\sum_{i:p_i>0}q_i\le1. \]
Applying Jensen gives
\[ \begin{aligned} -D(p\Vert q) &=\sum_{i:p_i>0}p_i\log\frac{q_i}{p_i}\\ &\le\log\!\left(\sum_{i:p_i>0}q_i\right)\\ &\le0. \end{aligned} \]
Thus \(D(p\Vert q)\ge0\). Equality requires the ratio \(q_i/p_i\) to be constant on the support of \(p\) and all of \(q\)’s mass to lie there. Since both distributions sum to one, this means \(p=q\). If some \(p_i>0\) but \(q_i=0\), the divergence is infinite, so the inequality still holds.
Individual observations may support either model, but a model expects nonnegative log evidence in its own favor when the averaging uses its probabilities. This does not establish which model describes reality, nor guarantee positive evidence in a particular sample. Even in our example, a run of B outcomes can produce a large loss.
The order generally matters. Reversing our two distributions gives
\[ D(\mathbb P^Y\Vert\mathbb P^X) \approx0.02041100\text{ nats}. \]
The ratios are inverted and the averaging probabilities change. KL divergence is therefore asymmetric; it is not an ordinary geometric distance between two points.
The financial meaning of the same calculation
Measure prices in units of a strictly positive traded reference investment, with distributions reinvested, so \(Y(T)=1\) in its own units. A nonnegative asset \(X\) with positive initial price and implied measure \(\mathbb P^X\) has gross return
\[ \frac{X(T)}{X(0)} =\frac{\mathbb P^X(\omega)}{\mathbb P^Y(\omega)}. \]
The likelihood ratios \(1.2\) and \(0.8\) are thus also the asset’s two wealth multipliers. Taking logarithms and averaging under \(\mathbb P^X\) gives
\[ \begin{gathered} \mathbb E^X\!\left[\log\frac{X(T)}{X(0)}\right]\\ =D(\mathbb P^X\Vert\mathbb P^Y). \end{gathered} \]
The value \(0.02013551\) is an expected log return. It is not the expected ordinary percentage return. In this example, the expected gross return under \(\mathbb P^X\) is
\[ 0.6(1.2)+0.4(0.8)=1.04, \]
so the expected simple return is 4%. Averaging returns and averaging their logarithms answer different questions.
The superscript on \(\mathbb E^X\) matters just as much as the logarithm. An investor may hold beliefs \(\mathbb P^M\) different from the asset’s implied measure. If that investor assigns probabilities \((0.4,0.6)\) instead, the very same asset has expected log return approximately \(-0.06096\) under her beliefs. The nonnegative divergence identity becomes her expected-performance statement only when \(\mathbb P^M=\mathbb P^X\).
Continuous observations use densities
For a continuous observation, an exact point usually has probability zero under both models. Dividing those point probabilities would give no useful ratio. Instead compare densities with respect to the same underlying variable:
\[ \begin{gathered} D(f_X\Vert f_Y)\\ =\int f_X(z)\log\frac{f_X(z)}{f_Y(z)}\,dz. \end{gathered} \]
This is the same instruction as before: take a density ratio, take its logarithm, and average under the first density. The normal model illustrates it explicitly with two bell-shaped curves and their likelihood-ratio price.
From expected performance to long-run growth
Why use expected log performance as a growth criterion? Repeat the same investment independently. Let \(p\) and \(q\) be the per-trial asset and market distributions, and write the gross return in period \(k\) as
\[ G(k)=\frac{p(\omega(k))}{q(\omega(k))}. \]
Reinvesting the proceeds compounds these returns:
\[ \begin{aligned} \frac{X(n)}{X(0)}&=\prod_{k=1}^{n}G(k),\\ \log\frac{X(n)}{X(0)}&=\sum_{k=1}^{n}\log G(k). \end{aligned} \]
The expected log return after \(n\) periods is \(nD(p\Vert q)\) under the asset model. More is true: if the observations are independent and identically distributed and the one-period log return is integrable, the strong law of large numbers gives
\[ \frac1n\log\frac{X(n)}{X(0)} \longrightarrow D(p\Vert q) \]
almost surely under \(\mathbb P^X\). This means convergence on a set of paths with probability one under the asset model. Relative entropy is the long-run log growth rate in this repeated experiment. The statement is conditional on the chosen model governing the observations; it does not establish that the model is correct.
The fair-game paradox
Under the market model, each return has mean one. Nevertheless, Jensen’s inequality gives
\[ \begin{aligned} \mathbb E^Y[\log G(1)] &\le\log\mathbb E^Y[G(1)]\\ &=0. \end{aligned} \]
For distinct positive \(p\) and \(q\), the inequality is strict. Averaging the same log ratios under \(q\) gives \(-D(q\Vert p)\). With the same independence and integrability assumptions, now under the market model,
\[ \frac1n\log\frac{X(n)}{X(0)} \longrightarrow -D(q\Vert p) \]
almost surely under \(\mathbb P^Y\). So typical market-model paths decay toward zero even though
\[ \mathbb E^Y[X(n)]=X(0) \]
at every finite date. Increasingly rare, very large outcomes sustain the arithmetic average. There is no contradiction: average wealth across many possible outcomes and long-run growth along a typical path are different quantities. If \(p=q\), every multiplier is one and there is no growth or decay.
In our example the two long-run log rates are about \(+0.02014\) under the asset model and \(-0.02041\) under the market model. Generate paths in the binomial model to see the distinction. The linear information flow page develops the common additive structure across model families.
When both distributions are positive, averaging the same log return under the market measure reverses the divergence:
\[ \begin{gathered} \mathbb E^Y\!\left[\log\frac{X(T)}{X(0)}\right]\\ =-D(\mathbb P^Y\Vert\mathbb P^X). \end{gathered} \]
In the example this is approximately \(-0.02041100\), even though the market’s expected gross return is one.
Throughout the finite-state construction, market state prices satisfy \(q_i>0\). Asset probabilities may be zero: the convention \(0\log(0/q_i)=0\) reflects the limiting value of that expression. If \(p_i=0\) while \(q_i>0\), the asset pays zero in that state. Its expected log return under a model assigning positive probability to that state is \(-\infty\); the corresponding reverse divergence is \(+\infty\). In independent repetitions under the market model, such a state is eventually encountered with probability one, and reinvested wealth then stays at zero. The finite-log-mean assumption for the displayed market strong law does not apply in this case.
For general density models, positive mass under the numerator outside the denominator’s support gives infinite divergence. Finite subtraction identities require suitable support and integrability assumptions; one must not subtract two infinities.
Source: Jan Vecer, Principles of Portfolio Choice: An Information-Theoretic, Likelihood-Based Perspective, Chapters 1–3. Continue to utility and risk aversion to see how an investor’s chosen payoff can differ from the belief-matching investment.