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In his 1998 retrospective, Imre Csiszár describes the method of types as a 1970s refinement of the typical set approach, with older counting ingredients. He credits his research group with turning those ingredients into a general tool, while explicitly declining to give a complete discovery history.^[ref] The useful shift is from an approximately typical collection to exact frequency classes: the latter retain enough structure to study how fast error probabilities decay.

For an attainable empirical distribution $q$ on a fixed finite alphabet of size $m$,Thus $\ln |T_n(q)|$ differs from $n{H}_e(q)$ by at most $m\ln (n+1)$, uniformly over attainable profiles. Information entropy is the leading logarithmic count; the polynomial factor matters at short lengths.^[ref]

The upper bound has a short explanation. Under an independent and identically distributed source whose law is $q$ itself, each string in $T_n(q)$ has probability $e^{-n{H}_e(q)}$. Multiplying by the number of such strings cannot give a probability above one. A corresponding polynomial lower bound follows from the multinomial coefficient; the Stirling's approximation gives sharper prefactors when needed.

For the balanced four-symbol class, $H_e(q)=\ln 2$, so the upper bound is $e^{4\ln 2}=16$, while the exact size is six. Information entropy is not an exact count for a short string. On the leading scale it becomes precise: the difference divided by $n$ tends to zero as $n$ grows with the alphabet fixed.

For a string of empirical distribution $q$, independence givesMultiplying this by the size of a type class gives its exact probability. The entropy bounds for a type class then implyThese bounds apply to one attainable empirical distribution, not automatically to a tail event containing many classes.^[ref] If $p(a)=0$ but $q(a)>0$, every string in that class is impossible and $D_e(q\Vert p)=+\infty$.

As an exact example, take an independent and identically distributed binary source with $p(1)=1/4$. Each balanced four-symbol string has probability $(1/4{)}^2(3/4{)}^2=9/256$. There are six such strings, soThe relative entropy is $D_e((1/2,1/2)\Vert (3/4,1/4))=\frac{1}{2}\ln (4/3)$. The exponential upper bound is $e^{-4D_e}=9/16=0.5625$, visibly larger than the exact probability. It captures the cost of an atypical frequency without supplying the exact finite-length prefactor.

For $n=40$, the exact probability of twenty ones iswhile the exponential upper bound is approximately $0.00317121$. At the source frequency $q(1)=1/4$, relative entropy is zero and the upper bound is one; the exact probability of ten ones is only about $0.144364$. A likely neighborhood of frequencies need not give a large probability to a single exact profile.

The striking cancellation is that there are about $e^{n{H}_e(q)}$ strings in a class, but each costs about $e^{-n[H_e(q)+D_e(q\Vert p)]}$. The information entropy terms cancel, leaving relative entropy as the rarity exponent. Summing over polynomially many classes preserves that leading exponential scale. This is the elementary counting mechanism behind finite-alphabet versions of Sanov theorem and many error exponents in hypothesis testing.

A type summarizes a string by how often each symbol occurs. Fix a finite alphabet $\mathcal{A}$ of cardinality $m$ and a positive length $n$. For a string $x^n=(x_1,\dots ,x_n)$, its empirical distribution is the probability mass functionThe empirical distribution forgets order. The strings $0011$ and $1010$ both assign probability $1/2$ to each binary symbol; $0001$ assigns $3/4$ to zero and $1/4$ to one. Thus a long string can have a short summary even though that summary does not reconstruct the original order.

For independent and identically distributed symbols, this loss of order has a useful consequence: all strings with the same empirical distribution have the same probability. The method of types uses this symmetry to turn questions about exponentially many strings into counting questions about relatively few frequency profiles.^[ref]