Draft:Method of types

Submission declined on 6 August 2026 by FuzzyMagma (talk). This draft provides insufficient context for those unfamiliar with the subject. Please see the guide to writing better articles for how to improve your writing. If you would like to continue working on the submission, click on the "Edit" tab at the top of the window. If you have not resolved the issues listed above, your draft will be declined again and potentially deleted. If you need extra help, please ask us a question at the AfC Help Desk or get live help from experienced editors. Please do not remove reviewer comments or this notice until the submission is accepted. Where to get help If you need help editing or submitting your draft, please ask us a question at the AfC Help Desk or get live help from experienced editors. These venues are only for help with editing and the submission process, not to get reviews. If you need feedback on your draft, or if the review is taking a lot of time, you can try asking for help on the talk page of a relevant WikiProject. Some WikiProjects are more active than others so a speedy reply is not guaranteed. How to improve a draft Wikipedia:Contributing to Wikipedia – a basic overview on how to edit Wikipedia. Help:Wikitext – how to use the markup Help:Referencing for beginners – how to include references Wikipedia:Article development – how to develop your article Wikipedia:Writing better articles – how to improve your article Wikipedia:Verifiability – make sure your article includes reliable third-party sources You can also browse Wikipedia:Featured articles and Wikipedia:Good articles to find examples of Wikipedia's best writing on topics similar to your proposed article. Improving your odds of a speedy review To improve your odds of a faster review, tag your draft with relevant WikiProject tags using the button below. This will let reviewers know a new draft has been submitted in their area of interest. For instance, if you wrote about a female astronomer, you would want to add the Biography, Astronomy, and Women scientists tags. Add tags to your draft Editor resources Easy tools: Citation bot (help) | Advanced: Fix bare URLs Declined by FuzzyMagma 8 days ago. Last edited by ArickGrootveld 60 minutes ago. Reviewer: Inform author. Resubmit Please note that if the issues are not fixed, the draft will be declined again.

Submission declined on 27 April 2026 by Ldm1954 (talk). This draft's references do not show that the neologism meets Wikipedia's criteria for inclusion. The draft requires multiple published secondary sources that: provide significant coverage: discuss the term in detail, not just brief mentions or routine use; are reliable: from reputable outlets with editorial oversight; are independent: not connected to the subject, such as promoted or sponsored content. Please add references that meet all three of these criteria. If none exist, the subject is not yet suitable for Wikipedia. Wikipedia is not a dictionary and we do not accept articles that are simply definitions of neologisms. Declined by Ldm1954 3 months ago.

• Comment: see Ldm1954 comments. Still needs more work. FuzzyMagma (talk) 20:32, 6 August 2026 (UTC)

• Comment: This page can probably be OK, but needs some improvement. This draft really only uses papers from Csiszar, and a non-obvious link to (probably) §13.3.3 of a book. Please:
  1. Look at WP:MOS for format details, e.g. bold of the title.
  2. Add a little broader context for the general reader.
  3. Use broader sources to better establish notability.
  4.Provide more encyclopedic information such as uses, currently this is an extended definition. Ldm1954 (talk) 12:24, 27 April 2026 (UTC)

The method of types is a tool in information theory and large deviation theory to analyze events from the perspective of the empirical distribution.^[1][2] The method of types classifies events as typical when the empirical distribution closely matches the true distribution.

It has been used to prove results in hypothesis testing, channel coding,^[2][3] and universal source coding,^[1][4] and it is used to prove Sanov's theorem in the finite support setting.^[5][6] It is often used to simplify complex probability expressions into more natural expressions involving the Kullback–Leibler divergence. The topic is considered a traditional method in information theory,^[7][8] is usually covered in introductory courses on information theory,^{[3][4][9][10][11]} and textbooks.^{[1][8][12][13]}

It was popularized by Imre Csiszár,^[14] with the first formal treatment given in the book "Information Theory: Coding Theorems for Discrete Memoryless Systems".^[8][12]. Csiszár claimed that the technique was originally motivated by the large deviation bound described in Hoeffding's seminal paper on multinomial hypothesis testing.^[15][2]

Let ${\displaystyle {\mathcal {X}}}$ be an alphabet with a finite number of elements. We start with a sequence of i.i.d. random variables following a finite support distribution ${\displaystyle Q}$, ${\displaystyle X^{n}=(X_{1},\dots ,X_{n})\sim Q}$, where ${\displaystyle X_{i}\in {\mathcal {X}}}$. For this sequence, let ${\displaystyle N(k|X^{n})=\sum _{i=1}^{n}\mathbf {1} \{X_{i}=k\}}$ denote the number of occurrences of the symbol ${\displaystyle k}$ in the sequence ${\displaystyle X^{n}}$.

Denote the empirical distribution of the samples with ${\displaystyle P_{X^{n}}}$, so that ${\displaystyle P_{X^{n}}[k]={\frac {N(k|X^{n})}{n}}}$. The sets of sequences having empirical distribution ${\displaystyle Q}$, are called the type classes ${\displaystyle T(Q)=\{X^{n}\in {\mathcal {X}}^{n}:P_{X^{n}}=Q\}}$. To describe the set of all empirical distributions possible from ${\displaystyle n}$ samples we use the symbol ${\displaystyle {\mathcal {P}}_{n}}$, which is a subset of the ${\displaystyle d-1}$ dimensional probability simplex ${\displaystyle {\mathcal {P}}}$.

The method of types is built on top of the following results:^[1]

• ${\displaystyle |{\mathcal {P}}_{n}|\leq (n+1)^{|{\mathcal {X}}|}}$

• ${\displaystyle Q[X^{n}]=2^{-n\left[H(P_{X^{n}})+D(P_{X^{n}}\|Q)\right]}}$

• ${\displaystyle {\frac {1}{(n+1)^{|{\mathcal {X}}|}}}2^{nH(Q)}\leq |T(Q)|\leq 2^{nH(Q)}}$

• ${\displaystyle {\frac {1}{(n+1)^{|{\mathcal {X}}|}}}2^{-nD(P\|Q)}\leq Q[T(P)]\leq 2^{-nD(P\|Q)}}$

Here ${\displaystyle H(P)}$ denotes the Shannon entropy, and ${\displaystyle D(P\|Q)}$ denotes the Kullback–Leibler divergence. The last equation is derived from the previous ones by noticing that every element of ${\displaystyle T(P)}$ has the same probability when the samples are taken i.i.d.

These bounds are tight in terms of the error exponent,^[16] but may give a loose bound in terms of ${\displaystyle |{\mathcal {X}}|}$ and the sub-exponential terms in ${\displaystyle n}$.^[17]

The method of types is used for hypothesis testing problems, when the underlying distributions have finite support^[18][2][19], because it provides a tight bound on the error exponent. It was also used to prove some initial conditional limit theorems^[20], which are special cases of the general class of limit theorems. The bound on the size of a type class can be used to prove tight bounds on the exact value on binomial coefficients^[1]. It is also used to prove the principle of maximum entropy.^[21]