Applying Bayesian Reasoning to Estimate the Type of a Feature Suppose I have a set of strings $S$ and I want to find out whether these strings have a certain type. To be more specific, I want to find out whether these strings are surnames. Suppose I have a large list $L$ of surnames from many regions in the world. ($L$ can be viewed as a set.) Now, there are likely many surnames not in $L$ - either because the list does not capture them, or because they are spelled in a way that is similar but not exactly as on the list (e.g. "Markov" vs "Markow" vs "Markoff" etc).
Suppose that for each entry in $S$ (or a sufficiently large sample thereof) I can provide a confidence or probability $p$ that this is a surname (1 if the surname matches some entry in $L$ exactly and less than 1 if there is only a fuzzy match; for those who want to know specifically, I am using the Python fuzzyset package that calculates that value on the basis of the Damerau-Levenshtein distance).
How can I calculate the probability that the entire set $S$ is a list of surnames? I thought of Bayesian reasoning, but I can't put the calculation together properly.
What I want is the probability that $S$ contains surnames given the confidence that I have about individual entries of $S$ being "fuzzily" in $L$.
Specifically, everything conditioned on the specific list $L$, I want to know $\Pr(A_S | B_s)$ where $A_S$ is the event that $S$ is indeed a set of surnames and $B_s$ is the event that $s \in S$ matches "fuzzily" some element in $L$; let $\Pr(B_s)$ be the confidence $p$.
I don't get how to proceed from here - or am I entirely on the wrong track?
 A: What you are missing is a likelihood function.  For that, you will need to go linguists.  My mother's maiden name, for example, contains two changes.  One letter was dropped and another letter substituted when my great grandparents arrived in America.  The original spelling has survived in other branches of the family.  Nonetheless, her family came from a country that uses Latin letters.
In your example, you chose Markov versus Markoff.  In Cyrillic, Markov is spelled Марков.  While all of those look like Latin characters, one need only encounter Ж and you are out of similar characters.  That presents a different challenge to immigration officials.
It becomes more complex when you encounter languages that don't use alphabets.  For example, 周 doesn't work at all for Latin languages without sounding things out.  In addition to alphabets for writing systems, the world uses abjads, abugidas, syllabic and lexographic systems.
In addition, some languages have sounds English lacks, such as Georgian or Russian.  In addition, some languages have a first surname and a second surname here in the United States.  For example, Tom Many Hides or Mary Runs With Deer have two and three surnames respectively.  Indeed, it could be one surname with spaces as letters.
Indeed, probably the only reason the Siksikaitsitapi in the United States uses English surnames is that it was a crime to speak their own language in the United States.
So your question becomes "what is the probability Smyth is a surname given that Smith is a known surname?"  To discover that, you need to know the transition rules from one language to another.  That is complicated by the transition to print and then a later transition to computers.
Giseldone became Gislandune which, over ten centuries, has become Islington, a borough in London.
Close languages, such as French or German suffer small changes.  As the distance increases, such as Italy or Greece, the changes become larger.  When larger transitions happen such as Russian, Swahili, or Chinese, the changes are greater.  The probabilities are different.
Printing and computers changed the rules a bit too.  Preprinting name variations such as Smith and Smyth came from the absence of a defined lexography.
Of the six known examples of William Shakespeare's signature, he signed his name Willm Shakp, William Shaksper, Wm Shakspe, William Shakspere, Willm Shakspere, and William Shakspeare.  Printing made people think that there should be a single solidified form.  There should be one correct way and the others are incorrect.
The printing press created a crisis that had not previously existed in Christianity.  There is not a single known version of the early bibles.  They don't agree with each other.  Nobody cared either.  Modern textual scholarship shows that there are 400,000 variant passages for the new testament, which has 33,000 verses.  About half of those are due to spelling differences, but the rest are differences in substance. The verses actually say different things. Prior to the printing press, that wasn't an important thing, but printing makes people think there is a solidified form that is right.
Computers created a new problem.  The longest surname in the United States is 53 characters long.  Old computer systems couldn't handle a 53 character name.  Also, there is a real surname called "Null," which creates its own nightmares.  Whereas printed names have some sense of being fixed or solidified, computerized storage means that there can be no changes.  If your name doesn't fit the computer's lexicography, well, it needs to be fixed until it does into a new solidified, computer-compliant form.
What you need from linguists is a probability of a transition happening.  That is what a likelihood function is.  It is the ability to map transitions to probabilities.  As I am not a linguist, I cannot help with that.  However, my suspicion is that there is a giant literature on this.  My other guess is that the Damerau-Levenshtein distance will perform well on French and German names and horribly on names originating in Tagalog.
To get to a Bayesian probability statement, you need to know $\Pr(X|\theta)$, what is the probability of a transition given that a specific transition rule applies.
