A Way to Apply Bayes' Rule to Conflicting Evidence

Here's an application of Bayes' Theorem that, IMO, deserves to be more widely known:

where A is a set of possibilities, E is a set of sources of evidence, and i_e is the event that a source of evidence indicated whichever possibility it indicated (as opposed to indicating some other possibility).  It's really a general formula for deriving probability from any number of conflicting, corroborating, and possibly biased sources of evidence.  So long as the sources of evidence are independent and they each indicate exactly one of a limited number of possibilities, this formula applies.  For instance, suppose you have several people making claims about something that happened, some agreeing and some disagreeing.  How do you weigh their testimony against each other?  If you can find a way to estimate how likely each person was to make the claim that he or she claimed given each of the possibilities, this formula provides you with a way forward.

Here's a math paper that states the formula as a theorem and proves it: Bayes' Theorem Under Conditional Independence.  It's Theorem 4.  (The paper expresses the formula with different symbols, but it's essentially the same thing.)  I haven't found it anywhere else in print or online. Since the numerator on the right-hand side is the basis for naive Bayes classifiers, maybe it's included in some AI course material somewhere.  I suppose it falls between two stools; it isn't one of the most basic applications of Bayes' theorem, so it wouldn't be in introductory material, but to a professional mathematician, it might seem too obvious to deserve mentioning.

Professor Kirke's House

The filming location for Professor Kirke's house in the BBC version of "The Lion, the Witch and the Wardrobe" (IMDb) is Llanvihangel Court.  Some of the rooms, especially the spare room that contains the wardrobe, look like they might be sets, but everything else appears to be filmed there.  For evidence, watch this YouTube copy of "The Lion, the Witch and the Wardrobe", specifically at these times, noting:

• 3:16 - the front of the house,
• 23:10 - the fountain,
• 48:31 - the garden house (guardhouse) next to a red tree and
• 49:01 - the stained-glass window,

and compare them to these 2017 photographs of Llanvihangel Court and the photo of the stained-glass window here: Elizabeth I from Elizabeth I and the Family of Charles I - work from Stained Glass in Wales (llgc.org.uk)

Unlike some other locations for the series, the house wasn't mentioned in closing credits.  At the time of filming, it was a private house and the owners may have not wanted publicity.  In recent years, the house appears to have hosted tours and events.  It has a web site: https://www.llanvihangelcourt.com/

I found the house by looking for recognizable objects in scenes set inside and outside the house and searching the Internet for them.  It took me a while to notice the stained-glass window, but eventually I did, and that was the key.  It depicts a lady that looks like Queen Elizabeth I.  On closer inspection, I saw that there is a gentleman on either side of the lady and one of them is laying down a cloak.  I recalled there was an incident where Sir Walter Raleigh laid down his cloak for Queen Elizabeth so she would not have to step in a puddle.  Googling "elizabeth", "cloak", "raleigh", and "stained glass" as required terms turned up the "Stained Glass in Wales" web page mentioned above, which identifies the location of the stained-glass window: Llanvihangel Court. 

As for the puddle incident, Wikipedia says it is probably apocryphal.  Another source flatly states that it never happened because the first known written reference to it was written 80 years later by a Thomas Fuller, who sometimes fabricated stories.  Of course, it is a non sequitur to go from "this person sometimes fabricates stories" and "this person was the first person we know of to write about it" to "we know this never happened", so I disagree.  You can read Thomas Fuller' account of the incident here.  Actually, it isn't so much an account as a mention.  Quoting Fuller: "his introduction into court is said to have born an elder date, from the time he spread his plush cloak for Her Majesty to step upon over a wet place".  It seems like Fuller is stating hearsay or repeating tradition.  As it is written, I don't know why he would have made it up himself.  For myself, I would rate the incident to be about as likely to be true as any bit of remembered information about a historical person, or any other unproven statement made in the book it was included in.