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.

"New Symbols for Base-16 and Base-256 Numerals" Bested on Reddit

In 2017, I co-authored a paper that presented a system of hexadecimal digits that could be combined as ligatures to form base-256 digits.  It's online here.  In 2019, someone posted a system of base-256 digits on Reddit that I and the other author agree is better: Made a surprisingly simple base-256 numerical system...

In our paper, we enumerated 9 desirable qualities for a set of hexadecimal digits in order to evaluate different sets.   This other system has/hasn't the qualities as follows:

• MNE: Yes
• STR: No
• LIG: Yes
• AMB: Yes
• DSP: Yes
• BIN: Yes
• 0: No
• 1: No
• TRN: Yes

This system has 6 of the 9 enumerated qualities.  This is 2 fewer qualities than our published system, but I nevertheless think it is a better system because 1) its base-256 "ligatures" are easier to write and 2) I can remember how these digits encode binary numbers much more easily than our published system, and therefore this system would be easier for me to use.  Well done, u/Tuckertcs, whoever you are.

On another topic, in our paper we suggested the name "sedecisedecimal" for base-256 numbers.  I still think it's the best Latin/Greek-based name for them.  Would anyone seriously prefer "duocentehexaquinquagesimal"?

Bill Watterson's Political Cartoons

While attending Kenyon College, Bill Watterson (the author of Calvin and Hobbes) was editorial cartoonist for the school newspaper, The Kenyon Collegian.  Most weeks during his tenure, they printed a cartoon of his, usually a political cartoon.  Since the Kenyon Collegian's archives are available online, you can see them all.  The following is a complete set of links to issues of the Kenyon Collegian that have these cartoons: 

• Volume CIV (1976-1977), issues 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27
• Volume CV (1977-1978), issues 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26
• Volume CVI (1978-1979), issues 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24
• Volume CVII (1979-1980), issues 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27

It's Rational to Believe in Jesus

So says Aron Wall (and me).  He has posted a fair amount of material on his web site that explains how believing what the Bible says about Jesus can be rational.  Several of those postings amount to as good a case for Christianity as I've seen expressed anywhere, so I'm posting links to them for your benefit (and my own; I will re-read them).  They are listed in an order that I think makes sense for them to be read if they were to be read in sequence.

---

There is good evidence that Jesus was resurrected from the dead and that he is the Messiah:

1. Can Religion Be Based on Evidence?
2. Theology: Less Speculative than Quantum Gravity
3. Christianity Is Based on Observations
4. Let Us Calculate
5. Christianity Is True

The Old Testament part of the Bible is not a fabrication:

• Some Comments on Biblical History

Christianity is more likely to be true than other religions:

1. Introduction
2. World Evangelism
3. Ancient Roots
4. Supernatural Claims
5. Historical Accounts
6. Early Sources

---

Note: I don't know Aron Wall personally; I've just read his blog and agree with his approach to Christian apologetics.

How to Transfer User TonePrints from One Windows PC to Another

This post is for anyone who has TonePrints for a TC Electronic effect pedal on one computer and wants to transfer them to another computer.

Version 4.1.06 of the TonePrint application stores user TonePrints in a file named "database.sqlite3" in the user's app data folder, which, in Windows 10, is a hidden folder named "C:\Users\[user name]\AppData\Roaming".  If you want to replace your user TonePrints on one computer with user TonePrints from another computer, just copy this file from the one computer to the other.

If you want to copy TonePrints from one PC to another, but you want to keep your old TonePrints, too, simply copying the file won't work, because it will overwrite TonePrints.  For that purpose, I wrote some scripts for Windows PC that will grab TonePrints from one PC and install them on another PC without deleting the TonePrints that are already there.  You can download them in a zip file here:

• TonePrint_transfer.zip

To use them, download the zip file, extract the contents to an empty folder, then follow the instructions in the file named "README.txt" that is included in the zip file.  They should work with version 4.1.06 of the PC TonePrint application.

I don't know how to transfer user TonePrints from a smartphone to a PC, or vice versa.  The smartphone app might have a database.sqlite3, but I have not been able to locate it on my phone.

UPDATE: the scripts won't work for new versions of the TonePrint application (e.g. 4.4.03).  The database file name apparently changed to "databaseV1.sqlite3" or something.

How to Build & Run Vampire ATP in Cygwin

Vampire is an Automated Theorem Prover.  It was apparently developed on Linux, but it can be run on Windows using Cygwin, which provides the build tools, software packages and operating system services that Vampire depends on.  Here are some instructions for building and running Vampire in 64-bit Windows 10.  A note on building Vampire with Cygwin can be found on Vampire's GitHub page, here.

Install Cygwin:

1. Install Cygwin and its default packages.
2. Install these Cygwin packages:
   
   • cygrunsrv
   • gcc-core
   • gcc-g++
   • make
   • zlib
   • zlib-devel
   • zlib-devel
   • zlib0
3. Move C:\cygwin64\bin to the front of the %PATH% environment variable.

Install Git Command-Line Tools

1. Install the Git command-line tools.  An installer can be downloaded from https://git-scm.com/download/win.

Install the Z3 Theorem Prover

1. Download a Win64 release of Z3 from https://github.com/Z3Prover/z3/releases.
2. Make z3.dll available from PATH, either by putting z3.dll from this release in some folder that is in PATH, or by adding the Z3 "bin" folder to PATH.

Build Vampire

1. Pull Vampire's Git repository from GitHub (https://github.com/vprover/vampire).
2. In Vampire's Makefile, add "-D_GNU_SOURCE" to CXXFLAGS and CCFLAGS.
3. Change all instances of "exit(0);" in the source code to "_Exit(0);".  This is needed to prevent the child processes forked by the main Vampire process from calling the main process' atexit handlers.
4. Create a folder named "include" in the Vampire repository's main folder.
5. Copy the contents of the Z3 "bin" folder to that "include" folder.
6. Compile and link vampire by issuing the command "make vampire_z3_rel" from a command prompt.

Start the Cygwin Windows Service

1. Start Cygwin's bash shell.
2. From the bash command prompt, issue the command "cygserver-config" to configure the Cygwin Windows service.
3. In Windows, right-click "This PC", select "Manage", then select "Services and Applications, then select "Services", then start the "CYGWIN cygserver" service.

 Run Vampire

1. Run Vampire from a command prompt.  For example:
   vampire_z3_rel_master_4055 --mode casc -t 300 "Problems\SET\SET159-6.p"
2. Vampire may not succeed in deleting the semaphores that it creates in the Cygwin server.  To delete them, issue the command "ipcs -s | grep '^s\\s' | cut -d' ' -f3|xargs -n 1 ipcrm -s" from a command prompt.  Alternatively, add a call to system() at the end of Vampire's main function that issues this command.