The beginning of the article is an excellent (though conventional) introduction to Bayes and Bayesian updating, but what I found original and have never seen expressed so clearly were the remarks about dishonest discussion tactics, especially the first one:

Formulate a vague hypothesis H1 so broad that it is not extremely unlikely (thus has non-negligible prior, P(H1) > 0), then in the updating step sneakily introduce a much more specific hypothesis H1’ that is far less likely a priori, but that yields a high probability for the evidence P(E|H1’).

If the audience doesn’t notice the bait and switch, it’ll come away with the impression that the evidence strongly supports H1/H1’ vis-à-vis H0, when it actually doesn’t.

Reminiscent of the Motte and Bailey fallacy.


Comment under the article:

  In particular, the geometry of the lottery tickets seems important. It is plausible that 55
  numbers are set up in a 7 x 8 matrix pattern, with one wildcard (to produce 56 objects). Imagine 
  that it is done as such:
   *,  1,  2, .. ,  7
   8,  9, 10, ..,  15
  16, 17, 18, … ,  23
  24, 25, 26, 27…, 31
  then multiples of 9 are the main diagonal. That would explain frequency in an easy way.
Most useful takeaway from this article: If you choose to play the lotto, don't use cute numbers. Odds are high that other people will too, and you'll have to split the pot.

> Part of the explanation surely lies in the unusually large number (433) of lottery winners

> But on the previous draw of the same lottery ...

> the unremarkable sequence of numbers {11, 26, 33, 45, 51, 55} were drawn ...

> and no tickets ended up claiming the jackpot.

This article started off really clear and straightforward, then it dove into stuff that I clearly wasn't going to be able to follow, and upon skipping it I arrived at the explanation which was was plainly obvious from common sense.
H''''': The lottery is run by corrupt officials, who want to cover their tracks. In order to do so they want to have many associates win so no single winner is subject to scrutiny. They need to communicate to this diverse group of conspirators the correct numbers to choose in a simple way, say: multiples of 9, or anti-diagonal of ticket.

Remark 4: The human mind is an amazing hypothesis generating machine. If it also knows about Bayesian statistics, it is capable of accounting for it. Paranoia is a bitch.

An alternative take via The Conversation [1]:

> Based on anecdotal evidence from other lotteries, this number may not at all be unusual. We also need to consider the many thousands of similar lotteries drawn around the world each year, almost all of which receive no international press. While such outcomes are highly improbable for any given draw, the huge number of total lotteries means it’s actually quite likely at least one of them will produce a remarkable outcome by chance alone.

[1] https://theconversation.com/433-people-win-a-lottery-jackpot...

There have been a few national American lotteries in the past handful of years with a jackpot so big ($1.5+ billion) that it was statistically advantageous to buy every number sequence; the total price of the lottery tickets was less than the jackpot. The problem with this lotto hack was that it was not humanly possible to buy that many tickets. Also, if even 1 other person won, the person would be in the hole some 1/2 a billion $ after splitting the winnings.
Interesting question for this lottery draw.

1 - How about past lottery dates. Is there also a high purchase of these same sequence of numbers?

2 - Who are the 433 winners? Is there an unusual clustering of the winners? Was it mostly purchased by a single entity?

A few other ways to thing about the lottery question. What are the chances this sequence would occur in the entire history of the lottery to date? What are the chances that any sequence of multiples of 9 would get drawn? What are the chances that any sequence of multiples of any number would be drawn? What are the chances that any simple mathematical progression of numbers of any kind would be drawn, again in the history of the lottery not just on one day?

I play an online game called Axis & Allies 1942 Online. It uses dice for resolving battles, and in a typical game you might roll many dozens of battles and several handfuls of virtual dice for each battle. We regularly see people go on Discord or the Steam forums to complain that some extraordinarily unlikely outcome happened to them. The thing is with thousands of people playing the game, typically in several games at a time, each rolling hundreds of handfuls of dice every day, one in 10,000 odds outcomes that seem extreme are going to happen on a daily basis to someone. Often several someones a day. Every now and them one of those people is likely to go and complain about it online, so IMHO this is an expected outcome. So far this argument doesn't seem to have convinced many of the 'victims' though.

The odds of N-1 numbers being “remarkable” diminishes for each number drawn. Unless remarkable can be conjured and retconned for every number combination or every Nth number drawn. Each combination is equally likely before the first number is drawn. And equally likely at the end of the draw. But odds change when you have a partial result that’s already ruled out possibilities for each drawing.

You can’t have 9, 18, 27, 36, 45, 54 if you draw an 8. The odds of that are zero.

There's a lot of math here, but what stands out for me is very simple. If you're going to play the lottery, play numbers that have the lowest odds of others playing them too. So any cute sequences, valid dates and so on are right out. You have the same chance of winning with truly random, weird numbers. But you increase your odds of not having to share.
For any six numbers that can exist or be drawn, once they are drawn can find something about them that is improbable.

Sooooo, that found something improbable about the numbers, once they have been drawn, means essentially nothing.

That reminds me that I finally need to start participating in lotto, just to have a skin in the game.

Not participating - zero chance of winning.

Participating - 1 in 30 million chance of winning 4m USD.

Maybe they just got the numbers from a fortune cookie?
ELI5 Bayesian theory. https://youtu.be/HZGCoVF3YvM
It’s an interesting read but the author could have just used Kolmogorov complexity to explain all this.
And yet what Mr. Tao misses is his own assumption that the drawing of each number is truly independent.