Chess will never be solved, here's why

It's a good question as an illustrative example. As always, prior beliefs matter (the general pattern is that the amount they matter should go to zero as the amount of data goes to infinity).

Pretend we haven't seen any data yet. A prior belief consists of a probability distribution on the unit interval, indicating how likely we think is is (before we have seen any data) that the coin is loaded in a way that would make it come up heads a particular fraction of the time.

For example one prior belief might be very specific - that there is a 98% chance that the coin is fair, a 1% chance that the coin has two heads and a 1% chance that the coin has 2 tails. This is an extreme state of belief because it is inconsistent with the possibility that the coin has a heavy side, so that it comes up heads 55% of the time, or whatever. It's generally best not to exclude any possibility that is not absolutely certainly not true.

At the other extreme, a very neutral prior belief (in one sense) would be a uniform distribution. The possibility that the coin is weighted so that heads comes up a proportion p of the time is independent of p. i.e. 90% heads is exactly as likely as 30% is exactly as likely as 50% heads and so on (it can be easier to think of this in intervals - eg. it is equally likely that p is between N/100 and (N+1)/100 for any N from 0 to 99.

It's better to pick something like that (there are sophisticated reasons to pick something slightly less uniform, but it's not a big deal).

Anyhow, then we do the Bayesian inference. We first work out the probability of the evidence given the prior probability for ever probability p. Then we apply Bayes rule.

The evidence was that we got 106 heads. Here is the probability distribution for p given that evidence compared to the distribution before the evidence (the red line which is flat - a uniform probability). What you see is that 106 elements of evidence is enough to be pretty sure the probability of a head is quite close to 1 (unlikely to be as low as 0.9, say).

We can actually estimate the probabilty of a head by completing the inference by averaging over all possible values of p. I have put the answer in the title of the graph:

So the answer to your question is that with these neutral assumptions, we would think there was just under 1% chance of the next result being a tail. This makes intuitive sense. If you do this in an analytical way, you can think like the following - it's rather neat:

First we have no data. So let's make a prior by pretending we have a tiny amount of data. Say half a sample which was a head and half a sample which was a tail.

Then when we have our real data, we add the data to our prior data and we have 106.5 heads and 0.5 tails, which we view as indicating the true probabilities - i,e, just over 99.5% chance of heads.

(What I did was more like starting with a prior of 1 head and 1 tail. For technical reasons 0.5 and 0.5 is considered a slightly better choice).

With this viewpoint, you can see that the more data you have, if it's all heads, the nearer the probability gets to 1. But it never gets there. This is appropriate as it is obvious that no amount of data excludes the possibility that there is a low but finite probability of tails. It just gets better at excluding larger probabilities of tails from being likely.

The sad thing is that @Optimissed has just got something half right and I had credited him for it.

I bet it is not a full mute he prob just swore on accident .. Long quash still hasn't gotten unmuted or come back

Half-right? That sounds pretty good - for him.
He may only be 24 hour robo-muted.
If so - he'll be 'tactically back' looking to exploit his mute.

Here's an illustration why lots of draws between Stockfish 16 with a monkey on its back against Stockfish 16 with a different monkey on its back does not prove much. [And the word "prove" is important].

Suppose Stockfish assesses 1. d4 as being 0.2 better than 1. e4 at its maximum search depth. Both players play 1. d4 all the time and each player plays well enough to draw. But meanwhile it was true that at a much greater depth, twice what Stockfish reached, and via some less than obvious choices, 1. e4 was a very difficult forced win. The evidence shed absolutely no light on this because every single game started 1. d4.

That is a simplified example, but the same applies to any winning line that the engine simply can't see at its maximum depth. This is not likely, but it is logically possible.

'Stockfish with a monkey on its back'
Hahahahaah.
I think it might be interesting to have these matchups between computers - where earlier generation engines play the latest engines ...
both with and without GM 'monkeys' involved.
If tygxc is 'right' or is to 'look right' then the weaker engines should be able to draw against the latest ones.
Its like comparing infinities.
Note that if the weaker engines continue to lose as the years pass -
that would Torpedo tygxc.
Hey how about its already the case?

Without mistakes that are Caught.
Without mistakes that are caught but insufficiently exploited for the Win.
Will engines prove that many famous draws between top players were actually wins? Or perhaps wins for either side depending on what point in the game is being considered?
Maybe that's already been done.
'Famous draws that were wins' ...
Doesn't Leap Out on google search just now.
Draws often don't get much attention.
People are interested in wins - not draws.

Totally false. The best chess games end with me winning. (Not worried about how long.)

translation of tygxc's posts: 'i have no evidence, these engines are just reallly strong, ok?'

"Say I toss a loaded coin 106 times and it lands heads 106 times.
If we now toss the loaded coin 30 more times, what is the probability it lands tails at least once?"

you just claimed earlier "The outcome of 106 ICCF WC Finals games is no random event,"

you are contradicting yourself, as per usual.

I didn't quite get round to answering his precise question, but principled Bayesian reasoning gives something like a probability of 0.15.

the fact that he even talks of probability is proof in itself that his assertions are fantasy.

@8670

"Stockfish 16 with a monkey on its back"
++ This is utterly disrespectful and shows zero understanding of correspondence play.
It is rather a human jockeying multiple engines.
Equestrian sports is not horses with humans on their backs.

It is not all Stockfish, he may also consult Komodo or another engine.

The human decides on the move after consulting his engines.
Stockfish is not Stockfish: there are tunable parameters like 'contempt'.
The human decides on the depth and width of the computer search: he can choose a deep search even up to the 7-men engine table base, or he can choose a broad search keeping all reasonable candidate moves.
The human decides how long to spend on a move. There are 50 days for 10 moves, but the human can decide to spend e.g. 41 days on one move and play the other 9 moves in 1 day.
The human can decide to declare up to 45 days leave (and still continue analysis).
The human selects the opening.
The human proposes linear conditionals.
The human proposes / accepts / declines draws.

All this makes ICCF correspondence much stronger than engine vs. engine.
If the strongest human is 2900 and the strongest engine 3600, then the strongest correspondence play is 4300.
You can about double the ICCF rating to convert to an equivalent over the board FIDE rating.

@8666

"just over 99.5% chance of heads"
++ The chance of heads lies between 1 and 106/107.

As for the 30 ongoing ICCF WC Finals games the chance of a decisive game lies between 0 and 17/136, based on human errors in the previous ICCF WC Finals.

That is for 1 loaded coin toss, or 1 ongoing ICCF WC Finals game.
For 30 loaded coin tosses:
1 - (106/107)^30 = 24.5%
For 30 ongoing ICCF WF Finals games:
1 - (119/136)^30 = 98.8%
So I expect at least 1 of the 30 ongoing ICCF WC games to end decisively because of some human error. Stay tuned, we will see how the 30 ongoing games end.

However, the most plausible explanation for the 106 draws out of 106 finished ICCF WC Finals games is an error distribution of 106 - 0 - 0 - 0.
I am willing to accept the possibility of 105 - 0 - 1 - 0. But then Chess is still a draw,
and the games still are part of a weak solution because of redundancy,
as more than one black way was found to draw after 1 e4, 1 d4, or 1 Nf3.

@8678

"even talks of probability"
++ Our world is probability. Thermodynamics and quantum mechanics are probability.
All you see, hear, feel, smell, or taste is probability.
Atoms, electrons, fotons act as they do because that is most probable.
The Schrödinger equation and the notion of Entropy are probabilistic.

But maybe he has that red telephone ...
Yes those engines are getting stronger and stronger ...
but I'm suggesting that todays engines would lose to engines five years ago ...
if you trace it back you're going to find that engines got stronger ...
obviously - 
but I wonder if the point is clear:
Engines keep proving that the previous engines weren't perfect.
Connect the dots ...
Another way to put it:
there's a big huge Generic side of maths and science.
And that generic side is the Primary side. Not the brand name stuff.
Generic.
I'll post internet definitions of a couple of terms.

ge·ner·ic /jəˈnerik/ 
adjective 
1. characteristic of or relating to a class or group of things; not specific.
"chèvre is a generic term for all goat's milk cheese" 
Similar: general common collective nonspecific inclusive
all-inclusive all-encompassing broad
comprehensive universal
cross-disciplinary interdisciplinary multidisciplinary
BIOLOGY relating to a genus. 
noun 
a consumer product having no brand name or registered trademark. 
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Point: No brand names.
Does 'scientific method' put a brand name on science?
Unfortunately many will act as if it does.
Does tygxc understand 'generic'?
In fairness I have to say that tygxc is not a science denier and good for him.
But even among those who do not deny science - when trying to 'box it' one could go astray as it were.
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Now 'Bayesian' is next.

"I am willing to accept the possibility of 105 - 0 - 1 - 0. But then Chess is still a draw"

thats not how proof works, sorry. that doesnt bring us any closer to solution.

Regarding 'Bayesian' and its origins
The term is named after Thomas Bayes who was born in 1701 and attended the university of Edinburgh.
"In probability theory and statistics, Bayes' theorem (alternatively Bayes' law or Bayes' rule), named after Thomas Bayes, describes the probability of an event, based on prior knowledge of conditions that might be related to the event.[1] For example, if the risk of developing health problems is known to increase with age, Bayes' theorem allows the risk to an individual of a known age to be assessed more accurately by conditioning it relative to their age, rather than assuming that the individual is typical of the population as a whole."
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Thomas Bayes was apparently to the mathematics of probability what Pythagoras was to geometry.
So then there's a slight issue ...
is it necessary to specify 'Bayesian' in the math being referred to?
Obviously there would be non-Bayesian math ...
but probability math continues to be what it is with or without Bayes being referred to.
And it continues to be what it is without Laplace being referred to either.
Notable that Laplace survived the french revolution which Lavoisier did Not.
And probability math and statistics continue to be what they are - without the word 'epistemological'. Yes there's 'ontological' too. And other terms.
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Point: the situation is a generic one.

MEGA is obviously well informed and educated - something like how Elroch and Dio and MARattigan and mpaetz are ...
But I think tygxc is getting a kind of 'advantage' with this '30 more games' business.
If he can make it 'about that' and create a kind of knife edge and the whole thing is to stand or fall on that then he's managed to create an artificial issue.
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Much more to the point would be whether engines of five years from now can and could or would or do beat today's engines.
Then don't his arguments go out the window?
But I can do even better than that.
The process is already established.
Engines of today would or do beat engines of decades ago.
Right?
Where's tygxc's evidence that that doesn't continue?
New stars are discovered constantly - does that mean that will stop tonight?
Point: the engines are imperfect - so draws don't prove.
tygxc could have been a kind of 'forensics lawyer'.
He could get up there in front of the jury and quote this or that postulation or finding by scientists - using their names too - and then say 'You see? The science evidence is Crushing and you Must Acquit'
I can see it now.