Hot hand theory

Contrary to previous opinions the hot hand theory may actually exist.
The Conversation - US: Momentum isn't magic – vindicating the hot hand with the mathematics of streaks. http://google.com/newsstand/s/CBIw6_aw2DQ

Streaks of hot and cold athletic performance? Sure! The human body isn't a constant and may be better or worse at a task at a particular time depending on many factors.

Streaks for gambling? Nothing but random coincidence.

Quote: Hunterhill

Contrary to previous opinions the hot hand theory may actually exist.
The Conversation - US: Momentum isn't magic – vindicating the hot hand with the mathematics of streaks. http://google.com/newsstand/s/CBIw6_aw2DQ

Quote from the paper:

The world HHHT, in which the researcher has fewer eligible flips besides the chosen flip, restricts his choice more than world HHHH, and makes him more likely to choose the flip that he chose. This makes world HHHT more likely, and consequentially makes tails more likely than heads on the chosen flip.

In other words, selecting which part of the data to analyze based on information regarding where streaks are located within the data, restricts your choice, and changes the odds.

The complete proof can be found in our working paper that’s available online. Our reasoning here applies what’s known as the principle of restricted choice, which comes up in the card game bridge, and is the intuition behind the formal mathematical procedure for updating beliefs based on new information, Bayesian inference.

In another one of our working papers, which links our result to various probability puzzles and statistical biases, we found that the simplest version of our problem is nearly equivalent to the famous Monty Hall problem, which stumped the eminent mathematician Paul Erdős and many other smart people.

An 11 percentage point relative boost in shooting when on a hit-streak is not negligible. In fact, it is roughly equal to the difference in field goal percentage between the average and the very best 3-point shooter in the NBA. Thus, in contrast with what was originally found, GVT’s data reveal a substantial, and statistically significant, hot hand effect.

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(1) The "mainstream science" view of people who are not very sophisticated in probability is "hot hand does not exist".

A lot of this nonsense comes from learning probability from K-12 and college math teachers who never got very far in probability & statistics (e.g. maybe they studied less than 1 semester of probability in their own college studies, or only learned probability as part of another class).

Learning from outdated material. Bayesian Networks were invented in 1985 by Judea Pearl at UCLA.
Your math teachers may have learned their material from before this time.

(2) For those who know a bit more about probability, "mainstream science" also has the "principle of restricted choice".
https://en.wikipedia.org/wiki/Principle_of_restricted_choice

(3) Bayesian networks show that most people trying to apply mathematical rules to "evidence" at multiple dates, usually mess up & argue & debate. Common issues in unusual situations are "ignoring some evidence", "double-counting evidence", etc...
https://en.wikipedia.org/wiki/Bayesian_network

With vast amounts of evidence required to predict diseases, make self-driving cars, etc... people use Bayesian nets to guide their use of probability & evidence.

(4) A lot of common paradoxes, and unusual problems discussed on this board
...should have been covered in a 1st semester (or 2nd semester) "introduction to probability" or "probability for non-mathematicians/non-scientists" class, or a 1st semester statistics class.

Godel's incomplete theorems ...are a possible 1st semester or 2nd semester undergraduate class.

By the time you are into 2nd year undergraduate probability/statistics classes, you should be well beyond these parlor game paradoxes & debates.

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...but popular press in the US is written for the 4th-6th grade level.
And the math/science background of most reporters is pretty damn low.

Even the scientific writers for Scientific American, who are pretty good, fall hook-line-and-sinker for the Academic politics games of the scientists they interview. Some of their stories are totally "hocus". Gobbledegook from the powerful "mainstream scientists" - often the results of 1-2 main groups, ignoring the other 3-4 less powerful groups.

This post may get me into very hot water with other people here.
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Hot Hand in coin flips.

(A) We flip a supposedly fair 50/50 coin 5 times and it lands Heads 5-times in a row. What are the odds of the next coin flip being Heads? Is it 50%?
(B) We flip a supposedly fair 50/50 coin 5 times & we don't know the results of the flips. What are the odds of the next coin flip being Heads? Is it 50%?

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I'm not going to answer for either case, ...which hopefully will get me into less "hot water". :-)

What I will ask is:
(1) What is the value of the information difference in the two situations? Does it affect the answer?
In case A, the coin has already landed "Heads 5-times in a row."
(2) Has situation A been cherry-picked out of many possible situations? Was it obtained by a true random test? Or not? This affects "a priori" knowledge, where two people arguing may have different "a priori" assumptions, thus different conclusions.
(3) Is this coin fair? We need to know the "a priori" possibilities that someone has slipped in an unfair coin. We need more information.

One of the main things which happens in math training at universities (undergraduate & graduate) is that people learn to break-down situations and systematically look at MORE and MORE of the information, whether stated, unstated, or assumed.

(4) Did the coin get damaged or changed in the process of the flips?
(5) Is it a coin which will land on the same side every time, after the first flip primes it? (Trick coin).
(6) Has some craps dice-setter practiced perfect flips with coins, and can control which side lands up?
(7) etc...

Higher math is about going to ridiculous lengths to classify MORE and MORE possible things about seemingly common situations.

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Consider these variations:
(C) We flip a supposedly fair 50/50 coin until we see 5 Heads in a row. What are the odds of the next coin flip being Heads? Is it 50%?
(D) This is a one-time experiment. We flip a supposedly fair 50/50 coin until we see 5 Heads in a row. What are the odds of the next coin flip being Heads? Is it 50%?
(E) I walk up to a person. We have not flipped any coins. We flip a supposedly fair 50/50 coin 5 times and it lands Heads 5-times in a row on the 1st five flips. What are the odds of the next coin flip being Heads? Is it 50%?
(F) I walk up to a person in NYC Central Park who has a table & is trying to make some money. We have not flipped any coins. We flip a supposedly fair 50/50 coin 5 times and it lands Heads 5-times in a row on the 1st five flips. I bet $5 on Heads. What are the odds of the next coin flip being Heads? Is it 50%?
(G) I walk up to a person in NYC Central Park who has a table & is trying to make some money. We have not flipped any coins. We flip a supposedly fair 50/50 coin 5 times and it lands Heads 5-times in a row on the 1st five flips. I bet $500 on Heads. What are the odds of the next coin flip being Heads? Is it 50%?
(H) We flip a supposedly fair 50/50 coin until we see 5 Heads in a row. We see 15 flips before five Heads appear. What are the odds of the next coin flip being Heads? Is it 50%?
(I) We flip a supposedly fair 50/50 coin until we see 5 Heads in a row. We see 847 flips before five Heads appear. What are the odds of the next coin flip being Heads? Is it 50%?
(J) We flip a supposedly fair 50/50 coin 500 times and it lands Heads 500-times in a row. What are the odds of the next coin flip being Heads? Is it 50%?

What is the information difference between situations A-J? Does it affect the answers?

If this were a course on Bayesian Networks, can you create a network which models all of situations A-J?
What are the unknown parameters & assumptions you need in situations A-J in order to answer the questions?

P.S. A probability/statistics person would probably not answer 50% or 0.5.
If they made a more complete answer, they might say "Under assumptions AA-EE, extrapolating missing information with guesses FF-ZZ, with X% confidence, I estimate this will occur with probability/likelihood Y%, given that I had time T to approximate the answer using a average/combination of the results from models AAA-FFF."

Quote: mamat

Hot Hand in coin flips.

(A) We flip a supposedly fair 50/50 coin 5 times and it lands Heads 5-times in a row. What are the odds of the next coin flip being Heads? Is it 50%?

Yes

Quote: mamat

(B) We flip a supposedly fair 50/50 coin 5 times & we don't know the results of the flips. What are the odds of the next coin flip being Heads? Is it 50%?

Yes

Quote: mamat

I'm not going to answer for either case, ...which hopefully will get me into less "hot water". :-)

I think you're safe, unless someone wants to stroll down the maturity of chances avenue. The answers are predicated on the assumption the coin is fair, as stated.