THE STATISTICAL TECHNIQUE OF BEING MAXIMALLY INFORMATIVE
Maximally Informative Methods are used in multiple statistical analyses.
The maximally informative method is the approach of solving a problem in the minimum possible steps regardless of the different possible conditions: worst or best-case scenarios. To understand the method precisely, we will be making the applications of this method in solving some puzzles.
Find the marked card.
I have picked 16 cards from the 52 cards of a deck, comprising Kings, Queens, Aces, and 2s of all suits. I have marked one of these cards without preference, and you need to find the marked card by asking the minimum number of yes/no questions.
Some of the questions you can ask are -:
Is the marked card Ace of hearts?
Is the marked card a King?
Is the marked card a red coloured card?
A maximally informative question will be the one that will provide you with the same amount of information regardless of my answer. In statistical language, both my answers ‘yes’ and ‘no’ must be equally likely. Here, 3rd question is a maximally informative question as it eliminates the possibility of cards either being red or being black hence eliminates half of all possibilities.
So, how many maximally informative questions will you need to ask?
Since the question can have only two possible answers asking a maximally informative question will eliminate half of all the remaining possibilities. So possibilities will reduce from 16 to 8 to 4 to 2 to 1. So you will have to ask four maximally informative questions to find the marked card.
WHY SHOULD A MAXIMALLY INFORMATIVE APPROACH BE VALUED HIGH?
On average, a maximally informative approach helps you reach a conclusion in the minimum possible number of steps. Returning to the previously mentioned questions, the 1st approach may conclude in just 1 question, but it is 15 times more likely to not conclude in just 1 question. On average, it will take 8.5 questions to conclude using that approach. Using the 2nd approach, it will take you 2.5 questions to reduce possibilities to 4 cards by finding if the marked card is a King, Queen, Ace or 2. While using the maximally informative approach, you will always find the marked card in 4 questions.
The most informative questions are the ones whose answers we are most uncertain about.
For 1st question, we know that 15 out of 16 times, the answer will be ‘No’.
For the 2nd question, we know that 3 out of 4 times, the answer will be ‘No’.
For 3rd question, we have maximised the uncertainty of the answer, and hence it is the most informative question.
FINDING THE FAKE COIN
There are four coins, out of which one may or may not be fake. You need to identify the counterfeit coin if any. You are given one original coin and a measuring scale to compare the weights of coins. What is the minimum number of times you need to weigh?
A counterfeit coin can either be heavy(+) or light (-). Let the coins be marked as 1,2,3, and 4. Then there can be the following possibilities -: 1+,1-,2+,2-,3+,3-,4+,4-, All original(A) making a total of 9 possibilities. (The notation 1+ means that there is a possibility that a coin marked as 1 is counterfeit and more weighted than original coins.)
There are 3 possible outcomes for a measuring scale -: either left more weighted or right more weighted, or both sides are equally weighted. If we use a maximally informative approach, all the 3 outcomes must be equally likely. Then, in the 1st weighting, you took 4 coins. Will you include the original coin provided or not?
Let’s weigh with 1 and 2 on the left platform and 3 and 4 (case-1) or 3 and original (case-2) on the right platform.
Case-1 Original Coin not included (Coin 1 and 2 on left and 3 and 4 on the right.)
Left weigh more : 1+,2+,3-,4-
Clarification on how we reduced the possibilities -: If left weigh more it implies that 1 and 2 combined are more weighted than 3 and 4 combined but only 1 of these are fake. If 1 is fake then 1 will be more weighted (1+), same goes for 2 and if 3 is fake then 3 will be less weighted (3-) and same goes for 4.
Right weigh more : 1-,2-,3+,4+
Equal: A
This is not a maximally informative case as all outcomes are not equally likely, or we are more certain about the results.
Case-2 Original Coin included (Coin 1 and 2 on the left and 3 and original on the right.)
Left weigh more: 1+,2+,3- (Weigh 1 and 2 to conclude)
Right weigh more: 1-,2-,3+ (Weigh 1 and 2 to conclude)
Equal: A,4-,4+ (Weigh original and 4 to conclude)
Including the original coins makes the equally likely situation for all the outcomes. Hence leading to a maximally informative case with each outcome leading to 3 different possibilities.
Using a maximally informative approach, in this case, we can reduce possibilities from 9 to 3 to 1. So, we will need to weigh just 2 times to find the counterfeit coin or verify that there are no counterfeit coins.
I hope this blog helped you develop a good understanding of the technique of being maximally informative. To better understand the concept, try the above puzzle with 40 coins. If you found the knowledge interesting and useful, share this with your friends.
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