I want you to imagine a world consisting of a sequence of independent and identically distributed random variables $X_{i}$ , and two computer programs.

The first program is called Reporter. As input, it accepts a bunch of the random variables $X_{i}$ . As output, it returns a list of sets whose elements belong to the domain of the $X_{i}$ .

The second program is called Audience. As input, it accepts the output of Reporter. As output, it returns a probability distribution.

Suppose the $X_{i}$ are drawn from the following distribution:

$P (X = x) = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ \begin{matrix} 1 / 2 & x = 1 1 / 4 & x = 2 3 / 16 & x = 3 1 / 16 & x = 4 \end{matrix}$

We can model drawing a sample from this distribution using this function in the Python programming language:

import random

def x():
    r = random.random()
    if 0 <= r < 1/2:
        return 1
    elif 1/2 <= r < 3/4:
        return 2
    elif 3/4 <= r < 15/16:
        return 3
    else:
        return 4

For compatibility, we can imagine that Reporter and Audience are also written in Python. This is just for demonstration in the blog post that I'm writing—the real Reporter and Audience (out there in the world I'm asking you to imagine) might be much more complicated programs written for some kind of alien computer the likes of which we have not yet dreamt! But I like Python, and for the moment, we can pretend.

So pretend that Audience looks like this (where the dictionary, or hashmap, that gets returned represents a probability distribution, with the keys being random-variable outcomes and the values being probabilities):

from collections import Counter

def audience(report):
    a = Counter()
    for sight in report:
        for possibility in sight:
            a[possibility] += 1/len(sight)            
    d = sum(a_j - len(a) for a_j in a.values())
    return {x: (a_i - 1)/d for x, a_i in a.items()}

Let's consider multiple possibilities for the form that Reporter could take. A particularly simple implementation of Reporter (call it reporter_0) might look like this:

def reporter_0(xs):
    output = []
    for x in xs:
        output.append({x})
    return output

The pairing of audience and reporter_0 has a Very Interesting Property! When we call our Audience on the output of this Reporter, the probability distribution that Audience returns is very similar to the distribution that our random variables are from!^[1]

>>> audience(reporter_0([x() for _ in range(100000)]))
{1: 0.500

...

Deception