Mathematician wrote two long, much appreciated comments to Uniformly and/or Randomly Driving Towards One Half. I'll reproduce them here, with only very slight formatting changes (I am not in favour of gaps between the final word in a question and the question mark). My response follows below.
Mathematician Comment 1:
I'm glad to see that you are still trying to solve mathematical problems, but I am a little bit disappointed to see that you are still not using words as they should be used and that you still fall into common misconception about probabilities.
Your whole point seems to be that the answer 1/2 is THE answer to Bertrand's paradox, and that other methods of choosing a chord are skewed. I'm not sure that I could explain why this is not a valid point directly, so I'm really going to use your own words to make you see where your errors are.
First, the wording of the problem. Apparently, you seem to think that the sentence:
> If you pick, at random, a line which passes through the circle
is equivalent to the initial formulation:
> If you pick, at random, a chord of the circle
You even say: "I deliberately used different and simpler language, but I don't think that my wording introduces or omits anything of consequence." and this is a major flaw in your reasoning. It has many consequences. I would go even further, the Bertrand "paradox" (which is no paradox at all, but that's another question) is ultimately an example to understand that some sets do not come with a natural parametrization. If you change the wording of the problem by identifying the given set with another new set, then is is entirely possible that the new set has a natural parametrization.
What you are doing with your rephrasing of the problem is that you inadvertently choose a particular parametrization of the set of chords. You are actually identifying the set of chords with the set of lines which passes through the circle. Now, this parametrization is useful and natural, but it is in no way unique. There are many other way to identify the set of chords to some other set. For example, I can identify the set of chords and the set of pairs of point on the circle. I can also identify the set of chords and the set of points inside the circle. Both identifications are perfectly natural and very useful.
Imagine that I ask you the problem in the following way :
> If you pick, at random, two points on the circle, what is the probability that segment between these two points will be longer than the sides of the equilateral triangle?
I deliberately used different and simpler language, but I don't think that my wording introduces or omits anything of consequence. (Does this sentence remind you of something?) Is this wording of the problem less or more natural than your own wording? I argue that both are natural, but both are not equivalent to the original wording ...
It might seem obvious to YOU that your wording is more natural. It might seem more obvious to ME that my wording is more natural. But in the end, there is absolutely no reason to think that one is better than the other.
It seems that I need to cut my comment in two half because it is too long ... so I will get back after a short break
Mathematician Comment 2:
Now after this wording of the problem, you argue that the answer should be 1/2, referring to Jaynes treatment. I agree with you, but I don't think that you understand the words you are using:
> Jaynes appears to be suggesting one, appealing to rotational and scale invariance.
Let me ask you a question. I give you a precise circle. The set of chord of this precise circle is a well-defined set. Why does it mean for a measure on this set to be "scale invariant"?
The "scale and rotational invariance" is meant to be applied to a measure on the set of all lines in the plane. This is a completely different set from the set of chord on a specific circle. Now, what Jaynes meant is that if we choose a parametrization of chords as lines that passes through the circle, then we should put a measure on the set of chords that comes from a measure on the set of lines. Moreover, the measure chosen on the set of lines should not depend on the placement of the circle (translationally/rotationally invariant) and on the size of the circle (scale invariant). And fortunately, there is a unique measure on the set of all lines in the plane that satisfies these properties.
But I can choose a different parametrization of the set of chords, for example by the two endpoints on the circle. Then there is absolutely no reason to think that the measure on the set of chords should come from a measure on the set of lines. And the expression "scale invariant" is meaningless here, because the circle is fixed, and the circle is NOT scale invariant. The only thing that would make sense is "rotationally invariant". It turns out that there is a natural measure on the set of pairs of points on the circle, and it gives you a DIFFERENT measure on the set of chord.
> However, my argument here is that the set of chords selected by the 1/3 method is skewed.
Skewness is a relative notion. In this problem, there is no point of reference which would be the "unskewed" result.
The fact that you can get back the 1/2 answer by pulling one of the endpoint of the chord towards infinity is rather neat. But it is in no way an indication that 1/2 is a better result ...
> This is precisely the problem (in my humble opinion) with the 1/4 method, because Cartesian co-ordinates are used within a circle.
What??? Again your misunderstanding of basic mathematics is surfacing. There is absolutely no reason to use cartesian coordinate to define a uniform probability distribution on the disc. And moreover, you can use cartesian coordinate without "skewing" the result.
But more importantly your method of "randomly choosing a point in che circle":
> we would select, at random, an angle 0 > θ > 2π from the x axis and a distance from the locus of the circle, r, where 0 > r > R
does not produce the result that you probably think it produces. The probability measure that is induced by this process is not the uniform probability measure on the disc ...
If you were familiar with polar coordinates, you would know that the usual area is given by "r dr dθ", but what you did was taking the measure given by "dr dθ" (I'm simplifying the argument because it seems irrelevant to make a full course on measure theory here). Of course, your mistake is "lucky" because it produced the result that you wanted : 1/2. But the mistake is real, and hence the result is relatively meaningless.
In conclusion, your main mistake is to think that there is only one natural parametrization of the set of chords. But even if there were a "best" parametrization, it is irrelevant in the context. You should understand that the Bertrand paradox is not about chords at all ...
Perhaps I should give you another example of a similar problem:
* Pick, at random, a right triangle inscribed inside a circle of diameter 1. What is the probability that one angle of the triangle is less than pi/6?
How would you answer the question?
First off, I have to repeat that I am not a professional mathematician and have no intention of spending another six years or more at university to become a Doctor of Mathematics. The last time I checked, the vast majority of the world's population, maybe even the population of the universe, are not maths docs. Therefore, I think it is not unreasonable that I don't always use the correct terminology agreed to within the mathematical cabal.
It is possible that you are intentionally sending a message along the line of "get a maths doctorate or shut the fuck up" but I don't think you are. Unintentionally, however, this is precisely the message you seem to be sending with some of your comments (some have been far worse, perhaps with intent). I hope you take this with in the spirit with which it is intended - I am curious, other people are curious, and we curious people don't need to be frapped down by supercilious experts for the most minor of infractions. If at all possible, it’s better to get to the meat of our misunderstandings. And I was not being sarcastic when I wrote that your comments are much appreciated.
You make a comment about lines and chords, as if I don't know the difference. Please see A Farewell to the Bertrand Paradox in which I think I make it pretty clear that I understand the difference (I repeatedly wrote "You now have two points on the circle, between which is a chord") - although in retrospect it seems like I don't understand the word "farewell". My implication, although not clearly expressed, is that a line passing through a circle defines a chord and a chord (of length greater than zero) uniquely defines a line. If this is fundamentally wrong (as opposed to just oddly phrased), please advise.
You then get into parameterisation (or perhaps parametrization, if there is a meaningful difference beyond our spelling preferences). To the extent that I understand you, I think I agree. The method you use to define your chords, to select your chords, makes a difference. What we differ on is whether there is a "right" and "wrong" way to select chords to satisfy the Bertrand Paradox, as stated. My position, which I accept may be fundamentally wrong, is that if your method doesn’t arrive at a uniform distribution of chords (by some reasonable measure), then your method isn't "at random".
The question that arises immediately is by what "reasonable measure" can I claim that the 1/2 method produces a uniform distribution of chords and the other two methods don't. Below I may go some way to explain what I had in my head, but first … invariance.
I suspect that we would agree that the distribution of chords in a circle is invariant in terms of rotation, translation and scale. To clarify, imagine a circle with a locus at (0,0), defined by radius of 1 and an orientation such that θ=0 aligns with the positive y-axis (let's set Point A to (0,1)=(1,0), if you get my little joke [and yes, I know it’s more conventional to take θ from the x-axis, but it's just a convention and my convention has the vertex of the triangle, and thus Point A, at the top]).
The distribution of chords in that circle (given the same parameterisation) is not affected if: we move the circle (to a locus at (random(x),random(y)); rotate the circle (to an orientation at random(θ) from the y-axis); or increase the size of the circle (to a radius of random(R)). I think we would agree on that, but I may be missing something.
So long as I am not missing something crucial, therefore, a circle of the type I suggested (locus at (0,0) and (r=1,θ=0) coinciding with (x=0,y=1)) can stand in for circles of any size and location and Point A at (0,1)=(1,0) can represent all possible positions on the circumference, because using any other position on the circumference is equivalent to this circle being rotated. If I am wrong about this, then the following probably falls apart.
Borrowing from myself (at reddit), my "reasonable measure" of a uniform distribution is such that if you drew a representative sample of the chords with an arbitrarily small width (I am aware that chords don't have widths), then the resultant density would be smooth throughout the circle.
Wikipedia has something close to what I am talking about, just before they get into the "classical solution". However, I limit my visualisation of it to one orientation of the circle, so
- 1) put Point 1 at the apex of a [notional] equilateral triangle, draw a set of chords with an arbitrarily small angular separation (say 360 of them 1 degree apart)
- 2) select a radius and extend it out to a diameter, draw the set of chords perpendicular to the diameter with an arbitrarily small separation, say 360 of them R/180 apart
- 3) take an arbitrarily large number of points equally separated within the circle, say 360 of them, draw the set of chords for which the points are their midpoint
I am pretty sure that only 2) will smoothly fill the circle (for example if all chords are drawn with a width of R/360). I am also pretty sure that there will be some arcane argument as to why this either doesn't matter, is not (sufficiently) stringent or is completely wrong-headed - but as I said, it is what I had in mind.
I don't want to spend a ridiculous amount of time on this, so for the purposes of showing this, I will use 16 rather than 360 chords and I am not going to fuss about making the images pretty:
To my mind, the distribution created by method 2 is uniform and smooth in a way that the others simply aren't. Someone did complain that my argument here is apparently based on aesthetics - the method 2 result looks nicer. That's not really my point, my point is that the density of chords is smooth, no matter where you look in the circle. In the other two, either you have a clumping of chords (in method 1 there are more chords surrounding a point directly below the top of the circle [it's actually worse than I've represented]) or (in method 3) chords crossing. I do understand that as you keep rotating the circle to obtain a new set of chords in method 1, as your number of unique sets approaches infinity, the gaps will disappear, but the clumping will remain at the rim of the circle (some seem to want to call it a disc, hopefully I don't confuse anyone by calling it a circle). Similarly, I understand that the gaps will disappear if we consider more and more chord midpoints in method 3, but this introduces a similar clumping effect at the rim of the circle - to a greater extent, by which I mean the density of chords at the rim will be even greater than produced by method 1.
You gave the following challenge: "Pick, at random, a right triangle inscribed inside a circle of diameter 1. What is the probability that one angle of the triangle is less than pi/6?"
I understand that this is the same question, because 2.cos(π/6) is √3, so to be consistent, my answer would have to be 1/2. This is, however, on the understanding that I am picking from an existent set of all right triangles, not a set of triangles created by bisecting the circle and then picking a point at random on the circumference then drawing chords between that point and the two ends of the diameter previously established.
I think you've helped me here though. If we think about the absolute maximum proportion of unique right triangles with one angle less than π/6 that we could draw in the circle, using any method, the answer comes to 1/2. We could, for example, draw the set of right triangles using the 1/2 method for chord selection, then draw a diameter from one end of the chord, then complete the triangle. This, I think you would agree, results in 1/2. Knowing this, it seems odd to me that you can walk back from this figure to 1/3.
I note that, when unconstrained by a circle, can you draw your triangle by first drawing the hypotenuse (say at length 2R), then drawing a line from one end at an angle to the hypotenuse chosen at random such that 0 < θ < π/2 and then completing the triangle by drawing the final side as required. The criterion will be satisfied in the ranges 0 < θ < π/6 and π/3 < θ < π/2, so 2/3 of the time.
Also, as you do this, the right angle vertex describes a circle - meaning that there is a problem with the 1/3 answer. I've drawn that below as well, hopefully it's sufficiently clear (note that I have tried to show pi/3 around the created circle, I am not trying to imply that pi/6 (the angle on the left in the triangle at this point) is pi/3).
Thinking about this did lead me to stumble over another way to create a set of chords: draw a diameter (just line segment of length 2R will do), then repeatedly draw circles around one end of the diameter (or line segment) at arbitrarily small increments until you reach of circle of radius 2R. With each circle, draw the tangent that intersects with the other end of the diameter (or line segment). The points at which these tangents touch each of this series of circles describe a semicircle.
The result obtained by this method is 1/2, despite not appearing to be a reformulation of the classic 1/2 method. Perhaps it is but I’ve simply not worked out how yet, but in a rough modelling (2000 data points) the distribution does not appear to be the same when viewed in histogram form.
On parameterisation, I did some thinking about this along the lines of saying that if you have a 1/3 answer, then it seems (to me) that your selection method must simply have missed some of the chords. In my way of thinking (standard caveat about the possibility of being wrong), if we are asked to select a chord "at random" then it follows that we would be selecting from a set of ALL chords, rather than from a specific subset, unless advised otherwise. Thought from this perspective, our first concern is making sure that we have ALL chords available to select from. The question then is how to express this properly. I'm probably going to mess this up in some obscure way, but if you can at least try to understand what I am saying (and criticise the best formulation of my argument, rather than the worst), it would be appreciated.
I suggest that an expression for ALL chords in a circle defined by x2+y2=1 (in units of R where R is the radius of the circle) goes something like this:
The infinite set S of all unique sets Si of points that fulfil the following criteria:
-1 > c > 1 (defining the y axis intercept of the chord)
0 > θ > 2π (defining the gradient of the chord)
-√((-cosθ)2+(c-sinθ)2) > r > √((cosθ)2+(c+sinθ)2)
(x,y) = (r.cosθ,r.sinθ+c)
Note: the combined effect of these two conditions is (or is intended) to include all and only points between intercepts of the line defined by (x,y) = (r.cosθ,r.sinθ+c) and the circle defined by x2 + y2 = 1, thus defining a chord. In other words a unique set Si is intended to define a unique chord.
When corrected in terms of mathematical terminology, etc, is this a parameterisation and, if so, does it establish or define a structure (per u/Vietoris) for which there is a defined probability measure (per u/Vietoris) or probability distribution (per u/overconvergent)? And, if so, what Bertrand Paradox related answer would be expected from this parameterisation and associated probability measure/distribution?
With luck, I have already addressed your other points either here, or in comments at reddit. If I have missed something key, please let me know.