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A Quiz from my past self

· #maths #probabilities #coding #perl

TL;DR

Where I solve a mystery left by my past self.

As the rare readers of this blog of fixations know, I’ve been lately interested into using the so-called rejection method to generate random dice outcomes from other (hopefully) random dice outcomes. It all started from A 4-faces die from a 6-faces die and no, I still don’t regret this.

It turns out that I gave a non-trivial amount of thinking a few years ago when I was over-engineering Ordeal::Model, a Perl module to shuffle things (that serves as the backend machinery for ordeal, my take at generating a random bunch of picture to prompt story creation.

Over-engineering

As Ordeal::Model main aim was to support ordeal, which is admittedly an crystal case of under-engineering a UI, I think that the Fisher-Yates shuffle as it can be found in List::Util::shuffle was way, way totally good for the task.

Alas, I also like to learn a thing or two in these pet projects, so I couldn’t just take it and move on. To be fair, there was one key thought that was bugging me a bit: is the shuffle biased in any sense?

I know that the Fisher-Yates shuffle is unbiased as a process, but it relies on a random source to generate integers within ranges and that was something that was bugging me a bit. If the process is fed with biased data, can we still say that the whole thing is unbiased?

I also didn’t want to look into the actual implementation in List::Util - why do this when there was this beautiful wheel waiting to be re-invented?

Enter Ordeal::Model::ChaCha20

To make a long story short, I implemented Ordeal::Model::ChaCha20 to provide an as unbiased as I can possibly tell source of randomness for an implementation of randomly drawing stuff from a list.

Under the hood, it uses code taken from Dana Jacobsen’s Math::Prime::Util::ChaCha, adapted to work inside an object and used to draw random bits and return them as a string of 0 and 1 by calling the _core function:

 1 sub _bits_rand ($self, $n) {
 2    while (length($self->{_buffer}) < $n) {
 3       my $add_on = $self->_core((int($n / 8) + 64) >> 6);
 4       $self->{_buffer} .= unpack 'b*', $add_on;
 5    }
 6    return substr $self->{_buffer}, 0, $n, '';
 7 } ## end sub _bits_rand

The main star

This source of random bits is then used by the very central function of Ordeal::Model::ChaCha20, i.e. int_rand ($low, $high), that takes a range from where to draw random integers and gives you one… in that range:

 1 sub int_rand ($self, $low, $high) {
 2    my $N = $high - $low + 1;
 3    my ($nbits, $reject_threshold) = $self->_int_rand_parameters($N);
 4    my $retval = $reject_threshold;
 5    while ($retval >= $reject_threshold) {
 6       my $bitsequence = $self->_bits_rand($nbits);
 7       $retval = 0;
 8       for my $v (reverse split //, pack 'b*', $bitsequence) {
 9          $retval <<= 8;
10          $retval += ord $v;
11       }
12    } ## end while ($retval >= $reject_threshold)
13    return $low + $retval % $N;
14 } ## end sub int_rand

Line 2 temporarily transforms the problem from any integer within $low and $high, included to any number between 0 included and $N excluded. This shift is reversed at the very end (line 13), so from now on we concentrate on the simpler range.

Line 3 gets the relevant parameters for our rejection-based method. In particular, $nbits represents how many random bits have to be drawn in order to generate a candidate for the rejection method, and $reject_threshold is an integer number from which the rejection will start.

Loop in lines 5 to 12 apply the rejection method:

  • get a sequence of $nbits bits (line 6)
  • turn them into an unsigned integer ($retval, lines 7 to 11)
  • apply the rejection method if this is too big (line 5).

The puzzle wrapping

I promised a mystery, right? It’s in the function that establishes the right parameters $nbits and $reject_threshold, i.e. the following:

 1 sub _int_rand_parameters ($self, $N) {
 2    state $cache = {};
 3    return $cache->{$N}->@* if exists $cache->{$N};
 4 
 5    # basic parameters, find the minimum number of bits to cover $N
 6    my $nbits = int(log($N) / log(2));
 7    my $M = 2 ** $nbits;
 8    while ($M < $N) {
 9       $nbits++;
10       $M *= 2;
11    }
12    my $reject_threshold = $M - $M % $N; # same as $N here
13 
14    # if there is still space in the cache, this pair will be used many
15    # times, so we want to reduce the rejection rate
16    if (keys($cache->%*) <= CACHE_SIZE) {
17       while (($nbits * $M / $reject_threshold) > ($nbits + 1)) {
18          $nbits++;
19          $M *= 2;
20          $reject_threshold = $M - $M % $N;
21       }
22    }
23    return ($nbits, $reject_threshold);
24 }

I keep a cache (line 2) that is a sort of a global variable; this is OK because the function actually only depends on $N, not on the specific object that calls it. For paranoid reasons this cache is limited in size (test in line 16) but this is a detail.

If we have the item in the cache then we’re done: just return the two numbers for $nbits and $reject_threshold.

Otherwise, we first calculate $nbits in a reasonable way: let’s find the closer power of 2 that is greater than, or equal to, the input number $N. Line 6 will calculate the number of bits for the power of 2 that is smaller than, or equal to, $N, so we do some adjustment in lines 8 to 11 to make sure that our power of 2 is greater than, or equal to, our target $N.

You might object that line 8 might be a if just as well. But I’m paranoid, so the while works fine here to protect my sleep from weird stuff in the floating point operation in line 6. I know, I know.

Now that we have $M, our power of 2, we can also calculate the threshold for the rejection (line 12). If you look hard into it, it might be also expressed as:

my $reject_threshold = $N;

but we’ll keep it as-is to make it the same as line 20 and be consistent in our way to calculate the threshold.

So far nothing misterious, but then we ask ourselves if we’re going to cache this value. This got me cursing back at the past me for not writing a comment about what I had in mind!

As anticipated above, we save the pair of values we found only if there is still space in the cache; in this case, it makes sense to save an optimized value that will be reused over and over. Hopefully.

How do we optimize $nbits and $reject_threshold though?

The puzzle

The values of $nbits, $M and $reject_threshold are the lowest possible ones that allow us to apply the rejection method. Higher values, though, would work fine as well: at the expense of getting more bits (i.e. higher values for $nbits) we draw numbers from a greater pool, but the amount of rejections is strictly capped so the probability of rejection generally decreases.

How can we cap the rejection? Let’s make an example and aim to generate integers below 5 (i.e. $N is 5):

  • the minimum number for $nbits is 3, which gives us $M equal to 8. Less bits would give us up to 4 which isn’t sufficient to generate what we are after.
  • with the minimum number of bits, we get a failure 3 times out of 8 (i.e. 5, 6, and 7)
  • with one more bit we draw number from a pool of 16 candidates, so it might seem that we are rejecting more. On the other hand, we can do the following mapping and only miss one value (15):
    • 0, 5, and 10 map onto 0
    • 1, 6, and 11 map onto 1
    • …
    • 4, 9, and 14 map onto 4

So, when we consider all whole blocks of $N items, we restrict the rejection only to the the final block, which by definition has a size that is strictly less than $N.

Hence, the number of rejected candidates is always strictly less than $N, but the probability of a rejection decreases as we add more bits because the rejected candidates get divided by $M.

So… adding bits decreases a probability of rejection. But where should we stop? Should we add one more bit or be done with the value we have?

The code does the optimization in this cycle, with the condition in line 17:

17       while (($nbits * $M / $reject_threshold) > ($nbits + 1)) {
18          $nbits++;
19          $M *= 2;
20          $reject_threshold = $M - $M % $N;
21       }

which has been a mistery for me when I re-read it after about two years.

And then… it came to me again! I almost certainly reasoned how follows.

What is the expected number of draws that I will have to perform with $nbits? All draws are assumed to be independent from one another, so it’s a geometric distribution where the probability of the event we’re interested into is:

\[P_{accept} = \frac{R_t}{M} = \frac{R_t}{2^k}\]

where $R_t$ is the $reject_threshold and $k$ is $nbits. In a geometric distribution, the expected value for the number of draws until we have a success is the inverse of this probability. Hence, if we draw $k$ bits at every attempt, how many bits are we expecting to use for each non-rejected value? We just have to multiply by $k$ and we will get $K$:

\[K = \frac{k}{P_{accept}} = k \cdot \frac{2^k}{R_t}\]

which is exactly the left hand side in the test of line 17.

At this point we can as ourselves: does it make sense to draw one more bit instead? Well, if the value $K$ is greater than $k+1$, then we can leverage the go to next power of 2 trick to reduce the probability of a rejection and hopefully get a better result. On the other hand, if the average number of bits for each successful (i.e. non-rejected) draw is less than that, or even equal to it, we don’t get an advantage (in average) so we can just stop expanding.

In our example where $N is 5, the probability of an accepted value is $\frac{5}{8}$, which means that we will have to do $\frac{8}{5} = 1.6$ draws for each successful (i.e. non-rejected) value. This means that, on average, we will spend $3 \cdot 1.6 = 4.8$ bits per number. Why not draw 4 bits then? On average we will in any case draw more!

In this case, with 4 bits we would have an acceptance probability of $\frac{15}{16}$ (remember that we can use all outcomes from 0 up to 14 included, and only reject 15), which means an average number of rolls equal to $\frac{16}{15} \approx 1.07$ and an average number of bits equal to $4 \cdot \frac{16}{15} \approx 4.27$, which is only slightly over 4 bits and in any case better than the 4.8 that we would get with 3 bits only.

Now, the 4.27 we got is not greater than 5, so getting 5 bits instead of 4 would not be advantageous in average, but only make us waste bits.

In conclusion, there we have our condition for taking the next bit:

\[k \cdot \frac{2^k}{R_t} > k + 1\]

i.e. the condition in line 17. Mystery solved!

My take away…

… is to document all these things in the future, at least put a hint of where the solution lies, so that I will have the mystery and the solution!!!

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