Last time in this series we saw that we could compute a continuous posterior distribution when given a continuous prior and a discrete likelihood function; I hope it is clear how that is useful, but I’d like to switch gears for a moment and look at a different (but also extremely useful) computation: the expected value.

I’ll start with a quick refresher on how to compute the expected value of a discrete distribution.

You probably already know what expected value of a discrete distribution is; we’ve seen it before in this series. But in case you don’t recall, the basic idea is: supposing we have a distribution of values of a type where we can meaningfully take an average; the “expected value” is the average value of a set of samples as the number of samples gets very large.

A simple example is: what’s the expected value of rolling a standard, fair six-sided die? You could compute it empirically by rolling 6000d6 and dividing by 6000, but that would take a while.

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Aside: Again, recall that in Dungeons and Dragons, XdY is “roll a fair Y-sided die X times and take the sum”.

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We could also compute this without doing any rolling; we’d expect that about 1000 of those rolls would be 1, 1000 would be 2, and so on.  So we should add up (1000 + 2000 + … + 6000) / 6000, which is just (1 + 2 + 3 + 4 + 5 + 6) / 6, which is 3.5. On average when you roll a fair six-sided die, you get 3.5.

We can then do variations on this scenario; what if the die is still fair, but the labels are not 1, 2, 3, 4, 5, 6, but instead -9, -1, 0, 1, 3, 30?  As I’m sure you can imagine, once again we can compute the expected value by just taking the average: (-9 – 1 + 0 + 1 + 3 + 30) / 6 = 4.

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Aside: It’s a bit weird that the “expected value” of a distribution is a value that is not even in the support of the distribution; I’ve never once rolled a 3.5 on a d6. Beginners sometimes confuse the expected value with the mode: that is, the value that you’d expect to get more often than any other value. Remember, the expected value is just an average; it only makes sense in distributions where the sampled values can be averaged.

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What if the die isn’t fair? In that case we can compute a weighted average; the computation is the value of each side, multiplied by the weight of that side, sum that, and divide by the total weight. As we saw in a previous episode:

public static double ExpectedValue(
    this IDiscreteDistribution<int> d) =>
  d.Support()
   .Select(s => 
     (double)s * d.Weight(s)).Sum() / d.TotalWeight();

And of course we could similarly define methods for discrete distributions of double and so on. Hopefully that is all clear.

The question I want to explore in the next few episodes requires us to make a small extension of the meaning of “expected value”:

• Suppose we have a distribution of outcomes d, in the form of an IWeightedDistribution<double>
• Suppose we have a function f from double to double which assigns a value to each outcome.
• We wish to accurately and efficiently estimate the average value of f(d.Sample()) as the number of samples becomes large.

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Aside: If we had implemented Select on weighted distributions, that would be the same as the expected value of d.Select(f)— but we didn’t!

Aside: We could be slightly more general and say that the distribution is on any T, and f is a function from T to double, but for simplicity’s sake we’ll stick to continuous, one-dimensional distributions in this series. At least for now.

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There’s an obvious and extremely concise solution; if we want the average as the number of samples gets large, just compute the average of a large number of samples! It’s like one line of code. Since we make no use of any weights, we can take any distribution:

public static double ExpectedValue(
    this IDistribution<double> d,
    Func<double, double> f) =>
  d.Samples().Take(1000).Select(f).Average();

In fact, we could make this even more general; we only need to get a number out of the function:

public static double ExpectedValue<T>(
    this IDistribution<T> d,
    Func<T, double> f) =>
  d.Samples().Take(1000).Select(f).Average();

We could also make it more specific, for the common case where the function is an identity:

public static double ExpectedValue(
    this IDistribution<double> d) =>
  d.ExpectedValue(x => x);

Let’s look at a couple of examples. (Code for this episode can be found here.) Suppose we have a distribution from 0.0 to 1.0, say the beta distribution from last time, but we’ll skew it a bit:

var distribution = Beta.Distribution(2, 5);
Console.WriteLine(distribution.Histogram(0, 1));

      ****                              
     *******                            
    *********                           
    *********                           
   ***********                          
   ************                         
   *************                        
   **************                       
  ***************                       
  ****************                      
  *****************                     
  ******************                    
 *******************                    
 ********************                   
 **********************                 
 ***********************                
 ************************               
***************************             
*****************************           
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It looks like we’ve heavily biased these coins towards flipping tails; suppose we draw a coin from this mint; what is the average fairness of the coins we draw? We can just draw a thousand of them and take the average to get an estimate of the expected value:

Console.WriteLine(distribution.ExpectedValue());

0.28099740981762

That is, we expect that a coin drawn from this mint will come up heads about 28% of the time and tails 72% of the time, which conforms to our intuition that this mint produces coins that are heavily weighted towards tails.

Or, here’s an idea; remember the distribution we determined last time: the posterior distribution of fairness of a coin drawn from a Beta(5, 5) mint, flipped once, that turned up heads. On average, what is the fairness of such a coin? (Remember, this is the average given that we’ve discarded all the coins that came up tails on their first flip.)

var prior = Beta.Distribution(5, 5);
IWeightedDistribution<Result> likelihood(double d) =>
  Flip<Result>.Distribution(Heads, Tails, d);
var posterior = prior.Posterior(likelihood)(Heads);
Console.WriteLine(posterior.ExpectedValue());

0.55313807698807

As we’d expect, if we draw a coin from this mint, flip it once, and it comes up heads, on average if we did this scenario a lot of times, the coins would be biased to about 55% heads to 45% tails.

So, once again we’ve implemented a powerful tool in a single line of code! That’s awesome.

Right?

I hope?

Unfortunately, this naive implementation has a number of problems.

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Exercise: What are the potential problems with this implementation? List them in the comments!

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Next time on FAIC: We’ll start to address some of the problems with this naive implementation.