Improve description of initial value behavior in estimator

Rather than describing this as "debiasing", instead make clear that we
are properly normalizing a weighted average with a sum of weights less
than 1.

src/state.rs

@@ -430,15 +430,19 @@ impl Estimator {
         self.smoothed_steps_per_sec =
             self.smoothed_steps_per_sec * weight + new_steps_per_second * (1.0 - weight);
 
-        // Get an unbiased estimate of `smoothed_steps_per_sec` to serve as the data source for the
-        // double smoothed estimate. See comment on debiasing in `steps_per_second` for details.
+        // An iterative estimate like `smoothed_steps_per_sec` is supposed to be an exponentially
+        // weighted average from t=0 back to t=-inf; Since we initialize it to 0, we neglect the
+        // (non-existent) samples in the weighted average prior to the first one, so the resulting
+        // average must be normalized. We normalize the single estimate here in order to use it as
+        // a source for the double smoothed estimate. See comment on normalization in
+        // `steps_per_second` for details.
         let delta_t_start = duration_to_secs(now - self.start_time);
-        let debias = 1.0 - estimator_weight(delta_t_start);
-        let debiased_smoothed_steps_per_sec = self.smoothed_steps_per_sec / debias;
+        let total_weight = 1.0 - estimator_weight(delta_t_start);
+        let normalized_smoothed_steps_per_sec = self.smoothed_steps_per_sec / total_weight;
 
         // determine the double smoothed value (EWA smoothing of the single EWA)
         self.double_smoothed_steps_per_sec = self.double_smoothed_steps_per_sec * weight
-            + debiased_smoothed_steps_per_sec * (1.0 - weight);
+            + normalized_smoothed_steps_per_sec * (1.0 - weight);
 
         self.prev_steps = new_steps;
         self.prev_time = now;
@@ -464,32 +468,35 @@ impl Estimator {
         let delta_t = duration_to_secs(now - self.prev_time);
         let reweight = estimator_weight(delta_t);
 
-        // Debiasing:
+        // Normalization of estimates:
         //
-        // Our exponentially weighted estimate is a single value (smoothed_steps_per_second) that
-        // is iteratively updated. At each update, the previous value of the estimate is
-        // re-weighted according its age. At any point in time, the raw value of this estimate
-        // reflects the assumption that it contains properly weighted sample values going back
-        // indefinitely in time. But this assumption is false.
+        // The raw estimate is a single value (smoothed_steps_per_second) that is iteratively
+        // updated. At each update, the previous value of the estimate is re-weighted according to
+        // its age. The resulting estimate is therefore a weighted average, where the weight of
+        // each sample is the difference W(t1) - W(t2), where t1 and t2 are the time deltas since
+        // the end and beginning of the sample (respectively), and W(t) is a function describing
+        // the appropriate total weight for data older than t, e.g. W(t) = 0.1 ^ (t/15).
         //
-        // The value is initialized with some value when the estimator starts. The raw value of the
-        // estimator treats this as an appropriately weighted sample average across all times
-        // before t=0. Of course, the value is actually arbitrary. In other words, because the raw
-        // estimate gives a positive weight to this initial value, the resulting estimate will be
-        // *biased* towards the initial value.
+        // Note that since t = Sum(t_n) implies W(t) = Prod(W(t_n)), the formula for the total
+        // weight of data older than t is just the normal iterative weighting function.
         //
-        // A debiased estimate is the result of correcting the raw estimate by assigning 0 weight
-        // to the initial value. We can do this with a simple trick: set the initial value to 0,
-        // and then divide the raw estimate by the estimator weight for all time *since* t=0.
+        // Since W(0) = 1, and Limit(W(t)) -> 0 as t -> inf, the sum of the weights in the weighted
+        // average will be less than 1 unless there are samples going back to t = inf. Since this
+        // isn't possible, we initialize the raw estimate to 0, meaning that we have a finite
+        // number of samples with a total weight less than 1.
+        //
+        // Therefore, the raw estimate must be normalized by dividing it by the sum of the weights
+        // in the weighted average. This sum is just W(0) - W(t_f), where t_f is the time since the
+        // first sample, and W(0) = 1.
         let delta_t_start = duration_to_secs(now - self.start_time);
-        let debias = 1.0 - estimator_weight(delta_t_start);
+        let total_weight = 1.0 - estimator_weight(delta_t_start);
 
         // Generate updated values for `smoothed_steps_per_sec` and `double_smoothed_steps_per_sec`
-        // (sps and dsps) without storing them. Note that we debias sps when using it as a source
-        // to update dsps, and then debias dsps itself before returning it.
-        let sps = self.smoothed_steps_per_sec * reweight / debias;
+        // (sps and dsps) without storing them. Note that we normalize sps when using it as a
+        // source to update dsps, and then normalize dsps itself before returning it.
+        let sps = self.smoothed_steps_per_sec * reweight / total_weight;
         let dsps = self.double_smoothed_steps_per_sec * reweight + sps * (1.0 - reweight);
-        dsps / debias
+        dsps / total_weight
     }
 }
 
@@ -713,7 +720,7 @@ mod tests {
 
         // The first level EWA:
         //   -> 90% weight @ 0 eps, 9% weight @ 1 eps, 1% weight @ 0 eps
-        //   -> then debiased by deweighting the 1% weight (before -30 seconds)
+        //   -> then normalized by deweighting the 1% weight (before -30 seconds)
         let single_target = 0.09 / 0.99;
 
         // The second level EWA: