Rewrite documentation some on ProbabilisticScorer

We had some user confusion on how the probabilistic scorer works,
especially in reference to the half-life parameter. This attempts
to clarify how the bounds work, and how they are decayed.

Author: TheBlueMatt

lightning/src/routing/scoring.rs

@@ -315,26 +315,39 @@ type ConfiguredTime = Eternity;
 
 /// [`Score`] implementation using channel success probability distributions.
 ///
-/// Based on *Optimally Reliable & Cheap Payment Flows on the Lightning Network* by Rene Pickhardt
-/// and Stefan Richter [[1]]. Given the uncertainty of channel liquidity balances, probability
-/// distributions are defined based on knowledge learned from successful and unsuccessful attempts.
-/// Then the negative `log10` of the success probability is used to determine the cost of routing a
-/// specific HTLC amount through a channel.
+/// Channels are tracked with upper and lower liquidity bounds - when an HTLC fails at a channel,
+/// we learn that the upper-bound on the available liquidity is lower than the amount of the HTLC.
+/// When a payment is forwarded through a channel (but fails later in the route), we learn the
+/// lower-bound on the channel's available liquidity must be at least the value of the HTLC.
 ///
-/// Knowledge about channel liquidity balances takes the form of upper and lower bounds on the
-/// possible liquidity. Certainty of the bounds is decreased over time using a decay function. See
-/// [`ProbabilisticScoringParameters`] for details.
+/// These bounds are then used to determine a success probability using the formula from
+/// *Optimally Reliable & Cheap Payment Flows on the Lightning Network* by Rene Pickhardt
+/// and Stefan Richter [[1]] (i.e. `(upper_bound - payment_amount) / (upper_bound - lower_bound)`).
 ///
-/// Since the scorer aims to learn the current channel liquidity balances, it works best for nodes
-/// with high payment volume or that actively probe the [`NetworkGraph`]. Nodes with low payment
-/// volume are more likely to experience failed payment paths, which would need to be retried.
+/// This probability is converted into a linear score and multiplied with the
+/// [`liquidity_penalty_multiplier_msat`] and [`liquidity_penalty_amount_multiplier_msat`]
+/// parameters to calculate a concrete penalty in milli-satoshis. See the documentation of those
+/// parameters for the exact formulas.
+///
+/// The liquidity bounds are decayed by halving them every [`liquidity_offset_half_life`].
+///
+/// Further, we track the history of our upper and lower liquidity bounds for each channel,
+/// allowing us to assign a second penalty (using [`historical_liquidity_penalty_multiplier_msat`]
+/// and [`historical_liquidity_penalty_amount_multiplier_msat`]) based on the same probability
+/// formula, but using the history of a channel rather than our latest estimates for the liquidity
+/// bounds.
 ///
 /// # Note
 ///
 /// Mixing the `no-std` feature between serialization and deserialization results in undefined
 /// behavior.
 ///
 /// [1]: https://arxiv.org/abs/2107.05322
+/// [`liquidity_penalty_multiplier_msat`]: ProbabilisticScoringParameters::liquidity_penalty_multiplier_msat
+/// [`liquidity_penalty_amount_multiplier_msat`]: ProbabilisticScoringParameters::liquidity_penalty_amount_multiplier_msat
+/// [`liquidity_offset_half_life`]: ProbabilisticScoringParameters::liquidity_offset_half_life
+/// [`historical_liquidity_penalty_multiplier_msat`]: ProbabilisticScoringParameters::historical_liquidity_penalty_multiplier_msat
+/// [`historical_liquidity_penalty_amount_multiplier_msat`]: ProbabilisticScoringParameters::historical_liquidity_penalty_amount_multiplier_msat
 pub type ProbabilisticScorer<G, L> = ProbabilisticScorerUsingTime::<G, L, ConfiguredTime>;
 
 /// Probabilistic [`Score`] implementation.
@@ -388,19 +401,28 @@ pub struct ProbabilisticScoringParameters {
 	/// uncertainty bounds of the channel liquidity balance. Amounts above the upper bound will
 	/// result in a `u64::max_value` penalty, however.
 	///
+	/// `-log10(success_probability) * liquidity_penalty_multiplier_msat`
+	///
 	/// Default value: 30,000 msat
 	///
 	/// [`liquidity_offset_half_life`]: Self::liquidity_offset_half_life
 	pub liquidity_penalty_multiplier_msat: u64,
 
-	/// The time required to elapse before any knowledge learned about channel liquidity balances is
-	/// cut in half.
+	/// Whenever this amount of time elapses since the last update to a channel's liquidity bounds,
+	/// the distance from the bounds to "zero" is cut in half. In other words, the lower-bound on
+	/// the available liquidity is halved and the upper-bound moves half-way to the channel's total
+	/// capacity.
+	///
+	/// Because halving the liquidity bounds grows the uncertainty on the channel's liquidity,
+	/// penalties for payments that are within the liquidity bounds will be decreased. See the
+	/// [`ProbabilisticScorer`] struct documentation for more info on the way the liquidity bounds
+	/// are used.
 	///
-	/// The bounds are defined in terms of offsets and are initially zero. Increasing the offsets
-	/// gives tighter bounds on the channel liquidity balance. Thus, halving the offsets decreases
-	/// the certainty of the channel liquidity balance.
+	/// For example, if the channel's capacity is 1 million sats, and the current upper- and lower-
+	/// liquidity bounds are 200,000 sats and 600,000 sats, after this amount of time the upper-
+	/// and lower- liquidity bounds will be decayed to 100,000 and 800,000 sats.
 	///
-	/// Default value: 1 hour
+	/// Default value:  hour
 	///
 	/// # Note
 	///