Shirabyoshi are analogous to geisha in FOTS. They generate money and the cheapest way they can train is to distract armies.
Experience and Leveling Overview
Shirabyoshi come into the game at rank 1. If you have a [[Taira Palace]] or equivalent built they come in at rank 2. They cost 500 gold to recruit, and up to five may be recruited. They gain 3 experience per turn of sitting in a town or army, 3 per failed action, and 15 per successful action. As far as I know, there are two retainers that increase their "Entertaining Nobles" function (the one that makes more money per turn): Ceremonial Flute and Ceremonial Drum, but I'm going to assume that I'm just unlucky and there are actually three.
There are two skills that increase "Entertaining Nobles", both of which can have two points invested, and they are "Courtier" at level 3, and "Cultured Lady" at level 5, for +4 points in total.
Passive Leveling
This is the exp gained by simply putting a shirabyoshi in a town and letting her gain experience passively. This takes an incredibly long amount of time to get to level 6, 135 turns, usually most of a campaign.
Using this formula: C=(ax+(nx(1+x)/2))*T, where C=initial cost of training a shirabyoshi (500 koku), n is the amount of wealth generated per turn, x is the number of turns while T is tax rate (30%), and a is the amount of wealth that the shirabyoshi had gained on her previous level (which we treat as a flat wealth thing we tack onto her new wealth modifier each time she levels) we can figure the amount of time for each type of geisha to pay for themselves.
A shirabyoshi trained from scratch and is lucky with retainers will take 7+
A shirabyoshi trained from scratch and is unlucky with retainers will take 7+
A shirabyoshi trained from a Palace and is lucky with retainers will take 14+
A shirabyoshi trained from a Palace and is unlucky with retainers will take 14+
Active Leveling
This is the exp gained from using the shirabyoshi every turn to distract or seduce enemies. This is much faster, but more expensive. The absolute cheapest action a shirabyoshi can take is to distract a single enemy unit, which costs 50 gold. A shirabyoshi which succeeds every single turn will hit level 6 from scratch in just 27 turns. In lucky circumstances, an enemy army will end up RIGHT outside a town, but not be able to reach it yet. Shirabyoshi inside the town can then distract it, while simultaneously generating wealth for the town and getting passive experience! An example of this would be the province of Kawachi: a rebellion there always spawns an army right next to the capital. In such circumstances, a shirabyoshi can reach level 6 in 23 turns. What's even crazier is that you can quite easily argue that the benefits she gets from level 6 are minimal and that she can call it quits after hitting level 5, in which case she'd be done in a blistering 13 turns.
Shirabyoshi start off with a fairly dismal chance of success, at about 35% success rate at level 1. Thankfully, at level 2 and 3 they gain skills that greatly increase their distraction proficiency as well as their chance to escape unsuccessful actions. They can bump their rate of success to about 75-90% by rank 3-5.
Shirabyoshi that distract every turn to get to level 6 effectively costs 1850 to train (1650 if distracting while in town); level 5 costs 1250 (1150 if distracting while in town). Palace-trained shirabyoshi save at least 2 turns only but bear in mind that the first level it grants drastically increases chances of success.
A shirabyoshi trained to level 6 from scratch and is lucky with retainers will take 59 turns.
A shirabyoshi trained from to 6 from scratch and is unlucky with retainers will take 64 turns.
A shirabyoshi trained to level 5 from scratch and is lucky with retainers will take 42 turns.
A shirabyoshi trained from to 5 from scratch and is unlucky with retainers will take 45 turns.
None of the numbers in these analyses look particularly impressive compared to the rapid return of investment of just sinking money into buildings, but bear in mind that a level 5-6 shirabyoshi is granting +8 to +12 wealth growth ''per turn'', which dwarfs the wealth growth per turn of practically all buildings. They are actually quite similar in function to the food store line of buildings, except that they can move and also distract/seduce enemies. After your five shirabyoshi have hit level 5 or six, put them either in towns with clan estates to maximize their wealth generation, or put them on frontier provinces to tie down enemy units.