When you say "a simple set of conditional and unconditional markets", what do you have in mind?

Unconditional: "Will China invade Taiwan by 2030?"; Conditional: "If China experiences a recession any time before 2030, will China invade Taiwan by 2030?"

Your language makes me think you are referring to 1, and I agree that this will plausibly be of limited value.

Isn't estimating $P\left(Y\right)$ from $P\left(X_i\right)$ most of the value of LVPMs?! 

I think the $\log P\left(\overrightarrow{X}|\theta \right)-\log P\left(\overrightarrow{X}|\varphi \right)$ part of my scoring counts as an AMM liquidity provision?

Yes, that's correct; if the market creator is willing to issue these payouts, then they are playing the role of AMM. 

The question is how to add third-party liquidity provision to this system, i.e. where users can inject and remove liquidity from the market to increase payouts for traders (ideally while being compensated for their efforts). 

Hm, this sounds like a task that would be relatively isolated from the rest of your codebase, and therefore something I could do independently without learning much of Manifold Market's code?

That's the idea. You can see some example code for our Uniswap-style AMM here, but honestly, any well-designed api would be fine. What I'd like is: 1. A typescript interface that defines the current state of a LVPM at any given point, 2. Betting function: Given a bet on some variable and the current market state, return the new market state and user position, 3. Resolution function: given the current market state, the final outcome of the market, and a list of user positions, return a list of all the user payouts.

Thanks for writing such a thorough article! I’d be interested in seeing how LVPMs work in practice, but I must admit I’m coming from a position of extreme skepticism: Given how complicated real-world situations like the Russia/Ukraine war are, I’m skeptical a latent variable model can provide any marginal price efficiency over a simple set of conditional and unconditional markets. 

My suspicion is that if a LVPM were created for a question like “Will China invade Taiwan by 2030?” that most of the predictive power would come from people betting directly on the latent variable rather than from any model-provided updates as a result of people betting on indicator variables. The number, type, and conditional dependency graph of indicator variables is too complicated to capture in a simple model and would function worse than human intuition, imho.

Other thoughts:

• For LVPM to be useful, you’d probably want to add/remove indicator variables in real-time. What would be the process for doing this be? What is the payout of someone who bet before certain indicator variables were added? (This is sort of similar to the problems Manifold has faced in determining payouts for free-response markets.)
• Price efficiency will suffer if you can’t sell your position before resolution or can’t provide liquidity in the form of a limit order or AMM liquidity provision. I think all of these problems are solvable in theory, but may require a good deal more mathematical cleverness.
• Making a compelling UI for a LVPM is a hard problem.
• I can't make any guarantees, but the first step to getting a prototype of this up and running on Manifold (or elsewhere) would be creating a typescript npm package with the market logic. You might be able to convince me to work on this at our next hackathon...