I have been pushing for latent variable markets to be added to Manifold. Yesterday, Nathan Young suggested I should design an interface so I've done some of the work and so that it's clearer what I want. Here is my attempt at doing so.

I have a few different examples that can be loaded:

- : One sort of thing that might be nice to use prediction markets for is something like "What will happen with the Ukraine war?". I.e. we've got this thing that is happening and we would like to know what is going to happen. However, a lot of things could happen! Any one market isn't going to capture much of it, and if we have multiple independent markets for different aspects of the outcome, we can't distinguish between "there will be a wishy-washy situation where Ukraine will win in some random places while Russia will win in other random places" vs "either Russia is gonna absolutely steamroll Ukraine or Ukraine is going to fight back hard enough that Russia collapses". The reason this fails is because independent markets don't capture the correlations in outcomes. However, latent variables are a very powerful and intuitive method for capturing correlations, which allows us to have an overall market on "How well will the conflict go for Ukraine?" with countless specific indicators whose correlations are correctly addressed.
- : There are numerous markets on Manifold on specific economic indicators like such as "Will there be X inflation in Y country?"; however, the global economy is an interconnected system, and so there will likely be correlations across different types of economic issues. Getting an overview of these correlations seems like it would be nice, both because the correlations will likely be *especially* impactful (e.g. downturn in many countries at once is more important than downturn in any one country), and because the correlations capture important information about the structure of the economy.

This is a latent variable market. It alllows you to make correlated bets over other markets, to capture the distribution of broader patterns of outcomes than just a single yes/no resolution.

The latent variable market works by having a continuum of possible latent outcomes, which you can see denoted by the bell curve in the "Distribution" tab above. For each latent outcome, the market has probabilities for all of the indicators, specifying how likely they are in the given outcome. Outcomes can be referred to with the term "σ", which refers to how far they are from the median outcome. 0σ refers to the mainline latent outcome where the latent variable goes as expected, while 2σ refers to the latent variable being at the high extreme of expectation, and -2σ refers to the latent variable being at the low end of the expectation.

If you think you have seen evidence that narrows down the distribution of outcomes relative to the market beliefs (i.e. something that rules out the most extreme "no" or "yes" positions to the market question), you might want to make bets to narrow down the latent variable distribution:

If you think the market is underestimating the possibility of extreme outcomes, you might want to make bets to expand the latent variable distribution:

If you think the market is biased with respect to the outcomes it considers, you might want to make bets that shift the latent variable distribution.

The latent variable market resolves based on a number of concrete observable indicator markets. So when you are making predictions on this market above, you are implicitly also making predictions either directly about the markets below, or indirectly making predictions about correlations in their resolution:

If you want to make sure that the latent variable is well-measured by the market, you might want to suggest additional markets for the creator of the latent variable to add to the predictions. Furthermore, if the latent variable is measuring a phenomenon that can exist to many different degrees, then it is best to have markets that cover all of these degrees, i.e. markets with very different resolution probabilities.

As a help to what sorts of markets might be worth adding to the latent variable, below you can see a plot of the signal:noise ratio for the different levels of the latent variable:

Of course the most important aspect of a prediction market is the scoring; you pay to buy shares, and then you get a payout based on whether your predictions were correct. So I've also implemented a VERY BASIC payout system, just to illustrate how the payouts for latent variables function. In the case of latent variables, you get paid out based on whether the overall constellation of indicators aligned with your latent bets. So there's some math (called 2PL Item Response Theory, in the implementation I used, though there are other options like Factor Analysis or Latent Class Analysis) which specifies the relationship between the latent and the indicator probabilities, and that relationship is used for the payout.

To play with the payout and get an intuition for it, you can use the checkboxes below. Your bets have currently cost proportional to 0 Mana, and if the outcome below happens, you get a payout proportional to 0 Mana, for a total profit proportional to 0 Mana.

There's also some fun things you can do once you have latent variables. For instance, you can randomly sample detailed outcomes from the posterior probability distribution, which gives you a much more concrete image of how the future could looks, versus if you just used independent markets for it.