Pros/cons of estimating parameters for missing observations?

Question

Some people are playing a game online. Every time a person plays, a new game board is generated randomly. On generation of a new board, the player can also choose a special weapon. (The choice of weapon does not affect the board generation.)

Over time, observed data are like this:

player weapon win
3      5      0    # player 3 used weapon 5 and lost
4      2      1    # player 4 used weapon 2 and won
8      1      1    # player 8 used no weapon and won

To find out if some players are better with some weapons, and others with other weapons, I've modeled this as a Bayesian logistic regression:

$logit(Pr(Win=1)) = \alpha_{p,w}$

$\alpha_{p,w} \sim N(\mu_{w},\sigma^2)$

So, $\mu_w$ is the effect of weapon $w$ across all players, and $\alpha_{p,w}$ is the effect of player $p$ with weapon $w$.

Because this is a Bayesian model, the prior for $\alpha_{p,w}$ is $\mu_w$, so in theory, I can estimate $\alpha_{p,w}$ for all combinations of $p$ and $w$ even if some players never use some weapons. (JAGS will do this with nested for loops over players and weapons, for example.) But is that a bad idea?

Also, is there a special name for this kind of model?

P.S. Please edit the title and tags if you can make it clearer.

Answer 1

1. No, it is not a bad idea. However, when you look at the posterior distributions for $\alpha_{p,w}$ you will see you haven't learned all that much about those $p,w$ combinations for which you have no data; they'll still be centered at 0, with a spread that's determined largely by the $p,w$ combinations for which you do have data (and the rest by your prior). You will have learned how spread out they might be, however, which may be good enough for your purposes.

2. I'd suggest a reformulation of your model to include a player effect:

   $\text{logit} (\text{P}(\text{Win})) = \mu_0 + \alpha_p + \gamma_w + \delta_{p,w}$

   where $\mathbb{E}\alpha_p = \mathbb{E}\gamma_w = \mathbb{E}\delta_{p,w} = 0$ in your priors. $\gamma_w$ is the weapon effect and $\delta_{p,w}$ is the player-weapon interaction.

This would be called something like a two-factor logistic model with an interaction term.