P(Tt −1) Unknown and so we chose it uniform.

P(T |Tt −1) Table, specified with a "prior" to prevent switching targets too often or simply learned. Uniform if there is no previous target.

P(S) Unknown and so uniform, it could be a prior on *Skills* (not uniform). P(Left _Value | Right *_Values*) All others are learnable tables. 

## Identification

If there were only perceived variables, learning the right conditional probability tables would just be counting and averaging. However, some variables encode combinations of perceptions and passed states. We could learn such parameters through the EM algorithm but we propose something simpler for the moment as our "not directly observed variables" are not complex to compute, we compute them from perceptions as the same time as we learn. In the following Results part, we did not apply learning but instead manually specified the probability tables.

## Questions

In any case, we ask our model:
P(S ∧T | hp1:n ∧ d1:n ∧ a1:n ∧ Δ*hp1:n* ∧id1:n ∧c1:n ∧ r1:n ) , which means that we want to know the distribution on S and T knowing all the state variables. We then choose to do the highest scoring combination of S ∧T that is available (skills may have cooldowns or cost more mana/energy that we have available). 

Using (Bayes rule) *P(S,T)=P(S|T).P(T)*, to decompose this question, we can ask:
P(T | hp1:n ∧ d1:n ∧ a1:n ∧ Δhp1:n ∧id1:n ∧*c1:n* )
which means that we want to know the distribution on T knowing all the relevant state variables, followed by (with the newly computed distribution on T)
€
P(S |T ∧ hp1:n ∧ d1:n ∧ a1:n ∧ Δ*hp1:n* ∧id1:n ∧c1:n ∧ r1:n )
in which we use this distribution T to compute the distribution on S with:
 
P(S = *skill*1 |…) = P( ∑T
S = *skill*1 ( |T ∧…).P(T))
We here choose to sum over all possible values of T. Note that we did not ask: 
P(S |T = most _ *probable*∧ hp1:n ∧…) but computed instead:
P(S |T ∧ hp1:n ∧ d1:n ∧ a1:n ∧ Δhp1:n ∧id1:n ∧*c1:n* ∧ r1:n )
T
∑
This computation has a high complexity (particularly when the sum has many terms, i.e. with a lot of targets), so we could choose not to do the sum and use and instantiate "most probable values", for instance of Target. Any such choice would lose information. There are possibly good combinations of S and T for a value of T that is not the most probable one. This downside may be so hard that we may want to reduce the complexity of computation by simplifying our model or its computation to be able to sum. We propose a solution in the discussion.