Fix stochastics

[timmens]

There seems to be something wrong about the stochastics in PyLCM.

In this PR, I want to validate and potentially fix the parts in PyLCM that work with stochastics. This will most likely close #39 and #63.

In particular, we have to answer the following questions:

• Mathematically, what do the state transition probabilities represent (joint probabilities or conditional probabilities)
• Given that we assume that stochastic state transitions can depend on other stochastic variables, does this interfere with our assumption that in a given period, the stochastic states are independent?

I believe here we must start form the mathematics again, and then work ourselves down to the implementation.

What we (most likely) need

• We need to improve the error handling of the transition matrices as well (BUG: Transition matrices with incorrect shapes do not throw error #63)
  Ideally, we check that the transition matrices and the model are consistent as early as possible; i.e., inside the solve(params) or simulate(params) call, but before any computation
• We absolutely need to refactor the stochastics related code; it is very hard to understand!
• Move computation of node weights from Q_and_F.py to solve_brute.py (helps in modularization, cleans up Q_and_F code)

[hmgaudecker]

I was going to write a quick reply because I do think these should be pretty clear, but without formulas it is really hard to grasp what you are after precisely. Could you maybe give an example in math?

In general, I guess a transition for some element $k$ of the state vector $s$ is always:

$$ s^k_{t+1} = g(\mu(s_t) + \varepsilon_t) $$

There might be cases where you scale $\varepsilon_t$ with $\sigma(s_t)$, but typically not when we are concerned with discrete states. The form of $g(\cdot)$ does not matter much here, for discrete states it will be that its index is above and/or below some cutoff.

When talking about independence etc, we are usually thinking about $\varepsilon$.

When we say "the next state depends on current state $m$", we are typically talking about the case that $\mu(\cdot)$ takes the form $\mu(s^m_t)$. In #39, we have the issue that we do not support the case where $\mu$ is a constant.