Nature acts twice

A convenient, alternative formulation can be given by allowing nature to act twice:

1. First, a nature action, $\theta \in \Theta$, is chosen but is unknown to the robot.
2. Following this, a nature observation action is chosen to interfere with the robot's ability to sense $\theta$.

Let $\psi$ denote a nature observation action, which is chosen from a nature observation action space, $\Psi(\theta)$. A sensor mapping, $h$, can now be defined that yields $y = h(\theta,\psi)$ for each $\theta \in \Theta$ and $\psi \in \Psi(\theta)$. Thus, for each of the two kinds of nature actions, $\theta \in \Theta$ and $\psi \in \Psi$, an observation, $y = h(\theta,\psi)$, is given. This yields an alternative way to express Formulation 9.5:

Formulation 9..6 (Nature Interferes with the Observation)  

1. A nonempty, finite set $U$ called the action space.
2. A nonempty, finite set $\Theta$ called the nature action space.
3. A nonempty, finite set $Y$ called the observation space.
4. For each $\theta \in \Theta$, a nonempty set $\Psi(\theta)$ called the nature observation action space.
5. A sensor mapping $h:\Theta \times \Psi \rightarrow Y$.
6. A function $L: U \times \Theta \rightarrow {\mathbb{R}}\cup \{\infty\}$ called the cost function.

This nicely unifies the nondeterministic and probabilistic models with a single function $h$. To express a nondeterministic model, it is assumed that any $\psi \in \Psi(\theta)$ is possible. Using $h$,

$$\Theta(y) = \{ \theta \in \Theta \;\vert\; \exists \psi \in \Psi(\theta) y = h(\theta,\psi) \} .$$ (9.27)

For a probabilistic model, a distribution $P(\psi\vert\theta)$ is specified (often, this may reduce to $P(\psi)$). Suppose that when the domain of $h$ is restricted to some $\theta \in \Theta$, then it forms an injective mapping from $\Psi$ to $Y$. In other words, every nature observation action leads to a unique observation, assuming $\theta$ is fixed. Using $P(\psi)$ and $h$, $P(y\vert\theta)$ is derived as

$$P(y\vert\theta) = \left\{ \begin{array}{ll} P(\psi\vert\theta) & ... ...heta,\psi)$. } 0 & \mbox{ if no such $\psi$ exists. } \end{array}\right.$$ (9.28)

If the injective assumption is lifted, then $P(\psi\vert\theta)$ is replaced by a sum over all $\psi$ for which $y = h(\theta,\psi)$. In Formulation 9.6, the only difference between the nondeterministic and probabilistic models is the characterization of $\psi$, which represents a kind of measurement interference. A strategy still takes the form $\pi: \Theta \rightarrow U$. A hybrid model is even possible in which one nature action is modeled nondeterministically and the other probabilistically.