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Higher-order conditional kernels

Consider the following conditional kernels $\varphi_{dice} = P(Addition \vert Die_0~Die_1)$ and $\varphi_{data} = P(X\vert Data)$ with

$$\varphi_{dice}.compute(addition~die_0~die_1) = \left\{ \begin{array}{ll} 1 & if~addition = add(die_0,die_1) \\ 0 & otherwise. \end{array}\right.$$ (4.1)

and

$$\varphi_{data}.compute(x~data) = Normal(x,f_{\mu}(data),f_{\sigma}(data)).$$

with $f_{\mu}:Data\rightarrow I\!\!R$ and $f_{\sigma}:Data\rightarrow I\!\!R$. Observe that for $\varphi_{dice}$ the search variable value $addition\in Addition$ is expressed as a function of the known variable values $\{die_1, die_2\} \in \{Die1,Die2\}$. Thus it is easy to check that

$$\varphi_{dice}.instantiate(die_0~die_1) = dirac(Addition,sum).$$

with $sum=add(die_1,die_2)$. That is, $\varphi_{dice}$ defines a family of dirac kernels. Similarly, for $\varphi_{data}$ we have that

$$\varphi_{data}.instantiate(data) = Normal(X,\mu,\sigma).$$

with $\mu = f_\mu(data)$ and $\sigma^2=f_\sigma(data)$. In this case $\varphi_{data}$ defines a family of normal distribution where the mean and the variance depends on the values of $data$.

Conditional kernels with similar structures to that of $\varphi_{dice}$ and $\varphi_{data}$ are very common in Bayesian modeling. In effect, several conditional kernels take one or more external functions (e.g. $add$, $f_\mu$ and $f_\sigma$) as arguments. Such conditional kernels are called higher-order conditional kernels.

Definition 25   Let $\omega_{k_{i=1,2...n}}\subseteq \omega_k\in\Omega_k$. A higher-order conditional kernel is a built-in conditional kernel $\varphi=P(\Omega_s\vert\Omega_k)$ that can take one or more external functions $f(\omega_{k_2}),f(\omega_{k_2})...f(\omega_{k_n})$ as arguments.

Higher-order conditional kernels are classified in: functional diracs and functional kernels. These classes were implicitly depicted by the two previous examples and are fully described in the following sections.