Conditional evaluation of random functions

Question

Motivation

Let $(Z(x))_{x\in \mathbb{R}^n}$ be a random function (also known as random field, or random process), i.e. a collection of random variables, but also $Z\in C(\mathbb{R}^n)$ almost surely. The construction of this is not completely trivial, but let us take this as a given.

Now if one were to simulate such a random function at a discrete number of points $x_1,\dots, x_n$, one is interested in the marginal distribution, i.e. we want to know $$ \mathbb{E}[f(Z(x_1), \dots, Z(x_n))] $$ for all $f$. What now happens in Bayesian Optimization, is that we sample our random function at some point $x_0$, i.e. we obtain $Z(x_0)$, then we want to know the conditional distribution of $Z(x)$ given $Z(x_0)$. I.e. $$ \mathbb{E}[f(Z(x))\mid Z(x_0)] $$ This is not a huge problem if we assume $Z$ is a Guassian random function (Gaussian process) with known covariance function. But using this conditional distribution, we tend to select our next evaluation point $X_1$. I.e. $X_1$ is a random variable which is $\sigma(Z(x_0))$ measurable. If we now want to know the distribution of $$ \mathbb{E}[f(Z(x))\mid Z(X_1), Z(x_0)] $$ things become much more tricky, as $Z(X_1)$ is a much more complicated object than $Z(x_0)$. My hypothesis is that $$ \mathbb{E}[f(Z(x))\mid Z(X_1), Z(x_0)] = \bigl(y\mapsto \mathbb{E}[f(Z(x))\mid Z(y), Z(x_0)]\bigr)(X_1) $$ as $X_1$ is completely measurable with regards to $\sigma(Z(x_0)$.

Hypothesis

More generally I expect for a random function in $Z \in C(\mathbb{R}^n)$ $$ \mathbb{E}[f(Z(x))\mid Z(X_n),\dots, Z(X_1),Z(x_0)] = \bigl((y_n,\dots, y_1)\mapsto \mathbb{E}[f(Z(x))\mid Z(y_n),\dots, Z(y_1),Z(x_0)]\bigr)(X_n,\dots,X_1) $$ for $X_n$ measurable with regards to $$ \mathcal{F}_{n-1} = \sigma(Z(x_0),Z(X_1)\dots, Z(X_{n-1})). $$

Progress

I have been able to prove the claim:

Let $Z$ be an almost surely random function, $X$ be $\mathcal{F}$ measurable, then $$ \mathbb{E}[f(Z(X))\mid \mathcal{F}] = \Bigl(y\mapsto \mathbb{E}[f(Z(y))\mid \mathcal{F}]\Bigr)(X) $$

using the fact that the evaluation function $e(z,y) = z(y)$ is continuous and therefore measurable for continuous functions, and regular conditional distributions (I can post a proof if requested).

Now if we were not in the continuous case, we could do the following $$ \begin{aligned} &\mathbb{P}(Z(x) = z \mid Z(X_1) = z_1, Z(x_0) = z_0)\\ &\overset{\text{Bayes}}= \frac{\mathbb{P}(Z(X_1)=z_1 \mid Z(x)=z, Z(x_0)=z_0)\mathbb{P}(Z(x)=z, Z(x_0)=z_0)}{\mathbb{P}(Z(X_1)=z_1\mid Z(x)=z)\mathbb{P}(Z(x)=z)} \end{aligned} $$ and then apply the proven statement above since $X_1$ is $\sigma(Z(x), Z(x_0))$ measurable as it is already $\sigma(Z(x_0))$ measurable. Afterwards we simply use Bayes backwards to get $$ \begin{aligned} &\mathbb{P}(Z(x) = z \mid Z(X_1) = z_1, Z(x_0) = z_0)\\ &=\Bigl(y\mapsto \frac{\mathbb{P}(Z(y)=z_1 \mid Z(x)=z, Z(x_0)=z_0)\mathbb{P}(Z(x)=z, Z(x_0)=z_0)}{\mathbb{P}(Z(y)=z_1\mid Z(x)=z)\mathbb{P}(Z(x)=z)}\Bigr)(X_1)\\ &= \bigl(y \mapsto \mathbb{P}(Z(x) = z \mid Z(y) = z_1, Z(x_0) = z_0)\bigr)(X_1) \end{aligned} $$ Unfortunately, it is not quite so easy to translate this to the continuous case. $Z(x_0)\in A$ does not work, since we need to know its precise value to know $X_1$. Well, since we have that $Z$ is continuous, it might work like this. But I am unsure. Such approximating arguments seem to be very troublesome (cf. Borel-Kolmogorov Pradox)

Does anyone already know the solution to this problem, or has any idea how to approach this? Since this is related to a common practical usecase, it seems like this should be something someone has already looked at.

Answer 1

The following is a proof for the absolutely continuous case (covering Gaussian random functions)

Lemma: Partial integration

Let $X,Z$ be random variables on polish spaces $(E_x, \mathcal{E}_x), (E_z,\mathcal{E}_z)$, $$ f:E_z\times E_x\to \mathbb{R} $$ a measurable function. Let $X$ be $\mathcal{F}$ measurable, $\kappa_{Z\mid \mathcal{F}}$ the regular conditional distribution, then we have for $\Pr$ almost all $\omega$ $$ \mathbb{E}[f(Z,X) \mid \mathcal{F}](\omega) = \int f(z, X(\omega)) \kappa_{Z\mid \mathcal{F}}(\omega, dz) $$ in the case $\sigma(\tilde{X})=\mathcal{F}$ this simplifies to $$ \mathbb{E}[f(Z,X) \mid \tilde{X}] = \int f(z, X) \kappa_{Z\mid \tilde{X}}(\tilde{X}, dz) $$

Proof

left to the reader (the usual argument starting with $f=\mathbf{1}_{A\times B}$)

Corollary: Conditional Sampling I

Let $Z$ be an almost surely random function, $X$ be $\mathcal{F}$ measurable, then $$ \mathbb{E}[f(Z(X))\mid \mathcal{F}] = (y\mapsto \mathbb{E}[f(Z(y))\mid \mathcal{F}])(X) $$

Proof

The evaluation function $$ e(z,x) = z(x) $$ is continuous on the space of continuous functions, and therefore measurable. $f\circ e$ is then also measurable. The application of [Lemma: Partial Integration] results in $$\begin{aligned} \mathbb{E}[f(Z(X))\mid \mathcal{F}](\omega) &=\mathbb{E}[f \circ e (Z, X) \mid \mathcal{F}](\omega)\\ &= \int f\circ e(z, X(\omega)) \kappa_{Z\mid \mathcal{F}}(\omega, dz)\\ &\overset{\text{Lem.}}= \Bigl(y\mapsto\int f\circ e(z, y) \kappa_{Z\mid \mathcal{F}}(\omega, dz)\Bigr)(X(\omega))\\ &= (y\mapsto \mathbb{E}[f(Z(y))\mid \mathcal{F}])(X)(\omega) \end{aligned}$$ Where we use Klenke (2014, Thm. 8.38) for the last equation and all equalities are true for almost all $\omega$.

Conditional sampling Lemma

Let $Z$ be a random function from $\mathbb{R}^d$ to $\mathbb{R}^m$. Assume that for all $\lambda$ we have that $Y, \tilde{X}, X, Z(\lambda)$ are random variables with joint densities on $\mathbb{R},\mathbb{R}^n, \mathbb{R}^d, \mathbb{R}$ such that $X$ is $\sigma(\tilde{X})$ measurable. Then we have $$ \Pr(Y\in A \mid Z(X), \tilde{X}) = \Bigl(\lambda \mapsto \Pr(Y\in A \mid Z(\lambda), \tilde{X})\Bigr)(X) $$ and $Y, \tilde{X}, Z(X)$ have joint density.

Proof

First we want to prove that $Z(X)$ also has a joint density with the other variables. For this we use the regular conditional distribution to get $$ \Pr(Z(X) \in A, (Y, \tilde{X})) = \int_B \underbrace{ \Pr(Z(X) \in A \mid (Y, \tilde{X}) = (y,x)) }_{=\mathbb{E}[\mathbf{1}_A(Z(X))\mid (Y, \tilde{X}) = (y,x)]} \Pr_{Y,\tilde{X}}(dy, dx) $$ which we can now apply the Corollary to get $$ \underbrace{\mathbb{E}[\mathbf{1}_A(Z(X)) \mid Y, \tilde{X}]}_{=: h(Y, \tilde{X})} = \Bigl( \lambda \mapsto \underbrace{ \mathbb{E}[\mathbf{1}_A(Z(\lambda))\mid Y, \tilde{X}] }_{=:h_\lambda(Y,\tilde{X})} \Bigr)(\underbrace{X}_{=:g(\tilde{X})}) = h_{g(\tilde{X})}(Y, \tilde{X}) $$ where we use the Doob-Dynkin factorization lemma repeatedly and the fact that $X$ is $\tilde{X}$ measurable. We therefore have for $(Y,\tilde{X})$ almost all $y,x$ $$\begin{aligned} h(y,x) &= h_{g(x)}(y,x) \\ &= \mathbb{E}[\mathbf{1}_A(Z(g(x)))\mid Y,\tilde{X} = (y,x)] \\ &= \int_A \varphi_{Z(g(x))\mid Y, \tilde{X}}(z\mid y,x) dz \end{aligned}$$ Putting things together we have $$ \Pr(Z(X) \in A, (Y, \tilde{X})) = \int_B \int_A \varphi_{Z(g(x))\mid Y, \tilde{X}}(z\mid y,x)dz \varphi_{Y,\tilde{X}}(y,x) dy dx. $$ This proves we have the joint density $$ \varphi_{Z(X), Y, \tilde{X}}(z,y,x) = \varphi_{Z(g(x))\mid Y, \tilde{X}}(z\mid y,x) \varphi_{Y,\tilde{X}}(y,x). $$ And thus the conditional density is also well defined and given by $$ \varphi_{Z(X)\mid Y, \tilde{X}}(z\mid y, x) = \frac{\varphi_{Z(X), Y, \tilde{X}}(z,y,x)}{\varphi_{Y,\tilde{X}}(y,x)} = \varphi_{Z(g(x))\mid Y, \tilde{X}}(z\mid y,x) $$ We now use this fact to reason with Bayes arguments, that $$\begin{aligned} \varphi_{Y\mid Z(X), \tilde{X}}(y\mid z, x) &= \frac{\varphi_{Y, Z(X), \tilde{X}}(y, z, x)}{\varphi_{Z(X), \tilde{X}}(z,x)} \\ &= \frac{ \varphi_{Z(X) \mid Y, \tilde{X}}(z \mid y, x)\varphi_{Y,X}(y,x) }{\varphi_{Z(X)\mid \tilde{X}}(z\mid x)\varphi_{\tilde{X}}(x)} \\ &= \frac{ \varphi_{Z(g(x)) \mid Y, \tilde{X}}(z \mid y, x)\varphi_{Y,X}(y,x) }{\varphi_{Z(g(x))\mid \tilde{X}}(z\mid x)\varphi_{\tilde{X}}(x)} \\ &=\varphi_{Y\mid Z(g(x)), \tilde{X}}(y\mid z, x) \end{aligned}$$ Which finally implies our claim $$\begin{aligned} \Pr(Y\in A \mid Z(X), \tilde{X} = (z,x)) &= \int_A \varphi_{Y\mid Z(X), \tilde{X}}(y\mid z, x) dy \\ &= \int_A \varphi_{Y\mid Z(g(x)), \tilde{X}}(y\mid z, x) dy \\ &= \Bigl(\lambda \mapsto \int_A \varphi_{Y\mid Z(\lambda), \tilde{X}}(y\mid z, x) dy\Bigr) (g(x)) \\ &= \Bigl(\lambda \mapsto \Pr(Y\in A \mid Z(\lambda), \tilde{X}=(z,x))\Bigr)(g(x)), \end{aligned}$$ i.e. $$ \Pr(Y\in A \mid Z(X), \tilde{X}) = \Bigl( \lambda \mapsto \Pr(Y\in A \mid Z(\lambda), \tilde{X}) \Bigr)(\underbrace{g(\tilde{X})}_{=X}). $$

Conditional Sampling Theorem

Let $Z$ be a random function from $\mathbb{R}^d$ to $\mathbb{R}^m$ with an existing density function. Let $x_1\in \mathbb{R}^d$ and define $$ \mathcal{F}_n = \sigma(Z(X_i): i\le n) $$ where $X_1=x_1$ and $X_{n+1}$ is $\mathcal{F}_n$ measurable. Then we have for measurable functions $f$ $$\begin{aligned} &\mathbb{E}[f(Z(X_{n+1}))\mid \mathcal{F}_n] \\ &= \Bigl( (y_1,\dots,y_{n+1})\mapsto \mathbb{E}[f(Z(y_{n+1})) \mid Z(y_1), \dots, Z(y_n)] \Bigr)(X_1,\dots, X_{n+1}) \end{aligned}$$ and for any $m\in \mathbb{N}$ the joint density of $Z(X_1),\dots, Z(X_{n+1}), Z(\lambda_1),\dots, Z(\lambda_m)$ exists for any $\lambda_i\in\mathbb{R}^d$.

Proof

Let us first address the density problem. By induction we have that $$ \underbrace{Z(X_1),\dots, Z(X_n), Z(\lambda_1),\dots, Z(\lambda_{m-1})}_{=:\tilde{X}}Z(\lambda_m) $$ has joint density. Then by the conditional sampling Lemma, $\tilde{X}, Z(X_{n+1})$ have joint density, as $X_{n+1}$ is $\sigma(\tilde{X})$ measurable. So we have $$ Z(X_1),\dots, Z(X_{n+1}), Z(\lambda_1),\dots, Z(\lambda_{m-1}) $$ have joint density. As $m\in \mathbb{N}$ was arbitrary, we have this for any $m-1$.

Now to the actual statement: As $X_{n+1}$ is $\mathcal{F}_n$ measurable, we first remove it using the Corollary, resulting in $$ \mathbb{E}[f(Z(X_{n+1}))\mid \mathcal{F}_n] = (y_{n+1}\mapsto \underbrace{\mathbb{E}[f(Z(y_{n+1}))\mid \mathcal{F}_n]}_{(*)})(X_{n+1}) $$ we now look at $(*)$ and define $Y:=f(Z(y_{n+1}))$ and $\tilde{X} = (Z(X_1), \dots, Z(X_{n-1}))$, we know that everything has joint density so by the conditional sampling Lemma $$ \mathbb{E}[Y\mid Z(X_n), \tilde{X}] = (y_n\mapsto \mathbb{E}[Y\mid Z(y_n), \tilde{X}])(X_n) $$ The remaining $X_i$ can be moved out with the same argument using $$ \tilde{X}= (Z(X_1),\dots Z(X_{i-1}),Z(y_{i+1}),\dots,Z(y_n)). $$