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I might be confused about something.

Consider doing inference on $Y'\mid X',Y,X$ using standard Gaussian Process Regression with 1d $Y$ and 1d $X$. Suppose $X$ is time-like (target is stationary or shift invariant and that the covariance grows like $\Delta X$. This would imply a good kernel is something like $k(x, x') \propto \frac{e^{- \frac{\alpha}{|x - x'|}}}{| x - x'|}$ yet we often see the squared exponential used as a default for stationary GPR kernels which is $e^{- \frac{(x - x')^2}{2 \sigma^2}}$.

Obviously the kernel doesn't make much sense because it tends to zero $x$ goes to $x'$. What is wrong with the reasoning? Why does "time like" mean using squared exponential?

Another way of asking this question is what kernel corresponds to the GPR being a brownian bridge density?

I might be confused about something.

Consider doing inference on $Y'\mid X',Y,X$ using standard Gaussian Process Regression with 1d $Y$ and 1d $X$. Suppose $X$ is time-like (target is stationary or shift invariant and that the covariance grows like $\Delta X$. This would imply a good kernel is something like $k(x, x') \propto \frac{e^{- \frac{\alpha}{|x - x'|}}}{| x - x'|}$ yet we often see the squared exponential used as a default for stationary GPR kernels which is $e^{- \frac{(x - x')^2}{2 \sigma^2}}$.

Another way of asking this question is what kernel corresponds to the GPR being a brownian bridge density?

I might be confused about something.

Consider doing inference on $Y'\mid X',Y,X$ using standard Gaussian Process Regression with 1d $Y$ and 1d $X$. Suppose $X$ is time-like (target is stationary or shift invariant and that the covariance grows like $\Delta X$. This would imply a good kernel is something like $k(x, x') \propto e^{- \frac{\alpha}{|x - x'|}}$ yet we often see the squared exponential used as a default for stationary GPR kernels which is $e^{- \frac{(x - x')^2}{2 \sigma^2}}$.

Am I confused or missing something obvious?

I might be confused about something.

Consider doing inference on $Y'\mid X',Y,X$ using standard Gaussian Process Regression with 1d $Y$ and 1d $X$. Suppose $X$ is time-like (target is stationary or shift invariant and that the covariance grows like $\Delta X$. This would imply a good kernel is something like $k(x, x') \propto \frac{e^{- \frac{\alpha}{|x - x'|}}}{| x - x'|}$ yet we often see the squared exponential used as a default for stationary GPR kernels which is $e^{- \frac{(x - x')^2}{2 \sigma^2}}$.

Obviously the kernel doesn't make much sense because it tends to zero $x$ goes to $x'$. What is wrong with the reasoning? Why does "time like" mean using squared exponential?

Another way of asking this question is what kernel corresponds to the GPR being a brownian bridge density?

I might be confused about something.

Consider doing inference on $Y'\mid X',Y,X$ using standard Gaussian Process Regression with 1d $Y$ and 1d $X$. Suppose $X$ is time-like (target is stationary or shift invariant and that the covariance grows like $\Delta X$. This would imply a good kernel is something like $k(x, x') \propto \exp\{-\alpha / |x - x'|\}$ yet we often see the squared exponential used as a default for stationary GPR kernels which is $\exp\{-(x - x')^2 / \sigma^2 / 2\}$.

Am I confused or missing something obvious?

I might be confused about something.

Consider doing inference on $Y'\mid X',Y,X$ using standard Gaussian Process Regression with 1d $Y$ and 1d $X$. Suppose $X$ is time-like (target is stationary or shift invariant and that the covariance grows like $\Delta X$. This would imply a good kernel is something like $k(x, x') \propto e^{- \frac{\alpha}{|x - x'|}}$ yet we often see the squared exponential used as a default for stationary GPR kernels which is $e^{- \frac{(x - x')^2}{2 \sigma^2}}$.

Am I confused or missing something obvious?