# Priors on Taylor Expansion series

I'm wondering what priors can i choose for a Taylor series as follows: $\theta_{1}+\theta_{2} (y-\alpha) + \theta_{3} (y-\alpha)^2$

What priors should I use for updating these parameters ($\theta_{1},\theta_{2},\theta_{3}$) in a Gibbs sampler to accommodate the conditions on Taylor expansion? Any help would be appreciated! Thank you so much in advance

I would most simply use a Normal distribution with mean 0 and covariance matrix $I_3$ as a prior. It works like a basic $L_2$ regularization with coefficient 1 (the non-Bayesian equivalent of a prior). It says "all coefficients are around 0 with variance 1".
But before doing so, you must normalize your input ($y-\alpha$) and output (whatever you call it) to be between say [-1;1] (or have mean 0 variance 1) to avoid "scale issues". You may also normalise $(y-\alpha)$ and $(y-\alpha)^2$ separately but it's unnecessary imo.
"Scale issues" happen when for example $y$ has order of magnitude $10^{-6}$ because some choice of units (measuring a microscopic distance in meters for example). Then a natural order of magnitude for $\theta_2$ would be $10^6$ times $\theta_1$ for the same effect on the output. For $\theta_3$ it would be $10^{12}$ times... The covariance matrix of the prior would need to take this into account which I find cumbersome. Normalization hopefully bypass all these problems.
• Thank you for your helpful answer dear @Benoit, how can i normalize $(y-\alpha)$ in a Gibbs sampler? Commented Oct 27, 2017 at 19:46
• I would normalize the dataset before running the sampler. Write $y'=a(y-\alpha)+b$ with the appropriate constants Same for the output. Then run the sampler on $y'$, get samples $(\theta_1',\theta_2',\theta_3')$ and convert to them back to $(\theta_1,\theta_2,\theta_3)$. Commented Oct 28, 2017 at 12:03