  We show that Thompson Sampling with Gaussian Prior as detailed by Algorithm 2
in (Agrawal & Goyal, 2013) is already differentially private. Theorem 1 show
that it enjoys a very competitive privacy loss of only $\mathcal{O}(\ln^2 T)$
after T rounds. Finally, Theorem 2 show that one can control the privacy loss
to any desirable $\epsilon$ level by appropriately increasing the variance of
the samples from the Gaussian posterior. And this increases the regret only by
a term of $\mathcal{O}(\frac{\ln^2 T}{\epsilon})$. This compares favorably to
the previous result for Thompson Sampling in the literature ((Mishra &
Thakurta, 2015)) which adds a term of $\mathcal{O}(\frac{K \ln^3
T}{\epsilon^2})$ to the regret in order to achieve the same privacy level.
Furthermore, our result use the basic Thompson Sampling with few modifications
whereas the result of (Mishra & Thakurta, 2015) required sophisticated
constructions.
