For a Bayesian approach you need a prior distribution and a likelihood.
A reasonable prior here is a Dirichlet distribution with equal probability for the 10 sides (the 10 parameters all equal to 1).
A reasonable likelihood is the multinomial.
Now just multiply the prior and the likelihood (and normalize) and you have your posterior. Since the Dirichlet and multinomial are conjugate the posterior will be a Dirichlet with new parameters (in this case the parameter for 2 will be 2 and for 8 will be 5 and all the others will remain 1). The mode of this distribution would be that the probability of a 2 is 2/15, an 8 is 5/15, and all others are 1/15.
Of course other priors and likelihoods could be used that would lead to other posteriors.