Let the \(X_{k}\) be i.i.d. random variables with unknown distribution which is to be inferred by a Bayesian procedure. Let the unknown probability distribution of \(X_1\) be assigned a prior \(\pi\in M_1(\Omega)\), with support \(\text{ supp} \pi\), where \(M_1(\Omega)\) is the space of probability measures on \(\Omega\). The posterior distribution, given the first \(n\) observations, is a function of the empirical measure \(L_{n}(\omega)=n^{-1}\sum_{k=1}^{n}\delta_{X_{k}}(\omega)\), where \(\delta_{X_{k}}\) denotes unit mass at \(X_{k}\) and is denoted by \(\pi^{n}(L_{n})\). Given a sequence \((X_{n})_{n\in N}\), let \((\nu_{n})_{n\in N}\) be a sequence of \(M_1(\Omega)\)-valued random variables such that \(\nu_{n}\) has distribution \(\pi^{n}(L_{n})\) for each \(n\geq 1\). Let \((b_{n})_{n\in N}\) be a sequence satisfying the conditions \(b_{n}/n\to 0\) and \(b_{n}^2/n\to\infty\) as \(n\to\infty\). For finite sets \(\Omega\) the authors prove that, on the set \((n/b_{n})(L_{n}-\mu)\to 0\), for any “regular” point \(\mu\) in the support of the prior, the sequence \((P((n/b_{n})(\nu_{n}-\mu)\in\cdot))_{n\in N}\) satisfies the large deviation principle on \(M_{b}(\Omega)\) - the set of finite signed measures on \(\Omega\) with speed \(n/b_{n}^2\) and the rate function \[ I(\nu)=\begin{cases} 2^{-1}\sum_{x\in \Omega}\nu^2(x)/\mu(x), & \text{if\quad \(\sum_{x\in \Omega}\nu(x)=0\)}\\ +\infty, & \text{otherwise}. \end{cases} \] Then the authors extend the result to Polish spaces for priors which are exchangeable with respect to a nested sequence of partitions. Pólya trees are considered as an illustrative example.