  The Markov chain Monte Carlo method (MCMC), especially the
Metropolis-Hastings (MH) algorithm, is a widely used technique for sampling
from a target probability distribution $P$ on a state space $\Omega$ and
applied to various problems such as estimation of parameters in statistical
models in the Bayesian approach. Quantum algorithms for MCMC have been
proposed, yielding the quadratic speedup with respect to the spectral gap
$\Delta$ compered to classical counterparts. In this paper, we consider the
quantum version of the MH algorithm in the case that calculating $P$ is costly
because the log-likelihood $L$ for a state $x\in\Omega$ is obtained via
computing the sum of many terms $\frac{1}{M}\sum_{i=0}^{M-1} \ell(i,x)$. We
propose calculating $L$ by quantum Monte Carlo integration and combine it with
the existing method called quantum simulated annealing (QSA) to generate the
quantum state that encodes $P$ in amplitudes. We consider not only state
generation but also finding a credible interval for a parameter, a common task
in Bayesian inference. In the proposed method for credible interval
calculation, the number of queries to the quantum circuit to compute $\ell$
scales on $\Delta$, the required accuracy $\epsilon$ and the standard deviation
$\sigma$ of $\ell$ as $\tilde{O}(\sigma/\epsilon^2\Delta^{3/2})$, in contrast
to $\tilde{O}(M/\epsilon\Delta^{1/2})$ for QSA with $L$ calculated exactly.
Therefore, the proposed method is advantageous if $\sigma$ scales on $M$
sublinearly. As one such example, we consider parameter estimation in a
gravitational wave experiment, where $\sigma=O(M^{1/2})$.
