Let $X$ follows a normal distribution with parameters $\theta$ and $\sigma^2$. Let a prior for $\theta$ be a Normal distribution with parameters $\mu$ and $b^2$. How can I find the Bayes risk of this task? I know how to find a Bayes estimator, but I am confused about how can I find the Bayes risk using this estimator.
$$\bar{\theta} = \frac{b^2}{b^2+\sigma^2} \times X + \frac{\sigma^2}{b^2+\sigma^2} \times \mu$$ $$Risk = \mathbb{V} \Bigl[ \frac{b^2}{b^2+\sigma^2} \times X + \frac{\sigma^2}{b^2+\sigma^2} \times \mu\Bigr] + \Bigl[ \theta - \mathbb{E}\Bigl( \frac{b^2}{b^2+\sigma^2} \times X + \frac{\sigma^2}{b^2+\sigma^2} \times \mu\Bigr)\Bigr]^2$$ Thank you!