I have a deterministic function $f(x;\theta)$ and measured observation $y=f(x;\theta) + \epsilon$. Here, $\theta$ is unknown, $x$ can be varied, $y$ is observation, and $\epsilon$ is a random error. I want to infer $\theta$ in a sample-efficient way but accurately. If I write posterior $$ P(\theta|\{y\}) = \frac{P(\{y\}|\theta)P(\theta)}{P(\{y\})} $$ where $\{y\}$ is a set of measurements with different values of $x$. To infer $\theta$ accurately I have to make the posterior as sharp as possible and single modal.
I see that I have to choose $x$ such that the denominator P({y}) is less likely and makes the error $\epsilon$ small to make the likelihood sharp. But how can I make the posterior single modal in a sample-efficient way? ( Sampling is expensive..)