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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..)

Kilean Hwang
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  • I mean if you get to choose $x$, then you can do whatever you want. Sample the same point over and over again – whpowell96 Feb 09 '24 at 20:12
  • You have a deterministic function with a random error? What does that mean? – joriki Feb 09 '24 at 22:42
  • @joriki Yes. The function is deterministic. However, there is a random measurement error. – Kilean Hwang Feb 11 '24 at 02:41
  • It’s not good style to answer “Yes” to someone pointing out an error as if there was no error and then edit the post to fix the error, making the criticism seem wrong. A more appropriate reply would have been “Thanks, I’ll fix that.” – joriki Feb 11 '24 at 07:04
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    @joriki You are right. Based on your point, I changed the wording in the post to make the meaning clearer. Thank you for pointing out. – Kilean Hwang Feb 12 '24 at 18:44

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