Hypothesis testing: Calculate beta without PDF of H1 I've found pages that show how to mechanically calculate beta as 1 minus alpha and/or show beta in the context of a PDF for the alternative hypothesis (H1 or Ha).  If you don't have a conjectured mean for H1, however, and/or don't have a PDF (gaussian or otherwise), how does one even conceptually approach the calculation of beta?
For example, suppose Ho was "The mean of the population is m" and H1 was "The mean of the population is not m".  There is no PDF for H1.

Annex: Inapplicable sources on calculating beta

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*how to calculate type II error $\beta$?

*https://towardsdatascience.com/understanding-alpha-beta-and-statistical-power-525b84453687

*https://www.statology.org/beta-level

*Calculation of beta error in case of two-tailed test of mean
 A: The power of a hypothesis test (beta) is a function of effect size (and SEM, and sample size, and test 'size' (alpha). Therefore to calculate power you need to choose values for all of those variables. The choices can be thoughtful (e.g. use estimates from preliminary experimentation) or arbitrary (e.g. use alpha=0.05 just because).
Given those choices, your example of not knowing the relevant distribution is ill-directed because once you make the choices and the relevant distribution will often become known. In the case you give of the test hypothesis being that the population mean=$m$, the alternative becomes that the mean is a specific chosen value rather than just 'not $m$.
A: Even if everything is Gaussian, you don't know the distribution under the alternative hypothesis. You pick the difference you want to be able to detect.
If you want to detect a large difference, you have a lot more power to detect it than if you want to detect a small difference; it's easier to catch that something is off my a meter than off my a nanometer.
This idea applies even when you don't have the ideal distribution (often, but not necessarily, Gaussian). In many cases, a decent sample size leads to near-convergence of the true test statistic distribution to the theorized test statistic distribution. In such a case, you might just move past the fact that your data lack the ideal distribution and just do the power calculation the usual way, knowing that your imperfet calculation will be close. If you have greater concerns, you might turn to simulation.
