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You may want t consider the von Mises distribution, aka Tikhonov distribution, and plays the role similar to the normal distribution in 1D statistics: $$p(\theta ; \alpha, \theta_0 ) = \frac{ e^{\alpha \cos (\theta -\theta_0)}} {2 \pi I_0(\alpha)}$$ For $\alpha=0$ it is uniform, for $\alpha >> 1$ the distribution is sharply peaked at $\theta_0$ C....

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There are alternatives, for example, you can use constrained optimization, or regularization. Notice however, that in most cases those approaches can be thought as Bayesian inference in disguise. For example, constraining range of the parameter during optimization, is the same as using flat prior over this range. Using $L_2$ regularization is the same as ...

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One way in which prior information can be incorporated into the estimator is through the likelihood (or model, depending on how you look at it). That is to say, when we build a standard parametric model, we are constraining ourselves to say that we are going to allow the model to follow a very specific form, that we know up to the values of the parameters ...

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The most obvious thing to do here would be to express the variable in polar coordinates and impose a prior on the angle and displacement. That is, you express your point $(x,y)$ as a vector $(\theta,r)$ where: \begin{aligned} x &= r \cos \theta, \\[4pt] y &= r \sin \theta. \\[4pt] \end{aligned} You can then impose a prior on $0 \leqslant \... 2 It is unusual, even with studentization, to see less variability for greater predicted values, and yet that's what you have here. There are three causes to consider: 1) misspecified mean-variance relationship, 2) omitted regressors or 3) undetected non-independence. They imply vastly different approaches to the problem of obtaining valid inference. 1) ... 2 You are right in the sense that in Bayesian statistics the space of parameters$\Theta$is endowed with the structure of a probability space$(\Theta, \mathcal{A},\pi)$where$\mathcal{A}$is a$\sigma$-algebra and$\pi$a probability (prior) so it make sense to ask questions of the form $$B\in \mathcal{A},\quad \int_B f(\theta)\pi(d\theta)$$ Where this ... 2 As also pointed out in the comments, you don't need prior since all you need is the posterior: $$P(\text{H_1 is true}|\mathbf{x})=P(\theta\leq0.5|\mathbf{x})=\int_0^{0.5} \pi(\theta|\mathbf{x})d\theta$$ Since this is Beta distribution,$0\leq\theta\leq1$, a uniform prior on$\theta$would be$\pi(\theta)=1$and you wouldn't notice it in the integration ... 1 I think it'd be easier to calculate the following:$$P(\Theta \leq c| Y = 0)=\int_{0}^c f_{\Theta|Y=0}(\theta)d\theta$$ where $$f_{\Theta|Y=0}(\theta)=\frac{P(Y=0|\Theta=\theta)f_\Theta(\theta)}{P(Y=0)}=\frac{e^{-\theta}\lambda e^{-\lambda\theta}}{P(Y=0)}$$ which is easy to integrate. 1 If you have a prior for the angle, I'd use it as the reference. E.g. I'd rotate all data so that the prior is at$180^{\circ}$and measure all angles on the scale$[0^{\circ}, 360^{\circ})$. I see no elegant solution to measuring distance between two angles,$\phi$and$\psi$. I'd calculate the differences$(\phi - \psi)$and$(((\phi + 180^{\circ}) \mod ...

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I think Einstein's use of the mollusc was simply a counterpoint to the concept of a "rigid body". If you were to rapidly accelerate a titanium anvil (from behind, as in a push) it would physically shorten an infinitesimal amount in the direction of travel. A jellyfish, on the other hand, undergoing the same process would be considerably more distorted... .

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