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Are the first two moments well defined for the horseshoe prior? I would say that the expectation is zero but the variance does not exist. Using the following argument.

Let $$\beta_i \mid \lambda_i, \tau \sim \mathcal{N}\left(0, \tau^2 \lambda_i^2 \right)\,, \qquad \lambda_i \overset{\mathrm{iid}}{\sim} \mathrm{C}^+(0, 1)\,, \qquad i = 1, \ldots, n\,,$$ where $\mathrm{C}^+(\cdot, \cdot)$ is the Half-Cauchy distribution, i.e., $\beta_i$ has the horseshoe distribution $\beta_i\overset{\mathrm{iid}}{\sim} \mathrm{HS}(0, \tau)$.

For $r = 1, 2$, we need to compute:

$$\int_{-\infty}^{\infty}\beta_i^r \int_0^\infty \;\mathrm{pr}(\beta_i \mid \tau, \lambda_i)\; \mathrm{pr}(\lambda_i)\;\mathrm{d}\lambda_i \;\mathrm{d}\beta_i.$$ The inner integral has no closed-form. I would be tempted to switch the order of integration, which would yield the expected result for the expectation, using the Fubini theorem.

But for the variance, we would end up with $$\tau^2 \int_{0}^{\infty}\lambda_i^2 \mathrm{pr}(\lambda_i)\;\mathrm{d}\lambda_i,$$ which doesn't exist (second moment of a Half-Cauchy distribution...).

Could you confirm that my arguments are correct?

Thank you very much.

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