Let (Theta_{0}=(0,1)) and let (P_{theta}) be the iid Bernoulli model (operatorname{Ber}(theta)). Let (mathcal{G}) be the additive group

Question:

Let $\Theta_{0}=(0,1)$ and let $P_{\theta}$ be the iid Bernoulli model $\operatorname{Ber}(\theta)$. Let $\mathcal{G}$ be the additive group of addition modulo one, so that $P_{g \theta}=\operatorname{Ber}(\theta+g)$ is the Bernoulli model with parameter $\theta+g$ modulo one. Explain why $\operatorname{Ber}(\theta) \mapsto \operatorname{Ber}(\theta+g)$ is not a group action on distributions in the sense of Sect. 14.3.2.

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