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PostsP(\theta)\thetaDP(\theta\vert D)DP(\theta\vert D)\pi(\theta)\pi(\theta) = P(\theta\vert D)=\dfrac{P(D\vert\theta)P(\theta)}{P(D)}P(D\vert \theta)P(D\vert \theta)P(\theta)\pi(\theta)P(D)P(D)\thetaP(D) = \int_{\Theta} P(D\vert\theta)P(\theta)\text{d}\theta\pi(\theta)\pi(\theta)\theta_1, \dots, \theta_N\theta\begin{split} P(D) &= \int_{\Theta} P(D\vert \theta)P(\theta)\text{d}\theta \\ &\approx \dfrac{1}{N}\sum_{i=1}^N P(D\vert\theta^{(i)})\end{split}N\pi(\theta)\theta(R, +, \cdot, 0, 1)R(+, \cdot)(0, 1)\bf{Ring}\bf{Rng}\it{I}: \bf{Ring} \rightarrow \bf{Rng}\it{A}: \bf{Rng} \rightarrow \bf{Ab}\it{A}\it{A}(S, \cdot, e)S\cdote\it{M}: \bf{Ring} \rightarrow \bf{Mon}\bf{Mon}\mathcal{X}(X, *, e)\mathcal{Y}(Y, *’, f)\it{\phi}: \mathcal{X} \rightarrow \mathcal{Y}\mathcal{X}\mathcal{Y}\begin{equation}\phi(a * b) = \phi(a) *' \phi(b), \forall a\; b \in \mathcal{X}\end{equation}\begin{equation}\phi(e) = f\end{equation}

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