I was wondering how to proof this formula, commonly used in Bayesian Prediction:
$ \mathrm{P}(x|\alpha) = \int_\theta \mathrm{P}(x|\theta)\mathrm{P}(\theta|\alpha) \, \mathrm{d}\theta$
The left hand side can be expressed as the following, through marginalizing:
$ \mathrm{P}(x|\alpha) = \int_\theta \mathrm{P}(x, \theta | \alpha) \, \mathrm{d}\theta \quad \quad \ldots \text{(1)}$
Expanding the right hand side,
$ \int_\theta \mathrm{P}(x|\theta) \mathrm{P}(\theta|\alpha) \, \mathrm{d}\theta = \int_\theta \frac{\mathrm{P}(x,\theta)}{\mathrm{P}(\theta)} \frac{\mathrm{P}(\theta,\alpha)}{\mathrm{P}(\alpha)} \, \mathrm{d} \theta \quad \quad \ldots \text{(2)}$
Note that in equation (1), there will be a $\mathrm{P}(x,\theta,\alpha)$ term, but in equation (2), I can't see how that term will appear.
Thanks.