# On the predictive distribution regarding Bayesian methods for pattern recognition

I am following Pattern recognition and machine learning by Byshop and I was trying to derive myself the predictive distribution resulting from a new data point. For the sake of clarity I post the piece of text in question:

where $w$ are the parameters of the model treated as a random variable since we are in the Bayesian setting.

Anyhow I was trying to derive this myself and what I would do is

$$p(t|x, \textbf{x}, \textbf{t}) = \int p(t|x, \textbf{x}, \textbf{t},w) dw = \int \frac{p(t,x, \textbf{x}, \textbf{t},w)}{p(x, \textbf{x}, \textbf{t},w)} dw = \dots$$

And then I am unsure how to proceed. How could this be done?

EDIT: I was trying to apply Bayes rule with multiple conditions as shown here but to no avail.

$$p(t|x, \textbf{x}, \textbf{t}) = \int p(t, w|x, \textbf{x}, \textbf{t}) dw$$
$$\int p(t, w|x, \textbf{x}, \textbf{t}) dw = \int p(t|w, x, \textbf{x}, \textbf{t})p(w|x,\textbf{x},\textbf{t})dw$$
Now, recognizing that $$t$$ does not depend on $$\textbf{x}$$ and $$\textbf{t}$$, and $$w$$ does not depend on $$x$$,
$$\int p(t|w, x, \textbf{x}, \textbf{t})p(w|x,\textbf{x},\textbf{t})dw = \int p(t|w,x)p(w| \textbf{x}, \textbf{t})dw$$