bayesian curve fitting

Question

On Bishop's Pattern Recognition and Machine Learning book, equation 1.68 says

$p(t|x,X,T) = \int p(t|x,w) p(w|X,T) dw$

Here t is the target value, (X,T) is training dataset. I do now understand how the RHS came from LHS. Intuitively it makes sense. My confusion is from $p(t|x,X,T) = \int p(t|x,w,X,T) dw$, should not RHS be $\int p(t|x,w) p(w)$? I know $w$ is dependent on $X$ and $T$, we have to estimate $w$ from training data. But how it came mathematically?

Answer 1

The predictive distribution $p(t|x,X,T)$ is the marginal distribution of $p(t|x,w) p(w|X,T) $.

The Marginal distribution is givne as:

$ p_U(u) = \int_y p_{U,V}(u,v) \, \mathrm{d}v = \int_v p_{U\mid V}(u\mid v) \, p_V(v) \, \mathrm{d}v (1) $

Using Conditional probability rules you can show that

$p(t|x,w) p(w|X,T)=p(t,w|x,X,T) (2)$

Comparing (1) and (2) we see the correspondence $t=u$ and $w=v$

$p(t|x,X,T)= \int p(t,w|x,X,T) dw$