some question from your paper

[biaohe]

Dear Prof Perdikaris :
Currently i'm interested in Gaussian process and have read your paper"Nonlinear information fusion algorithms for data-efficient multi-fidelity modeling ".
Whats frustrating is that i have some questions about equation(2.14) in your paper:
latex $$ \int{p(f_t(x_*,f_{*t-1})|y_t,x_t.x_*)p(f_{*t-1}(x_*))}dx_*)$$
I think it should be
$$\int{p(f_t(x_*,f_{*t-1})|y_t,x_t.x_*)p(f_{*t-1}(x_*))}df_{*t-1}(x_*)$$ ,
because
$$p(A)=\int{p(A,B)dB}$$ .
It give me a little trouble.
Thanks for your help!

[paraklas]

Yes, I think you are right. Thanks for spotting the typo.

[biaohe]

Thanks for your reply!Another question is also about equation(2.14):
latex $$ \int{p(f_t(x_*,f_{*t-1})|y_t,x_t.x_*)p(f_{*t-1}(x_*))}dx_*)$$
In your paper,you had figure out the distribution is not Gaussian,but in your code,you have used GPR to get the mean and var for$$p(f_t(x_,f_{t-1})|y_t,x_t.x_)$$ and$$p(f_{t-1}(x_))$$. It implied that $f_t(x_,f_{t-1})$ and $f_{t-1}(x_)￥ is Gaussian(or approx gaussian) random variable because "In a Gaussian process, every point (predict point or observed data)in some continuous input space is associated with a normally distributed random variable".
Additionally. I think it may be :
$$f_t(x_,f_{t-1}) \sim N(\mu_{f_t|f_{t-1}},\sigma_{f_t|f_{t-1}}),\mu_{f_t|f_{t-1}}=Z_{t-1}(f_{t-1}(x_))$$
$$f_{*t-1} \sim N(\mu_{|f_{t-1}},\sigma_{f_{t-1}})$$

[biaohe]

add follows in line 89,example_1D,we can see the same result of mean and var .

my mean and variance

    my_mean,my_var = m2.predict(np.hstack((Xtest,mu1)))

as bayesian linear regression for predict p(f*)=\int{p(f*|w)p(w)dw},p(f*)=p(f*|mu(w)),here,p(w)=N(w;mu(w),sigma(w))
i think this is the reason