Suppose that each of $k$ statisticians has his own prior distribution for a certain parameter $\theta$, and let $f_{i}$ be the g.p.d.f. which statistician $i$ assigns to $\theta$, $i = 1, ..., k$. Suppose also that an executive forms his opinion about $\theta$ from the opinions of the $k$ statisticians and that he assigns to $\theta$ the g.p.d.f. $f^{*}$ defined, at each point $\theta \in \Theta$, as follows:
$$f^{*}(\theta) = \alpha_{1}f_{1}(\theta)\ + ... +\ \alpha_{k}f_{k}(\theta)$$
Here $\alpha_{1},\ ...,\ \alpha_{k}$ are weights such that $\alpha_{i} \geq 0$, $i = 1, ..., k$, and $\alpha_{1} +...+\alpha_{k} = 1$. The value of $α_{i}$ reflects the relative weight that the executive gives to the opinion of statistician $i$. Suppose further that the $k$ statisticians and the executive observe together the value of a random variable $X$ whose conditional g.p.d.f., given $\theta$, is $f(·|\theta)$. Show that the posterior g.p.d.f. of the executive will again be a linear combination of the posterior g.p.d.f.’s of the $k$ statisticians, with new weights $\beta_{1}, ..., \beta_{k}$ which will depend on the observed value of $X$. Also, discuss the conditions under which the weight $\beta_{1}$ in the posterior g.p.d.f. will be greater than the weight $\alpha_{1}$ in the prior g.p.d.f.
Attempt:
By the Bayes theorem, we have:
$$f(\theta|X) = \frac{f(X|\theta)f(\theta)}{\int_{\Theta} f(x|\theta ')f(\theta ') d\theta'} \propto f(X|\theta)f(\theta)$$
Thus,
$$f_{i}(\theta|X) \propto f(X|\theta)f_{i}(\theta)$$
See that
\begin{align} f^{*}(\theta|X) \propto f(X|\theta) f^{*}(\theta) &= f(X|\theta) [\alpha_{1}f_{1}(\theta) + ... + \alpha_{k}f_{k}(\theta)]\\ &= \alpha_{1}f(X|\theta)f_{1}(\theta) + ... + \alpha_{k}f(X|\theta)f_{k}(\theta)\\ & \propto \beta_{1}f_{1}(\theta|X) + ... + \beta_{k}f_{k}(\theta|X) \end{align}
Then the posterior g.p.d.f of the executive is linear combination of the posterior g.p.d.f's of the $k$ statisticians. My attempt is correct? Someone can help to solve the second question about the weight $\beta_{1}$? Thank you in advance.