= Solution
Let $M=\sum_{i=1}^n m_i$ be the total exposure, and define the <prior distribution> <expected value>, prior <variance> and average binomial noise by
$$
\eta=\mathbb E\theta,\qquad v=\operatorname{Var}(\theta),\qquad w=\mathbb E[\theta(1-\theta)].
$$
Since the <prior distribution> has a <probability density function> on $(0,1)$, $v>0$ and $w>0$. All relevant <moments> exist because the parameter and observations are bounded. The <binomial distribution> gives
$$
\mathbb E[X_i\mid\theta]=\theta,\qquad
\operatorname{Var}(X_i\mid\theta)=\frac{\theta(1-\theta)}{m_i}.
$$
The <law of total expectation>, <law of total variance> and <law of total covariance> consequently yield
$$
\mathbb E X_i=\eta,\quad
\operatorname{Var}(X_i)=v+\frac{w}{m_i},\quad
\operatorname{Cov}(X_i,X_j)=v\ (i\ne j),\quad
\operatorname{Cov}(\theta,X_i)=v.
$$
In particular, <conditional independence> of the annual observations does not imply unconditional <independence>: they share the uncertain parameter.
To derive the optimal <credibility estimate> among <affine functions> of the observations, put $A=\sum_i a_i$ and write $X_i=\theta+\varepsilon_i$. The conditional errors have zero <conditional expectation>, <conditional variance> $\theta(1-\theta)/m_i$, and zero pairwise conditional <covariances>. They are also uncorrelated with every integrable function of $\theta$. Therefore the joint <mean squared error> is
$$
\begin{aligned}
L(a_0,a_1,\ldots,a_n)
&=\mathbb E\left[\big((1-A)\theta-a_0-\sum_i a_i\varepsilon_i\big)^2\right]\\
&=((1-A)\eta-a_0)^2+v(1-A)^2+w\sum_i\frac{a_i^2}{m_i}.
\end{aligned}
$$
For fixed slopes, the first term is uniquely minimized by $a_0=(1-A)\eta$. For fixed sum $A$, the <Cauchy-Schwarz inequality> gives
$$
A^2=\left(\sum_i\frac{a_i}{\sqrt{m_i}}\sqrt{m_i}\right)^2
\le M\sum_i\frac{a_i^2}{m_i},
$$
with equality exactly when $a_i=A m_i/M$. Thus the remaining optimization is the strictly convex quadratic
$$
v(1-A)^2+\frac{w}{M}A^2.
$$
Its derivative vanishes at $A=Mv/(Mv+w)$. Consequently the <Bühlmann–Straub credibility estimate> is
$$
\boxed{\widehat\theta=Z\frac{\sum_i m_iX_i}{M}+(1-Z)\eta,\qquad
Z=\frac{Mv}{Mv+w}=\frac{M}{M+w/v}.}
$$
The individual coefficients are $a_i=Zm_i/M$ and $a_0=(1-Z)\eta$. The strict quadratic minimizations establish global optimality and uniqueness, not just necessary equations. Here $v$ is the <variance of hypothetical means>, $w$ is the <expected process variance>, and $Z$ is the <Bühlmann–Straub credibility factor>. More exposure gives more weight to the observed claim fraction. Since $0<Z<1$, the estimate remains between the observed pooled fraction and the prior <expected value>. If one separately permits a degenerate prior, $v=0$ leads to the constant estimate $\eta$, interpreted as $Z=0$.
For the <uniform distribution> prior on $(0,1)$,
$$
\eta=\frac12,\qquad v=\frac1{12},\qquad
w=\int_0^1\theta(1-\theta)\,d\theta=\frac16.
$$
With two years, put $M=m_1+m_2$ and $s=Y_1+Y_2=m_1X_1+m_2X_2$. Then $Z=M/(M+2)$ and
$$
\boxed{\widehat\theta=\frac{s+1}{M+2}
=\frac{m_1X_1+m_2X_2+1}{m_1+m_2+2}.}
$$
To compare with the <Bayes estimator under squared error loss>, use <conditional independence> to multiply the two binomial <likelihood functions>. Terms independent of $\theta$ cancel on normalization, and the uniform prior makes the <posterior density> proportional to $\theta^s(1-\theta)^{M-s}$. Hence <Beta-binomial conjugacy> gives
$$
\theta\mid(Y_1,Y_2)\sim\operatorname{Beta}(s+1,M-s+1),\qquad
\mathbb E[\theta\mid Y_1,Y_2]=\frac{s+1}{M+2}.
$$
Finally, for an arbitrary reported value $d$, conditional <quadratic loss> decomposes as
$$
\mathbb E[(\theta-d)^2\mid Y_1,Y_2]
=\operatorname{Var}(\theta\mid Y_1,Y_2)
+\big(d-\mathbb E[\theta\mid Y_1,Y_2]\big)^2.
$$
The unique minimum is the <posterior mean>. Therefore \b[the affine <credibility estimate> and the exact Bayesian estimate coincide] in this case. The agreement illustrates <exact beta-binomial credibility with unequal exposures>; for a general <prior distribution>, minimizing over <affine functions> of the observations need not recover the unrestricted <posterior mean>.
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