= Solution
For independent observations in a <natural exponential family>, the log <likelihood> is
$$
\ell_n(\theta)=n\{\theta^T\bar Y-\kappa(\theta)\}+\text{constant},\quad U_n(\theta)=n\{\bar Y-\nabla\kappa(\theta)\},\quad j_n(\theta)=n\nabla^2\kappa(\theta).
$$
In a <minimal exponential family> the <Observed Fisher information> matrix is positive definite, so the log <likelihood> is <strictly concave>. Any finite solution of the <likelihood> equation $\nabla\kappa(\widehat\theta)=\bar Y$ is therefore the unique <global maximum> <likelihood> estimator. Existence is a separate issue: the observed sufficient-statistic mean must belong to the image of the mean parameter map $\nabla\kappa$.
For a full regular minimal <natural exponential family> this image is the interior of the <convex support of an exponential family>, $C=\overline{\operatorname{conv}(\operatorname{supp}(h\nu))}$. The necessity follows from a supporting hyperplane argument. If a finite parameter had its mean on a boundary face, some nonzero $v$ would satisfy $v^TY\leq b$ <almost surely> and $\mathbb E(v^TY)=b$. Equality of the <expectation> forces $v^TY=b$ <almost surely>. <Exponential tilting> preserves the base-measure <null sets>, so this contradicts minimality.
For sufficiency, let $\bar Y$ be an interior point of $C$. Choose finitely many points $y_1,\ldots,y_m$ of the support whose <convex hull> contains a ball about $\bar Y$, say of radius $2\varepsilon$. Such points exist because $\bar Y$ is interior to the <closed convex hull>; one may approximate the vertices of a small simplex surrounding it by finite <convex combinations> of support points. Take support neighbourhoods of radius at most $\varepsilon$ with positive finite base-measure masses $a_j$. These masses can be taken finite by restricting to bounded neighbourhoods and using finiteness of the normalizing integral at any fixed interior parameter. For every vector $\theta$, at least one $j$ has $\theta^T(y_j-\bar Y)\geq2\varepsilon\|\theta\|$. Integrating over that neighbourhood yields
$$
\kappa(\theta)-\theta^T\bar Y\geq\varepsilon\|\theta\|+\min_j\log a_j.
$$
Thus the negative log <likelihood> grows to infinity as $\|\theta\|\to\infty$. At a finite boundary point of the full open <natural parameter space>, the normalizing integral is infinite, since otherwise that point would itself belong to the full domain. <Fatou's lemma> makes $\kappa(\theta)$ diverge on approach to such a boundary. A <minimizing sequence> for $\kappa(\theta)-\theta^T\bar Y$ therefore stays in a bounded subset away from the boundary; it has an interior limit where the minimum is attained. Its <gradient> vanishes, giving the required <likelihood> equation. <Strict convexity> gives uniqueness.
Hence, under these full, regular and minimal conventions,
$$
\boxed{\text{a finite unique MLE exists}\ \Longleftrightarrow\ \bar Y\in\operatorname{int}C.}
$$
This is <maximum likelihood existence in a full regular natural exponential family>. Regularity alone does not make every sample admissible: in a <Bernoulli distribution> an all-zero or all-one sample has its <sample mean> on the boundary of $C=[0,1]$, and the canonical maximum is approached only as $\theta\to-\infty$ or $+\infty$. The usual probability parameter then has a boundary maximum in an enlarged model, but no finite canonical maximizer. A regular restriction of the <natural parameter space> requires membership in its actual mean-map image, rather than merely in $\operatorname{int}C$. Without minimality, affine redundancies may prevent uniqueness of the canonical parameter even when the fitted <probability distribution> is unique.
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