Likelihood Ratio and Sufficient Statistics

Question

I am not very experienced with statistics, so I apologize if this is an incredibly basic question. A book I am reading (Examples and Problems in Mathematical Statistics - Zacks) makes the following claim that I cannot understand

1. $X = (X_{1},\ldots, X_{n})$ has joint CDF belonging to $\mathcal{F}$ which is parametrized by $\theta\in \Theta$. $\mathcal{F}$ has some dominating measure through which all densities will be defined.
2. A likelihood function defined over $\Theta$ is any function $L(\theta | X)$ that is equal to the density $f(x | \theta)$ up to multiplication by a function depending only on $x$. That is, $L(\theta | x) = \alpha(x)f(x | \theta)$. Let $L^{T}$ be a likelihood of a statistic $T(X)$.
3. The Neyman Fischer factorization theorem says that $T(X)$ is sufficient iff there exists some nonnegative functions $a,b$ so that $f(x | \mu) = a(x)b(T(x),\mu)$
4. Claim: If $T(X)$ is a sufficient statistic, then the likelihood ratio $$L(\theta | X)/L^{T}(\theta | T(X))$$ is constant with respect to $\theta$.

Unfortunately, I don't see why this is true. Let $h$ be the density of $T(X)$ (with respect to any appropriate dominating measure)

$$\frac{L(\theta | X)}{L^{T}(\theta | T(X))} = \alpha(x)\frac{f(x | \theta)}{h(t(x) | \theta)} = \alpha(x)\frac{a(x)b(T(x),\theta)}{h(T(x)|\theta)}$$

Why is this constant with respect to $\theta$ if I don't know the dependence of $b$ or $h$ on $\theta$?

Answer 1

If $T(\cdot)$ is sufficient, then $L(\theta)$ depends on $y$ only through $T(y)$. (This follows from the definition of sufficiency, as another $y'$ such that $T(y)=T(y')$ would lead to an equivalent likelihood $L(\theta;y')$.)

That is, there is a $g$ such that $$ L(\theta)\propto g(T(y);\theta) $$ From the definition of the likelihood, we also know that $L(\theta)\propto f(y;\theta)$.

Hence, $f(y;\theta)/g(T(y);\theta)=:h(y)$ does not depend on $\theta$. This is so because $$L(\theta)\propto f(y;\theta)\Rightarrow L(\theta)=c\cdot f(y;\theta)$$ and $$L(\theta)\propto g(T(y);\theta)\Rightarrow L(\theta)=c'\cdot g(T(y);\theta),$$ such that $c'\cdot g(T(y);\theta)=c\cdot f(y;\theta)$ or $$f(y;\theta)/g(T(y);\theta)=c'/c$$