A Does Maximizing Likelihood Over Truncated Support Increase Probability?

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Maximizing likelihood over truncated support can lead to a greater probability than evaluating at a specific parameter vector, as indicated by the proposition discussed. The conditions for this statement to hold include having more elements in the intersection of sets defined by certain thresholds than outside of them. Clarification is sought on how to present this proposition more effectively, suggesting a more verbal explanation alongside the mathematical notation. A question arises regarding the scenario where the maximizing value equals the initial parameter vector, prompting further exploration of implications. Overall, the discussion emphasizes the relationship between parameter optimization and probability outcomes in statistical contexts.
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Suppose ##\mathbf{X}## is a random variable with a finite support ##\Omega## and with some pdf ##f(\cdot; \mathbf{v}_0)## where ##\mathbf{v}_0## is the parameter vector. Define, ##\mathcal{A}:= \{\mathbf{x}:S(\mathbf{x}) \geq \gamma\} \subset \Omega## and ##\tilde{\mathbf{x}}:=S(\tilde{\mathbf{x}}) = \gamma##, ##\mathcal{B}:=\{\mathbf{x} \geq \tilde{\mathbf{x}}\}## where ##S:\mathbf{x} \mapsto \mathbb{R}^n##. Moreover, suppose that,

$$\#(\mathcal{B} \cap \mathcal{A}) > \#(\neg \mathcal{B} \cap \mathcal{A})$$
and
$$!\exists \mathbf{x}^*>\tilde{\mathbf{x}}:=\arg \max_{\mathbf{x}}S(\mathbf{x}).$$

Then, it implies that,

\begin{align}
\max_{\mathbf{v}}\sum_{\mathbf{x} :S(\mathbf{x}) \geq \gamma} f(\mathbf{x};\mathbf{v}) > \sum_{\mathbf{x} :S(\mathbf{x}) \geq \gamma} f(\mathbf{x};\mathbf{v}_0)
\end{align}

My questions:

1) Is the statement true?;
2) How could I improve the presentation of this proposition?;
3) What are the mildest possible conditions under which (1) will hold?
 
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kullbach_liebler said:
2) How could I improve the presentation of this proposition?;
I don't know what a statistical journal would want, but for the purpose of getting an answer on an internet forum, you could give a more verbal statement of the proposition before presenting it using only notation.

\begin{align}
\max_{\mathbf{v}}\sum_{\mathbf{x} :S(\mathbf{x}) \geq \gamma} f(\mathbf{x};\mathbf{v}) > \sum_{\mathbf{x} :S(\mathbf{x}) \geq \gamma} f(\mathbf{x};\mathbf{v}_0)
\end{align}

This appears to say the maximum value of a function ( which is a summation rather than an integration) when taken over a set is greater than the value of that function evaluated at the particular element ##v_o## in that set. Is that the general idea?
 
kullbach_liebler said:
1) Is the statement true?
Let ## \mathbf{v_max} ## be a value of ## \mathbf{v} ## which maximizes the sum. What happens when ## \mathbf{v_0} = \mathbf{v_max} ##?
 
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