- #1

kullbach_liebler

- 1

- 0

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?

$$\#(\mathcal{B} \cap \mathcal{A}) > \#(\neg \mathcal{B} \cap \mathcal{A})$$

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?

Last edited: