x and y are common shorthand for inputs and outputs respectively. Saying "y=LLM(x)" denotes that the input tweets (x) are fed to a black-box function (LLM) that returns a classification (y) for that input. Given that we're talking about binary classifiation it's expected for y to be a value between 0 and 1, which leads to defining LLM(x) as "a function that receives a tweet x as input, uses an LLM, and returns a value in the [0, 1] range representing its classification as ironic or not".
The point about beta (which I think the equation doesn't actually reflect) is just to indicate that this approach risks nothing because, worst case scenario, the weights you assign to the model can simply return the original LLM prediction. Keep in mind that 0*inf=0 and that LLM(x) only returns 0 or 1, so it's not accurate to say that the sigmoid would always return 1.
The function I() is the indicator function [1] which, in this case, returns 1 if the argument is true and 0 otherwise. It's only there to convert booleans to integers because summing booleans is not defined.
The point about beta (which I think the equation doesn't actually reflect) is just to indicate that this approach risks nothing because, worst case scenario, the weights you assign to the model can simply return the original LLM prediction. Keep in mind that 0*inf=0 and that LLM(x) only returns 0 or 1, so it's not accurate to say that the sigmoid would always return 1.
The function I() is the indicator function [1] which, in this case, returns 1 if the argument is true and 0 otherwise. It's only there to convert booleans to integers because summing booleans is not defined.
[1] https://en.wikipedia.org/wiki/Indicator_function