I use a neural network to classify the sentiment of some articles per day. Possible results are [1,2,3,4,5] (1=very negative, ..., 5 = very positive). Using one article as input the network gives me as output, a number 1-5, describing the sentiment and a the corresponding probability (a softmax probability), e.g. sentiment = 2, probability = 0.65.
If I calculate the mean sentiment of today's articles, is there a way to assign to this mean (the average of the articles sentiments) a probability that will take into account the softmax probabilities assigned to each article?
Assume that you have n articles, $X_1,...,X_n$ with mean sentiment $\bar X$. A probability that takes into account the softmax probabilities is
$$\sum_{x_i\in\{1,2,3,4,5\}\\\bar x_i=\bar X} P(X_1=x_1)*P(X_2=x_2)*...*P(X_n=x_n)$$
It is probably better to allow some leeway for the average, with l being a number between 0 and 4, 0 being the most exact (giving you the smallest probability and equivalent to the expression above) and 4 being the most forgiving (giving you a probability of 1).
$$\sum_{x_i\in\{1,2,3,4,5\}\\|\bar x_i-\bar X|\le l} P(X_1=x_1)*P(X_2=x_2)*...*P(X_n=x_n)$$