In the paper by Holbrook et al., 2011 (Division of labor increases with colony size in the harvester ant Pogonomyrmex californicus), there is a formula for calculating the mutual entropy between individual and tasks.
This involves the terms
p_j: the probability that any individual performed the jth task
p_i: the probability that the ith individual performed any task and
p_ij: the joint probability that the ith individual performed the jth task.
The formula for calculating mutual entropy is I_indiv,tasks = Σ p_ij * log(p_ij / (p_i * p_j))
If I have a individual x task data matrix where each entry is the number of times an individual performed a particular task, I would calculate
p_i as the number of tasks performed by individual_i / total number of tasks
p_j as the number of individuals performing task j / total number of individuals.
I'm understanding p_ij as the number of tasks performed by individual_i, if they performed task_j. Also, joint probability p_ij, in my mind is p_i multiplied by p_j.
if that's the case, wouldn't the term p_ij / (p_i * p_j) in the formula always be 1?
Or talk to me like I'm 5 and someone please let me know how i calculate p_ij?
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