Equation 1-a is the Bayesian update rule for updating location probabilities. The left hand side represents the belief of being at location l . The right hand side numerator is a multiplication of the probability of observing sensor reading s at location l based on the sensor model, and the probability of being at location l based on the action model. The right hand side denominator is the normalization constant, which is the summation of the maximum likelihood estimates of the sensor reading s over all locations.
Equation 1-b shows how the probability of observing a sensor reading s , which is an array of signal strength values from W Wi-Fi access points, is calculated. It is a summation of the results of Gaussian functions ( G ) for the specified location l and sensor i , at the sensor values x i .
Equation 1-c shows how the probability of being at location l given the previous action a t-1 is calculated. To obtain the left hand side of the equation, the product of a) the probability of being at l given that the previous action was a t-1 and the previous location was i , and b) the prior probability of being at location l , is summed over all values of i ranging from 1 to N , where N is the number of locations.
This should be in the right column.