I’m working on a model where it is convenient to use a triangular matrix for calculation of different log-likelihoods for different observed data. I can then increment these log-likelihoods by the number of observations that correspond to each log-likelihood in the modelling step and sum them to obtain the total log-likelihood. It turns out this operation has a name: the Frobenius inner product.
For example, in my transformed parameters block I wind up with a T x T matrix with entries [LL_11, LL_12, LL_13, …, LL_1T] in the first row, [0, LL_22, LL_23, … , LL_2T] in the second row on down to the final row which is T - 1 zeroes with LL_TT in the final position. In my transformed data block, I gather my data such that I have a matrix of counts: [n_11, n_12, …, n_1T], [0, n_22, … , n_2T], etc.
In pseudocode these are:
matrix[T, T] log_likeli; matrix[T, T] the_Ns;
So my question is: what’s the most efficient way to calculate the Frobenius inner product, given that a little over half of my matrix consists of zeroes? I can think of three ways.
Sum the matrix resulting from element-wise multiplication:
target += sum(log_likeli .* the_Ns);
Vectorise to a row_vector and vector and multiply:
target += to_row_vector(log_likeli) * to_vector(the_Ns);
Transpose and loop over rows/columns omitting zero terms through vector slicing:
for (i in 1:T)
target += log_likeli[i,i:t] * the_Ns[i, i:t]’;
I’ll be doing some testing, but I’d curious to hear if this has already been considered and what the best way might be for those who have already walked down this path.