Idea about Bayes

The problem with Bayes is that one must try all probability distributions with all possible parameters. I still think there should be a way to avoid this. What about this one?

What Bayes is actually doing is comparing how low is the cost in the actual configuration, compared with how low it could have been. We might do it by finding the ratio of the cost in the actual configuration, over the average cost for all configurations. This average cost might be computed along each neuron, or at a random sample in the whole space.

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Warning: Perhaps all wrong in neuron branching

In principle, our way of finding the optimal diameter was wrong: Murray’s law is satisfied by optimal systems, but is not a recipe to find the optimum. To do so, one must know the cost parameters (relation between the two terms and relative ‘importance’ of each branch), and then calculate the cost. When we just fix two branches and calculate the third with Murray’s law, we are implicitly changing the parameters of the cost (we go to a bifurcation whose ‘best fitting’ parameters are different than the initial ones). If we calculate the new cost with the old parameters, in some cases it may increase instead of decrease. For example:

>> d1=2; d2=1; d3=1;
>> C=L1*d1^2 + L1/d1 + L2*d2^2 + .5*L2/d2 + L3*d3^2 + .5*L3/d3
C =
12.7500
>> d1^3-d2^3-d3^3
ans =
6
>> d2=(d1^3-d3^3)^(1/3);
>> d1^3-d2^3-d3^3
ans =
1.7764e-015
>> C=L1*d1^2 + L1/d1 + L2*d2^2 + .5*L2/d2 + L3*d3^2 + .5*L3/d3
C =
16.3810

However, I think it is possible to do it correctly. For example, one can first calculate reasonable parameters for the network, and then calculate the optimum for each branch, with respect to those parameters. Also, do not think that the results found so far are completely meaningless: Although they are strictly incorrect, they do indicate a trend to larger deviations for shorter branch lengths.

New Figure 3 (Bayes)

>> fig_pap_bayes_09(1,zeros(100))