We continue our discussion on determining a model to represent the system.The example to summarize the text made use of a neural net model. A model articulates how a system behaves quantitatively. 
An inverse model is a mathematical model that fits experimental data. It aims to provide a best fit to the data. 
There are two ways by which we can select the appropriate model. The first is by observing trend lines which correspond to some well known mathematical formula. The second is on the observation of underlying physical processes which contribute towards the system. These physical interpretations contribute to model parameters.
 In order to fit the data points, a model may use least squares of errors. the errors called residuals may be both positive or negative which result in inaccurate measure. Instead the squares of the errors can be minimized to give a better fit.
We used the least squares error minimization to fit the data points. Another way to do this is using Maximum likelihood estimation. This derives inspiration from the irony that we seem to be asking whether our model parameters are correct. In reality, as the authors of the biomedical modeling explained, there is only one correct parameter set. This is Mother Nature since the experimental data is the only corroboration of any assumption. The experimentation results are also not the source of truth if we are subjecting it to our mode of questioning but if we let the data come in without assumptions, then they are at least one correct parameter set. A model is our imagination which we tether to data. While there may be great reward for believing our model is correct, the community specifically cautions against this. If we go by the data we know that the model will be more reliable. the other way around of fitting the model has the problem that we could be self-gratifying ourselves. There has to be a trade-off in what is right and the community and peer review of the model helps validate this effort. But in order to bring it up to them, we have to make our model speak for itself and support it with our own validations.
The maximum likelihood estimation attempts to resolve this concern by asking "Given my set of model parameters, what is the probability that this data set occurred ?"  This is translates as likelihood for the parameters given the data.
With the help of this measure, a model for the system can be accepted as representative of the assumptions. The inverse modeling is therefore also called "maximum likelihood estimation".   The chi-square error measure and maximum likelihood estimation have a relation between the two.  For a Gaussian distribution, it is easy to see that the probability of the data set coming from the model parameters involves minimizing the negative natural log of probability which is the chi-square function of weighted residuals. Furthermore, if the variance is uniform, then the chi-square function yields the sum of squared residuals defined earlier.

#codingexercise 
yesterday we talked about a helper method to evaluate expressions involving single digit numbers and without brackets.
int eval (List <char> exp)
{
int result = 0;
bool add = true;
for (int I =0; i < exp.count; i++)
{
if  (exp [i] == '-') {
add = false;
continue;
} else if (exp  [i] == '+'){
add = true;
continue;
} else {
if (add) result += Convert.toInt (exp [i]);
else result -= Converter.toInt (exp [i]);
add = true;
}
return result;
}