I am used to fitting data to estimate parameters in a model f(x,A) where x is my data point and A is the parameter(s) to be determined. To achieve this I usually use chi-squared fitting, so I am minimizing

[TEX]\chi^2 = \sum \frac{(y_i - f(x_i,A))^2}{\sigma_i^2}[/TEX].

I have recently been given an almost identical task except that my data is now succeed/fail as opposed to a continuous value. So my data is now a 1 or a 0 and my model predicts the probability of success p(x,A). I know that this is a Bernoulli trial and I expected to be able to use the maximum likelihood method without too much difficulty. I thought I had done but I have a question. I have determined the following expression for the Bernoulli maximum likelihood which I think is correct

[TEX] L(p,y) = \Pi^{n}_{i} p^{y_i}(1-p)^{1-y_i}[/TEX].

The problem that I have is that I don't have lots of trials to find the probability. I have a single trial at a number of different data points. What I ideally want is an expression of the form

[TEX] L(p....p_n,x....x_n,y....y_n) = \Pi^{n}_{i} (p(x_i ,A))^{y_i}(1-(p(x_i ,A)))^{1-y_i}[/TEX]

so that my probability also depends on the data point not just the outcome. Is this also a valid expression? If so is there an easy way to turn the product into a sum? It seems harder in the second equation.

Thanks.

[TEX]\chi^2 = \sum \frac{(y_i - f(x_i,A))^2}{\sigma_i^2}[/TEX].

I have recently been given an almost identical task except that my data is now succeed/fail as opposed to a continuous value. So my data is now a 1 or a 0 and my model predicts the probability of success p(x,A). I know that this is a Bernoulli trial and I expected to be able to use the maximum likelihood method without too much difficulty. I thought I had done but I have a question. I have determined the following expression for the Bernoulli maximum likelihood which I think is correct

[TEX] L(p,y) = \Pi^{n}_{i} p^{y_i}(1-p)^{1-y_i}[/TEX].

The problem that I have is that I don't have lots of trials to find the probability. I have a single trial at a number of different data points. What I ideally want is an expression of the form

[TEX] L(p....p_n,x....x_n,y....y_n) = \Pi^{n}_{i} (p(x_i ,A))^{y_i}(1-(p(x_i ,A)))^{1-y_i}[/TEX]

so that my probability also depends on the data point not just the outcome. Is this also a valid expression? If so is there an easy way to turn the product into a sum? It seems harder in the second equation.

Thanks.

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