- #1

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I am studying a problem related to Bayesian probability, and I came across two equations, which as far as I can tell can only be solved numerically, but as I'm no expert I would like to hear your opinion, please.

The first one is:

[itex]P(a) \cdot \left[ 1 - \Phi \left( \frac {x - \mu_a} { \sqrt 2 \cdot \sigma_a} \right) \right] = (1 - P(a) ) \cdot \Phi \left( \frac {x-\mu_b} { \sqrt 2 \cdot \sigma_b} \right) [/itex]

where:

[itex] \Phi (y) = \frac {1} {\sqrt {2 \pi}} \cdot \int_{- \inf}^y {e^{-t^2 / 2}} \, dt = \frac 1 2 \cdot \left[ 1 + {erf} \left( \frac {y} {\sqrt 2} \right) \right] [/itex]

P(a) is a probability, thus a real (?) number between 0 and 1, and I need to solve for x.

Initially I had no doubt that this could

*not*be solved analytically for x. But then as I was reading something about statistical power, in an example they showed how you can invert Φ using a 'probit' function, so I wondered if it's possible after all. I would have thought the inversion required Φ to be 'isolated', and this didn't seem possible here. But I'd be glad to be proven wrong!

The second one is:

[itex]N \cdot P(a) = \sum_{i=1}^N {\frac {P(a) \cdot A_i} {P(a) \cdot A_i + [1-P(a)] \cdot B_i} } [/itex]

where P(a) is as above, N is a positive integer and:

[itex] A_i = 1 - \Phi \left( \frac {x_i - \mu_a} { \sqrt 2 \cdot \sigma_a} \right) [/itex]

[itex] B_i = \Phi \left( \frac {x_i - \mu_b} { \sqrt 2 \cdot \sigma_b} \right) [/itex]

and I need to solve for P(a).

If I understand correctly, Φ has the property:

[itex] \Phi (-x) = 1 - \Phi (x) [/itex]

but I don't see if/how that helps me in this case.

Any idea?

Thanks

L