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Please note that my question is not about coding.

I am now learning Bayesian classification and I think I understand it in a discrete case. I have trouble understanding it for multivariate continuous data. My problem is in calculating the posterior.

Example:

Assume that we have a data set with two classes. The first class (D) is the patient with a disease. The second class (H) is healthy patients. Assume that I have 25 patient. 15 of patients are suffer from the disease. Now we can estimate the posterior probability that the new patient is healthy (using Bayesian theorem) as:

$P(Glass= H | X=x)= \frac{\pi_{H} f_{h}(x)}{\pi_{H}f_{H}(x)+\pi_{D}f(_{D}(x)} \rightarrow {(1)} $.

Where x is a realisation of the random vector X (the patient), $f_{H}, f_{D}$ are the density of each class respectively.

Assume that I fit a copula model to this data. Assume further that I calculate the density of each class. Using copula, I will have this:

This is just an example not a real data (for a simplification I assumed that u is the data.)

 set.seed(123)
u <- rCopula(15, frankCopula(3))
f_D <- dCopula(u, frankCopula(2.5))
f_D
 [1] 1.3446937 1.4909407 0.6999004 1.6392250 0.4032475 2.2401484 1.0616001 1.1444106 0.6963312 1.1229705
[11] 1.0431500 1.1350960 1.1060839 1.1336352 0.6921801

Now the density values of the first class is stored in f_D.

As we can see there are 15 values in f_D.

My question is (assume the prior of each class is 0.5):

For the Bayesian theorem, how can I calculate the posterior in (1)? Here, I have 15 values for f_D. Then, I should multiply them by 0.5 (the prior). However, I will not get a single value for the posterior! So, do I have to sum the values in f_D or product them?

Any help is appreciate.

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  • 1
    $\begingroup$ I'm not clear on why you don't use a direct probability model such as the binary logistic model. And is the disease really all-or-nothing, i.e., is there not an underlying continuous or ordinal severity of disease measure? I'm also unclear on the use of a copula here. Copulas are typically used for multivariate outcomes while your outcome is univariate. $\endgroup$ – Frank Harrell Apr 24 at 11:31

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