使用 python 的条件和联合问题

Question

I have a table of joint prob P(x,y) and I want to creat a function the calculate the conditional probabilities given a the joint prob list. For example I have the following list XY=np.array([[.1,0,0],[.1,.3,.2],[.1,0,.2]]) XY.shape= (3,3)

I want to get the conditional probibility for each index so to find the conditional probabilty of x1cony1=XY[0,0]/y1=1 x2cony1=XY[0,1]/y1=.1

Is there is a way to write a function for that to iterate through the matrix and provide me with a table of all the conditional prob?

Thanks

Answer 1

I don't understand your notation. I will try to answer your question with my notation and hopefully, you can extrapolate to yours. Let's say that the matrix XY contains the join probabilities, ie such that

P(x=x_i, y=y_j) = XY[i, j].

Well now, the conditional probability is

P(x=x_i | y=y_j) = P(x=x_i, y=y_j)/P(y=y_j) = XY[i,j]/(XY[0,j]+XY[1,j]+XY[2,j])

The denominator can be calculated easily using the NumPy sum over the first axis:

XY.sum(axis=0)

Now you should divide the first column (which contains the probabilities P(x=x_i, y=y_0)) by the first value of the sum, which is the probability P(y=y_0). And analogous for the other two columns. This can also be done straightforward because in NumPy the product of a (3,3) matrix with a (3) vector is done by columns, which is just what we want. Therefore this should give you the matrix of conditional probabilities:

XY = np.array(...)  # XY[i,j] is the probability of x=x_i, y=y_j.
# The probability of x_i conditioned to y_j is given by the [i,j] element of:
cond_prob = XY/XY.sum(axis=0)