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Convert a pdf into a conditional pdf such that mean increases and std dev falls

Data Science Asked by claudius on October 30, 2020

Let success metric(for some business use case I am working on) be a continuous random variable S.
The mean of pdf defined on S indicates the chance of success. Higher the mean more is the chance of success. Let std dev of pdf defined on S indicates risk. Lower the std deviation lower the risk of failure.

I have data,let’s call them X, which affects S. Let X also be modelled as bunch of random variables.

P(S|X) changes based on X.
The problem statement is I want to pick Xs such that P(S|X) has mean higher than P(S) and std deviation lower than P(S).

Just to illustrate my point I have taken X of 1 dimension.
Scatter plot between X(horizontal) and Y(on vertical):

enter image description here

You can see that P(S|X) changes for different values of X as given in the below plot:
enter image description here

For X between 4500 and 10225, mean of S is 3.889 and std dev is 0.041 compared to mean of 3.7 and std dev of 0.112 when there is no constraint on X.

What I am interested in is given S and bunch of Xs… pick range of Xs such that resulting distribution of P(S|X) has higher mean and lower standard deviation… Please help me find a standard technique that would help me achieve this.

Also I don’t want to condition on X such that number of samples are too small to generalise.I want to avoid cases such as on the left most side of tree where number of samples is 1.

One Answer

Just apply an optimization to search for the X values that satisfy the criteria you're looking for. Here's a simple demo:

set.seed(123)
mu_x_true = 1e4
mu_y_true = 3.75
n = 1e2

x <- rpois(n, mu_x_true)
y <- rnorm(n, sqrt(mu_y_true))^2

plot(x, y)

# conditions:
# E[Y|X] > E[Y]
# std(Y|x) < std(Y)

mu_y_emp = mean(y)
sd_y_emp = sd(y)

objective <- function(par, alpha=0.5){ 
    if (par[1]>par[2]) par = rev(par)
    ix <- which((par[1] < x) & (x < par[2]))
    k <- length(ix)
    if (k==0) return(1e12)
    mu_yx <- mean(y[ix])
    sd_yx <- sd(y[ix])

    alpha*(mu_y_emp - mu_yx) + (1-alpha)*(sd_yx - sd_y_emp)
}

init <- mean(x) + c(-sd(x), sd(x))
test <- optim(objective, par=init)

ix <- which((par[1] < x) & (x < par[2]))

mean(y[ix]) > mean(y) 
# TRUE

sd(y) > sd(y[ix])
# TRUE

Answered by David Marx on October 30, 2020

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