## logistic regression – Derive the conditional distribution \$ P (X | Y) \$ from a logit model

Let's say we have two random variables $$X_1 sim N ( mu, sigma ^ 2)$$ and $$X_2 sim Bern (0.5)$$. The binary result variable $$Y$$ is generated from
begin {align} P (Y = 1 | X_1, X_2) = frac {e ^ { beta_0 + beta_1X_1 + beta_2 X_2}} {1 + e ^ { beta_0 + beta_1X_1 + beta_2 X_2}} end {align}
$$Y sim Bern (P (Y = 1 | X_1, X_2))$$

Is there a simple way to calculate the conditional distributions of $$P (X_1 | Y = 1)$$,$$P (X_1 | Y = 0)$$,$$P (X_2 | Y = 1)$$,$$P (X_2 | Y = 0)$$? I think both $$X_1$$ and $$X_2$$ will keep the original distributions (Normal is still normal, Bernoulli is still Bernoulli).

We can make the conditional independence assumption if necessary. that is to say., $$P (X_1, X_2 | Y) = P (X_1 | Y) P (X_2 | Y)$$.

## The composition of the logit function and its inverse is not numerically invariant with respect to the decimal place

I have defined the following two functions:

``````logit[x_] : = Module[{},
Log[x/(1 - x)]
];
invLogit[x_] : = Module[{},
E^x/(1 + E^x)
];
``````

One is the reverse of the other. however,

``````In[37]: = logit[invLogit[34.55555]]Outside[37]= 34.6574
``````

Is it possible to increase the accuracy of calculations?