Binary bayesian classification

How do you construct a Bayes classifier for a binary target where it is assumed: $p(y=1)=\alpha$ and $p(x|y)$ both multivariate gaussian?

• This does not make sense. – Michael Chernick Apr 17 '17 at 3:51
• What do you mean that "... a and p(x|y) both multivariate gaussian"? a seems like a scalar. Do you mean that you want a normal prior for a? What do you mean by "multivariate"? You seem to have only 2 variables. Etc. – gung Apr 17 '17 at 12:32
• I vote to re-open this question. It is not well-phrased, but I interpret it as asking for a decision rule to decide between two hypotheses: $Y=1$ and $Y=0$ with prior probabilities $\alpha$ and $1-\alpha$ respectively, based on the observation $X$ where the conditional distribution of $X$ given $Y$ is a multivariate Gaussian distribution, both for $Y=1$ and $Y=0$. The decision rule is, of course, easy to state, but implementation is messy. The case when $X$ is bivariate Gaussian under both hypotheses is discussed in my answer to this question – Dilip Sarwate Apr 17 '17 at 14:35

Assumption: I assume that this question is asking for the Bayes classifier given that $Y$ is a Bernoulli random variable with parameter $\alpha$ and the conditional distributions of the observation $X$ are multivariate Gaussian distributions.
If both conditional distributions of $X$ given $Y$ are non-degenerate multivariate Gaussian distributions, that is, they possess densities $f_i(x) = f_{X\mid Y=i}(x\mid Y=i), i = 0, 1$, then the Bayes classifier can be expressed as $$\frac{\alpha}{1-\alpha}\cdot \frac{f_1(x)}{f_0(x)}~~\begin{array}{c}\hat Y=1\\\gtrless\\\hat Y=0\end{array}~~ 1.$$ The decision boundary can also be stated as the hypersurface defined by the equation $${\alpha}\cdot{f_1(x)} - {1-\alpha}\cdot{f_0(x)} = 0$$ which does not have a simple solution in general. The two-dimensional case is treated in this question where the hypersurface is found to be a conic section.