# Solved – Likelihood function of a Linear probability model

What is the Likelihood function of a linear probability model?

I know the likelihood function is the joint probability density, but how to construct the likelihood function when we only have the probability \$P(Y_i=1|X_i)\$ and \$P(Y_i=0|X_i)\$?

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In your example you simply don't have a likelihood function, because you defined just a probability model rather than a statistical one. If you know the probabilities \$\$ P(Y_i = 1 | X_i) = p_i quad mbox{and} quad P(Y_i =0| X_i) = 1-p_i, \$\$ then you have a "unique" conditional probability model. First you have to understand the differences between probability and statistical models. Please, see this post.

In order to have a statistical model, the above probabilities must be unknown. Typically, some relation is imposed: \$\$ P_theta(Y_i = 1 | X_i=x_i) = p_i(theta),\$\$ where \$theta\$ is the unknown parameter vector and \$p_i(theta) in [0,1]\$ for \$i=1, ldots, n\$. Now, we have a parametric statistical model, since for each \$theta\$ we have a probability model.

The likelihood function is \$\$ L(theta) = prod_{i=1}^n P_theta(Y_i = y_i | X_i=x_i) = prod_{i=1}^n p_i(theta)^{y_i}(1-p_i(theta))^{1-y_i}. \$\$

Usually, the shape of \$p_i(theta)\$ is commonly specified as:

1. \$p_i(theta) = frac{exp(eta_i(theta))}{1 + exp(eta_i(theta))},\$
2. \$p_i(theta) = F(eta_i(theta)),\$

where \$eta_i(theta) = alpha + beta x_i\$ and \$F\$ is a cumulative distribution function. You can find a good shape for \$p_i\$ by looking at the data behavior (plots, data dispersion and so forth).

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