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I was asked in an interview why do we use the binomial distribution in logistic regression and how is it related to the class that we are predicting?

Could anyone explain, without any mathematical equations, why do we use binomial instead on any other distribution?

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  • $\begingroup$ I would argue that we aren’t predicting a class. Logistic regression gives a probability (log-odds that we can convert to probability), and that probability is a parameter in a binomial distribution. $\endgroup$
    – Dave
    Commented Aug 10, 2020 at 11:15

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From wikipedia:

..., the binomial distribution with parameters n and $\rho$ is the discrete probability distribution of the number of successes in a sequence of n independent experiments, each asking a yes–no question, and each with its own boolean-valued outcome: a random variable containing a single bit of information: success/yes/true/one (with probability $\rho$) or failure/no/false/zero (with probability $\rho = 1 − \rho$).

So if you know that logistic regression is performed in order to model a binary output variable to some modelling question (i.e. to give 0 or 1, yes or no, etc.), it would make sense to base any probabilistic assumptions on a distribution, which shares this feature. Therefore, a binomial distribution may make sense compared to a continuous distribution, such as a Gaussian or Cauchy.

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Assume that you have a time variable and you observe at each time and at a certain bus stop if there is a bus arriving or not. Let the probability that a bus arrives at a bus stop at time $t$ be denoted as $p(t)$. This essence of success/failure is a binomial distribution and Logistic regression computes/predicts $p(t)$ by shifting and stretching the logistic curve.

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logistic regression involves the maximum likelihood method. One looks at the results or the observations of a random experiment and considers which of several possible causes could have led to it most probably ("maximum likelihood").

The regression coefficients are interpreted differently than in linear regression. there is no linear relationship. But depending on the sign, if it is positive, then an increase in the independent variable in question is produced (maximum likelihood). Conversely, there is a decrease. This is exactly the same thing that happens in the binomial distribution.

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