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I am working through McElreath's book on statistical rethinking. One of the problems is the following:

Using grid approximation, compute the posterior distribution for the probability of a birth being a boy. Assume a uniform prior probability. Which parameter value maximizes the posterior probability?

We are given 2 vectors, each of length 100. The components of each vector correspond to a particular family and takes on value 1 if the child is a boy and 0 if it's a girl. The first vector is for the first-born child for each family while the second vector is for the second-born child for each family.

To construct the likelihood, I consider each family as an independent sample. Within the family, the distribution of the sex of the children is binomial. Thus, if $p$ is the probability of being a boy, and $x_1,\dots,x_{100}$ represent the number of boys for each family, I get that the likelihood is $\binom{2}{x_1}p^{x_1}(1-p)^{2-x_1}\cdot\cdots\cdot \binom{2}{x_{100}}p^{x_{100}}(1-p)^{2-x_{100}}$.

I looked at the solutions and another approach considered was to pool the samples across all the families and consider it as a binomial distribution. The likelihood therefore is $\binom{200}{x_1+\dots +x_{100}} p^{x_1 + \dots + x_{100}} (1-p)^{200-(x_1 + \dots + x_{100})}$.

Mathematically, except for the constants, they're the same expression. (Or are the constants even equal?) From a modeling perspective, is there a difference between these 2 likelihoods? I'm surprised that the functional forms are almost the same.

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This is an illustration of sufficiency in action within Bayesian statistics: when given a sample $x_1,\dots,x_{100}$ from a $\mathcal B(2,p)$ distribution, the posterior distribution of $p$ given that sample is the same as the posterior distribution of $p$ given the sum $$x_1+\cdots+x_{100}$$ because this sum is sufficient (meaning that the likelihood functions are proportional functions of $p$).

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