Dr. Mine Dogucu

The notes for this lecture are derived from Chapter 4 of the Bayes Rules! book

**Bechdel Test**

Alison Bechdel’s 1985 comic Dykes to Watch Out For has a strip called The Rule where a person states that they only go to a movie if it satisfies the following three rules: - the movie has to have at least two women in it; - these two women talk to each other; and - they talk about something besides a man.

This test is used for assessing movies in terms of representation of women. Even though there are three criteria, a movie either fails or passes the Bechdel test.

Let \(\pi\) be the the proportion of movies that pass the Bechdel test.

Below there are three different people with three different priors about \(\pi\).

optimist | clueless | feminist |

Beta(14,1) | Beta(1,1) | Beta(5,11) |

Plot their priors.

**Informative prior:** An informative prior reflects specific information about the unknown variable with high certainty (ie. low variability).

**Vague (diffuse) prior:**

A vague or diffuse prior reflects little specific information about the unknown variable. A flat prior, which assigns equal prior plausibility to all possible values of the variable, is a special case.

`library(bayesrules)`

has`bechdel`

data frame. Randomly select 20 movies from this dataset (`seed = 84735`

).Based on observed data, update the posterior for all three people.

Calculate the summary statistics for the prior and the posterior for all three.

Plot the prior, likelihood, and the posterior for all three.

Explain the effect of different priors on the posterior.

Morteza, Nadide, and Ursula – all share the optimistic Beta(14,1) prior for \(\pi\) but each have access to different data. Morteza reviews movies from 1991. Nadide reviews movies from 2000 and Ursula reviews movies from 2013. How will the posterior distribution for each differ?

In a sequential Bayesian analysis, a posterior model is updated incrementally as more data comes in. With the introduction of each new piece of data, the previous posterior model reflecting our understanding prior to observing this data becomes the new prior model.

\(\pi \sim Beta(14,1)\)

\(\pi \sim Beta(15,1)\)

\(\pi \sim Beta(15,6)\)

Time | Data | Model |
---|---|---|

before the analysis | NA | Beta(14,1) |

at the end of 1970 | Y = 1, n = 1 | Beta(15,1) |

at the end of 1971 | Y = 0, n = 5 | Beta(15, 6) |

at the end of 1972 | Y = 1, n = 3 | Beta(16,8) |

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Time | Data | Model |
---|---|---|

before the analysis | NA | Beta(14,1) |

1972 | Y = 1, n = 3 | Beta(15,3) |

1971 | Y = 0, n = 5 | Beta(15,8) |

1970 | Y = 1, n = 1 | Beta(16,8) |

<>

Time | Data | Model |
---|---|---|

before the analysis | NA | Beta(14,1) |

1970 | Y = 1, n = 1 | |

1971 | Y = 0, n = 5 | |

1972 | Y = 1, n = 3 | |

Total | Y = 2, n = 9 |

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Let \(\theta\) be any parameter of interest with prior pdf \(f(\theta)\). Then a **sequential analysis** in which we *first* observe a data point \(y_1\) and *then* a second data point \(y_2\) will produce the same posterior model of \(\theta\) as if we *first* observe \(y_2\) and *then* \(y_1\):

\[f(\theta | y_1,y_2) = f(\theta|y_2,y_1)\;.\]

Similarly, the posterior model is invariant to whether we observe the data *all at once* or *sequentially*.