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status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
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repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | suggested re-reading day | |||

started reading on | finished reading on |

status | not read | reprioritisations | ||
---|---|---|---|---|

last reprioritisation on | suggested re-reading day | |||

started reading on | finished reading on |

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

The graph with edges removed is known as the manipulated graph

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

In causal graphs, causation flows along directed paths.

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
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repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

that expression is in the post-intervention world where the intervention do(𝑡) occurs. For example, 𝔼[𝑌 | do(𝑡), 𝑍 = 𝑧] refers to the expected outcome in the subpopulation where 𝑍 = 𝑧 after the <span>whole subpopulation has taken treatment 𝑡 . <span>

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

Whenever, do(𝑡) appears after the conditioning bar, it means that everything in that expression is in the post-intervention world where the intervention do(𝑡) occurs.

status | not read | reprioritisations | ||
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last reprioritisation on | suggested re-reading day | |||

started reading on | finished reading on |

It might seem like consistency is obviously true, but that is not always the case. For example, if the treatment specification is simply “get a dog” or “don’t get a dog,” this can be too coarse to yield consistency. It might be that if I were to get a puppy, I would observe 𝑌 = 1 (happiness) because I needed an energetic friend, but if I were to get an old, low-energy dog, I would observe 𝑌 = 0 (unhappiness).

status | not read | reprioritisations | ||
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last reprioritisation on | suggested re-reading day | |||

started reading on | finished reading on |

dog,” so both correspond to 𝑇 = 1 . This means that 𝑌(1) is not well defined, since it will be 1 or 0, depending on something that is not captured by the treatment specification. In this sense, <span>consistency encompasses the assumption that is sometimes referred to as “no multiple versions of treatment.” <span>

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
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repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

However, we do have conditional exchangeability in the data. This is because, when we condition on 𝑋 , there is no longer any non-causal association between 𝑇 and 𝑌 . The non-causal association is now “blocked” at 𝑋 b

status | not read | reprioritisations | ||
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last reprioritisation on | suggested re-reading day | |||

started reading on | finished reading on |

SUTVA is satisfied if unit (individual) 𝑖 ’s outcome is simply a function of unit 𝑖 ’s treatment. Therefore, SUTVA is a combination of consistency and no interference (and also deterministic potential outcomes)

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
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repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

ty-Unconfoundedness Tradeoff Although conditioning on more covariates could lead to a higher chance of satisfying unconfoundedness, it can lead to a higher chance of violating positivity. As we <span>increase the dimension of the covariates, we make the subgroups for any level 𝑥 of the covariates smaller. <span>

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
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repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

As we discussed in Section 4.2, the graph for the interventional distribution 𝑃(𝑌 | do(𝑡)) is the same as the graph for the observational distribution 𝑃(𝑌, 𝑇, 𝑋) , but with the incoming edges to 𝑇 removed.

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
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repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

The Bayesian network factorization is also known as the chain rule for Bayesian networks or Markov compatibility.

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
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repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

When there is an association between A and Y, even if A has a null causal effect, a zero causal effect on Y, then we say that there is bias under the null.

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
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repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

We have seen that confounding is a systematic bias when we are conducting causal inference research.