The other half of the notes, on the cricket simulator, are found here.

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Approximate Bayesian Computation (ABC) is a method of estimating the distribution of parameters when the likelihood is very difficult to find. This feature makes ABC applicable to a great variety of problems.

The classical ABC
method can be summarized as:

Step 1: Select a prior joint distribution F(theta) for the parameters of interest.

Step 2: Observe a
sample s from the population you wish to characterize.

Step 3: Compute T(s), a statistic, or set of statistics, from the sample.

Step 4: Repeat the
following many times

4A: Generate random
parameter values p from F(theta)

4B: Use the values
p to simulate a sample s'

4C: Compute the
distance between T(s') and T(s), the statistics taken from the
simulated and observed samples, respectively.

4D: If the
difference is less than some tuning value, epsilon, then accept the
values p.

Step 5: The
collection of values p from all accepted values is taken as the
posterior distribution of the parameters of interest.

Step 6: If the
parameters of interest are continuous, a smoothing method is applied
to this collection to produce a continuous posterior distribution.

The distance between
two statistics can be determined by Manhattan or Euclidean distance,
or a weighted distance if deviations between some statistic values
are more important than others.

The tuning parameter
epsilon determines the balance between accuracy and computational
efficiency. A small epsilon will ensure that only generated samples
that are very similar to the observed sample will be used. That is,
only samples that produce statistics very close to the real sample's
statistics. However, many samples will be rejected.

A large epsilon will
ensure that many samples will be produced, but some of those samples
will have statistics that deviate from the observed sample. Since the
process of producing new samples could be time-consuming, depending
on the application, this may be a worthwhile trade-off.

Compared to
classical Approximate Bayesian Computation method, I made some
modifications.

- Instead of having
an accept-reject rule, I opted to use every sample but assign weights
to the parameters behind each sample based on the distance between
T(s') and T(s).

- The weights and
smoothing mechanism are determined through a Kernel Density Estimator
(KDE)

Specifically, every
generated sample is gets treated as a point mass in a space across
the statistic values and parameter values.

Consider the example in
the following MS-Paint diagrams.

In Figures 1-3, we have a sample with a
single statistic value, T(s), vertical, and a single parameter of
interest, theta, horizontal. We don't know the parameter value of the
observed sample, so we represent the observed sample as a blue
horizontal line.

Points
A and B represent the statistic and parameter values from each of two
generated samples (also A and B, respectively). These are points instead
of lines because these samples are simulated using known parameter
values.

The
gray area at the bottom of each figure is the conditional density of
the parameter of the observed sample. The thin lines around points A and
B are the contour lines of the probability density, with peaks at the
points and decreasing outwards.

Figure 1: A generated sample that's far from the observed sample. |

In
Figure 1, we see the effect of generated sample A on the parameter
estimate. The statistic from sample A is far from the statistic of the
observed sample, so point A is far from the line. As a result, the
probability density does increase or decease much along the line, and
therefore the conditional density of the parameter estimate is diffuse,
as shown by the shallow gray hill.

Figure 2: A generated sample that's close to the observed sample. |

Figure 3: The combined estimate from both samples. |

In
Figure 3, we see the cumulative effect of both samples. The conditional
density from both resembles that from generated sample B. This is
because sample B's effect was given more weight. Generated samples that
more closely resemble the observed sample contribute more to the
parameter estimate.

I used this method
on two applications

1. A theoretical
network based on a network of sexual partners in Africa.

2. A real network of
precedence citations between decisions of the Supreme Court of
Canada.

The theoretical
dataset was based on a respondent driven sample. An initial seed of
people was selected, presumably by a simple random process. Members
of this seed were polled about the number of sexual partners they had
in the previous year, their HIV status, and their sexual orientation.
For each reported partner, contact information was taken in order to
recruit these partners into the sample as well. If the recruitment
pool was exhausted before the desired number of respondents, new seed
members were found.

Like the basis
dataset, we assumed that the sample we were given contained the
mentioned covariates of those polled, including the number of
partnerships, and we assumed that only the partnerships that led to
successful recruitment were available. This implies that large parts
of the network could be hidden to us, such as partnerships that 'lead
back' into the sample already taken.

Consider these two
figures.

Figure 4 is from
a sample of a network with identifying information about every
reported partnership is given. Figure 5 is from
the same sample, where only the partnerships that led to recruitment
are known. Notice that Figure 4 includes lines going out to points that we did not sample directly. Notice also that Figure 4 includes cycles, in which you could go from one point to itself by following the lines and without backtracking.

We don't see these cycles in Figure 5, the recruitment-only network, because a person cannot be recruited if they are already in the sample. A cycle might be a triangle between persons J, K, and L, where all three are connected. Person J can recruit both K and L through those connections, but K cannot recruit L, so we never see the K-L connection. What we see is one person, J, with two distinct partners instead of the whole picture.

Socially, the K-L connection is key information, and mathematically this lack of cycles makes estimation of anything about this network a lot harder.

Figure 4: A network with identifying information from the sample. |

Figure 5: A network with only recruitment information from the sample. |

Nevertheless, I used ABC to estimate some features (parameters) of the population that this sample came from.

The parameters of
interest were:

- The number of
members in the population, (prior: geometric, mean 2000, truncated
minimum 400)

- The probability of
an HIV infection passing along a partnership, (prior: uniform 0 to
0.3)

- The proportion of
this network that was already infected before any of these
partnerships (prior: uniform 0 to 0.5)

- The 'closeness
preference', a parameter governing the propensity to select partners
that are similar to oneself. Zero is no preference, 10 is a strong preference, and a negative number represents a general dislike for those similar to you. (prior: uniform -2 to 10)

I conducted two rounds of Approximate Bayesian Computation by producing 500 and 2500 samples, respectively.
The results are here. The yellow and red represent regions of high conditional probability. ABC allows us to narrow down the likely value of the real parameters to these yellow and red regions.

Parameter sets at points all across these graphs were tried, but only those in the non-blue regions produced samples that were anything like the observed sample.

Figure 6: Coloured contour maps of conditional densities, social network data. |

From these, we can calculate marginal means to get our parameter estimates.

Parameter | True Value | Mean (Initial Round) | Mean (Final Round) |

Population Size | 1412 | 2049 | 2094 |

Initial Infection Rate | 0.467 | 0.399 | 0.374 |

Transmission Chance | 0.093 | 0.175 | 0.201 |

Closeness Preference | 7.298 | 6.753 | 9.352 |

The second
application was on the precedence citations of decisions in the
Supreme Court of Canada (SCC). Because this is real data, I am not
able to find the ground truth with which to compare my results, but I
can showcase how the method behaves with real data.

When this data was
collected, there were 10623 documents produced by the SCC, 9847 of
which are cases (e.g. disputes between two parties). These decisions
are among the more than 3,000,000 cases and other documents recorded
in the Canadian Legal Information Institute (CanLII). Figure 7 shows part of the network of SCC cases.

Figure 7: Induced subgraph of cases and citations in Supreme Court of Canada |

Cases span from 1876
to 2016, with more recent cases represented as points near the top.
The larger points are those that have been cited often in Canadian
law. Lines between points represent citations between SCC cases. Only
1000 cases are shown here, so the true network between SCC cases has
roughly 10 times as many nodes and 100 times as many links.

My general interest in this dataset is to identify the features that make a law prone to being obscure over a long time.

I modelled each SCC case as 'vendor' that was trying to 'sell' citations to future Canadian legal cases. Each SCC case had an attractiveness a, where

I modelled each SCC case as 'vendor' that was trying to 'sell' citations to future Canadian legal cases. Each SCC case had an attractiveness a, where

and where

β

_{irrel}**:**Parameter determining the decay in attractiveness over time (per 5 years without a citation).
β

_{pa}**:**Parameter determining a preferential attachment factor, assuming that a case becomes more well-known whenever it is cited.
β

_{corp}**,**β_{crown}**,**β_{dis }**:**Parameters determining increased/decreased attractiveness based on whether a case involved a corporation, the crown, or dissent between Supreme Court Judges, respectively.
what I found was less clear that in was in the synthetic data case, as shown in Figure 8.

Figure 8: Coloured contour maps of conditional densities, Supreme Court of Canada data. |

It's impossible to read the variable names in this figure, I'm afraid. The jist, however, is that...

- a case that is not cited in five or more is less and less likely to receive citations over time.

- a case that is cited often is more likely to receive citations in the next five years.

- a case that involves a corporation is less likely to be cited.

- a case that involves the crown is more likely to be cited.

- a case that had dissenting opinions by some of the judges is more likely to be cited.

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