Likelihood Function

Instance of: joint probability

AKA: Likelihood

Distinct from:

English: The joint probability of the data, conditional on the parameters parameters. How likely are we to observe these data, given this paramater is the True one.

(Etz 2017)

Likelihood is not probability, but is proportional to a probability.
Likelihoods are relative, scaled by an arbitrary constant, and need not sum to one.
Under likelihood the data are fixed, and the hypothesis vary.

Formalization:

\[ \mathcal{L}(\theta|X) \]

Where \(\theta\) are the parameters, and \(X\) is the evidence.

Also written in terms of a probability of observing X given the a parameter value \[ \mathcal{L}(\theta) = K \times P(X|\theta) \]

Where \(\theta\) are the parameters, and \(X\) is the evidence, and \(K\) is an arbitrary scaling constant.

Cites: Wikipedia ; Wikidata ; Wolfram

Code

Examples:

library(DBI)
# Create an ephemeral in-memory RSQLite database
#con <- dbConnect(RSQLite::SQLite(), dbname = ":memory:")
#dbListTables(con)
#dbWriteTable(con, "mtcars", mtcars)
#dbListTables(con)

#Configuration failed because libpq was not found. Try installing:
#* deb: libpq-dev libssl-dev (Debian, Ubuntu, etc)
#install.packages('RPostgres')
#remotes::install_github("r-dbi/RPostgres")
#Took forever because my file permissions were broken
#pg_lsclusters
require(RPostgres)

Loading required package: RPostgres

# Connect to the default postgres database
#I had to follow these instructions and create both a username and database that matched my ubuntu name
#https://www.digitalocean.com/community/tutorials/how-to-install-postgresql-on-ubuntu-20-04-quickstart
con <- dbConnect(RPostgres::Postgres())

import torch

References

Etz, Alexander. 2017. “Introduction to the Concept of Likelihood and Its Applications.” PsyArXiv. https://doi.org/10.31234/osf.io/85ywt.