Exchangeable

Instance of: Property of random variables

AKA: exchangeable sequence of random variables, interchangeable

Distinct from:

English: The joint probability distribution of a sequence of random variables does not depend on the order of those variables, e.g. if you switch two rows in a dataset it does not alter the result of OLS. IID guarantees exchangeability, but it is not required for it. The random variables can be drawn from different distributions, and even influence each other, they ordering just cannot alter the joint distribution such that if you sample from jar 1 and then jar 2 you get a different joint probability than if you sample from jar 2 and then jar 1.

Formalization:

\[ \]

Cites: Wikipedia) ; Wikidata

Code

Imports and spin up toy data objects and databases.

toy_vector_numeric <- c(1,2,3,4,5)
toy_vector_character <- c('a','b','c','d','e')
toy_matrix <- matrix(1:9, nrow=3,ncol=3)
toy_list <- list('a','1',T,c('red','green'))
toy_df <- data.frame(id=c('unit1','unit2','unit3'), y=c(1,2,3), x= c(3,2,1))

toy_dirty_df <- data.frame(id=c('','NA','NaN','inf'), y=c(1,2,3,4), x= c(0,NA,NaN,Inf)) #can't explicitly include NULL 

library(data.table)
toy_dt <- data.table(id=c('unit1','unit2','unit3'), y=c(1,2,3), x= c(3,2,1))
toy_dirty_dt <- as.data.table(toy_dirty_df)
library(tidyverse)

── Attaching packages ─────────────────────────────────────── tidyverse 1.3.2 ──
✔ ggplot2 3.4.0      ✔ purrr   0.3.5 
✔ tibble  3.1.8      ✔ dplyr   1.0.10
✔ tidyr   1.2.1      ✔ stringr 1.5.0 
✔ readr   2.1.3      ✔ forcats 0.5.2 
── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
✖ dplyr::between()   masks data.table::between()
✖ dplyr::filter()    masks stats::filter()
✖ dplyr::first()     masks data.table::first()
✖ dplyr::lag()       masks stats::lag()
✖ dplyr::last()      masks data.table::last()
✖ purrr::transpose() masks data.table::transpose()

library(arrow)

Attaching package: 'arrow'

The following object is masked from 'package:utils':

    timestamp

import numpy as np
toy_vector_numeric = np.array([1,2,3,4,5])
toy_vector_character = np.array(['a','b','c','d','e'])
toy_list = ['a','1',True,['red','green']]
toy_dictionary = { 'a':1 , 'b':2, 'c':3}

from jax import numpy as jnp
toy_vector_numeric_jax = jnp.array([1,2,3,4,5])
#toy_vector_character_jax = jnp.array(['a','b','c','d','e']) #only numeric is allowed in jax

WARNING:jax._src.lib.xla_bridge:No GPU/TPU found, falling back to CPU. (Set TF_CPP_MIN_LOG_LEVEL=0 and rerun for more info.)

import pandas as pd
toy_df = pd.DataFrame(data={'id': ['unit1','unit2','unit3'], 'y': [1, 2, 3], 'x': [3, 2, 1]})

import torch

import tensorflow as tf

import pyarrow as pa

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)
# 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_Postgres <- dbConnect(RPostgres::Postgres())

DROP TABLE IF EXISTS toy_df;

CREATE TABLE IF NOT EXISTS toy_df (
  id varchar(5),
    y INTEGER,
    x INTEGER
);

INSERT INTO toy_df (id, y, x)
VALUES
    ('unit1',1,3),
    ('unit2',2,2),
    ('unit3',3,1);
    

#install.packages("duckdb")
library("DBI")
con_duckdb = dbConnect(duckdb::duckdb(), ":memory:")

#pip install duckdb==0.6.0
import duckdb
con_duckdb = duckdb.connect()