In ctgrubb/lemur.pack: Package for Team Lemur functional codebase

knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "##",
  eval = TRUE,
  cache = TRUE
)

quick_eval <- FALSE
options(scipen = 6)
set.seed(318937291)

library(extraDistr)
library(lemur.pack)
library(MASS)
library(mvtnorm)
library(latex2exp)
library(microbenchmark)
library(tidyverse)
library(gridExtra)
library(cowplot)

Populations

We are working with various finite populations which take on different values, ranging from simple discrete-valued categorical variables, to continuous-valued variables. We focus on three scenarios:

(1) Categorical -- $Y_i \in {a, b, c, \dots}$ (2) Discrete Numeric -- $Y_i \in {1, 2, 3, \dots}$ (3) Binned Numeric -- $Y_i \in {(-\infty, a], (a, b], \dots}$ (4) Continuous Numeric -- $Y_i \in \mathbb{R}$

Synthesis

We employ Markov chain Monte Carlo (MCMC) methods to sample from the posterior of our populations, and focus largely on jittering populations via symmetric proposals. Our posterior is constructed the canonical way, via

[ \pi(y|x) \propto \ell(x|y) p(y) ]

The two major components, our prior distribution $p(y)$, and likelihood function $\ell(x|y)$.

Likelihood

Because we are working with finite populations, $\ell(x|y)$ often cannot take on the form of sampling distribution. To illustrate, imagine a population...

[insert example here]

Prior

[still need to talk about priors... but we have the discrete case]

Comparisons

Categorical Populations

Binary

Binomial vs Hypergeometric

Brownian Bridge using Integers

Many Categories

Multinomial vs Multivariate Hypergeometric

Problems as number of categories increases

Brownian Bridge using Integers (cont'd)

Dirichlet Approx using Integers

Discrete Numeric Populations

Problems with Multinomial & Multivariate Hypergeometric

Brownian Bridge

Dirichlet Approx

Problems as number of unique values increases? -> Motivation for fixed interpolation points

Continuous Populations (bounded)

Brownian Bridge

Interpolated

Dirichlet Approx

Equal probability on endpoints vs 1/2 probability on

Continuous Populations (unbounded)

???

Parametric Populations

Normal "lock-step" fixed intervals in inverse-cdf space

Changing from "lock-step" -> i.i.d.

Binned data informing likelihood

ctgrubb/lemur.pack documentation built on May 7, 2023, 4:13 a.m.