A tutorial on tidy cross-validation with R
Analyzing NetHack data, part 1: What kills the players
Analyzing NetHack data, part 2: What players kill the most
Building a shiny app to explore historical newspapers: a step-by-step guide
Classification of historical newspapers content: a tutorial combining R, bash and Vowpal Wabbit, part 1
Classification of historical newspapers content: a tutorial combining R, bash and Vowpal Wabbit, part 2
Curly-Curly, the successor of Bang-Bang
Dealing with heteroskedasticity; regression with robust standard errors using R
Easy time-series prediction with R: a tutorial with air traffic data from Lux Airport
Exporting editable plots from R to Powerpoint: making ggplot2 purrr with officer
Fast food, causality and R packages, part 1
Fast food, causality and R packages, part 2
For posterity: install {xml2} on GNU/Linux distros
Forecasting my weight with R
From webscraping data to releasing it as an R package to share with the world: a full tutorial with data from NetHack
Get text from pdfs or images using OCR: a tutorial with {tesseract} and {magick}
Getting data from pdfs using the pdftools package
Getting the data from the Luxembourguish elections out of Excel
Going from a human readable Excel file to a machine-readable csv with {tidyxl}
Historical newspaper scraping with {tesseract} and R
How Luxembourguish residents spend their time: a small {flexdashboard} demo using the Time use survey data
Imputing missing values in parallel using {furrr}
Intermittent demand, Croston and Die Hard
Looking into 19th century ads from a Luxembourguish newspaper with R
Making sense of the METS and ALTO XML standards
Manipulate dates easily with {lubridate}
Manipulating strings with the {stringr} package
Maps with pie charts on top of each administrative division: an example with Luxembourg's elections data
Missing data imputation and instrumental variables regression: the tidy approach
Modern R with the tidyverse is available on Leanpub
Objects types and some useful R functions for beginners
Pivoting data frames just got easier thanks to `pivot_wide()` and `pivot_long()`
R or Python? Why not both? Using Anaconda Python within R with {reticulate}
Searching for the optimal hyper-parameters of an ARIMA model in parallel: the tidy gridsearch approach
Some fun with {gganimate}
Split-apply-combine for Maximum Likelihood Estimation of a linear model
Statistical matching, or when one single data source is not enough
The best way to visit Luxembourguish castles is doing data science + combinatorial optimization
The never-ending editor war (?)
The year of the GNU+Linux desktop is upon us: using user ratings of Steam Play compatibility to play around with regex and the tidyverse
Using Data Science to read 10 years of Luxembourguish newspapers from the 19th century
Using a genetic algorithm for the hyperparameter optimization of a SARIMA model
Using cosine similarity to find matching documents: a tutorial using Seneca's letters to his friend Lucilius
Using linear models with binary dependent variables, a simulation study
Using the tidyverse for more than data manipulation: estimating pi with Monte Carlo methods
What hyper-parameters are, and what to do with them; an illustration with ridge regression
{disk.frame} is epic
{pmice}, an experimental package for missing data imputation in parallel using {mice} and {furrr}
Building formulae
Functional peace of mind
Get basic summary statistics for all the variables in a data frame
Getting {sparklyr}, {h2o}, {rsparkling} to work together and some fun with bash
Importing 30GB of data into R with sparklyr
Introducing brotools
It's lists all the way down
It's lists all the way down, part 2: We need to go deeper
Keep trying that api call with purrr::possibly()
Lesser known dplyr 0.7* tricks
Lesser known dplyr tricks
Lesser known purrr tricks
Make ggplot2 purrr
Mapping a list of functions to a list of datasets with a list of columns as arguments
Predicting job search by training a random forest on an unbalanced dataset
Teaching the tidyverse to beginners
Why I find tidyeval useful
tidyr::spread() and dplyr::rename_at() in action
Easy peasy STATA-like marginal effects with R
Functional programming and unit testing for data munging with R available on Leanpub
How to use jailbreakr
My free book has a cover!
Work on lists of datasets instead of individual datasets by using functional programming
Method of Simulated Moments with R
New website!
Nonlinear Gmm with R - Example with a logistic regression
Simulated Maximum Likelihood with R
Bootstrapping standard errors for difference-in-differences estimation with R
Careful with tryCatch
Data frame columns as arguments to dplyr functions
Export R output to a file
I've started writing a 'book': Functional programming and unit testing for data munging with R
Introduction to programming econometrics with R
Merge a list of datasets together
Object Oriented Programming with R: An example with a Cournot duopoly
R, R with Atlas, R with OpenBLAS and Revolution R Open: which is fastest?
Read a lot of datasets at once with R
Unit testing with R
Update to Introduction to programming econometrics with R
Using R as a Computer Algebra System with Ryacas

Maximum likelihood estimation is a very useful technique to fit a model to data used a lot in econometrics and other sciences, but seems, at least to my knowledge, to not be so well known by machine learning practitioners (but I may be wrong about that). Other useful techniques to confront models to data used in econometrics are the minimum distance family of techniques such as the general method of moments or Bayesian approaches, while machine learning practitioners seem to favor the minimization of a loss function (the mean squared error in the case of linear regression for instance).

When I taught at the university, students had often some problems to understand the technique. It is true that it is not as easy to understand as ordinary least squares, but I’ll try to explain to the best of my abilities.

Given a sample of data, what is the unknown probability distribution
that *most likely* generated it? For instance, if your sample only contains 0’s and 1’s, and
the proportion of 1’s is 80%, what do you think is the most likely distribution that generated it?
The probability distribution that *most likely* generated such a dataset is a binomial distribution
with probability of success equal to 80%. It *might have been* a binomial distribution with probability
of success equal to, say, 60%, but the *most likely* one is one with probability of success equal
to 80%.

To perform maximum likelihood estimation, one thus needs to assume a certain probability distribution,
and then look for the parameters that maximize the likelihood that this distribution generated the
observed data. So, now the question is, how to maximize this likelihood? And mathematically speaking,
what is a *likelihood*?

First of all, let’s assume that each observation from your dataset not only was generated from the same distribution, but that each observation is also independent from each other. For instance, if in your sample you have data on people’s wages and socio-economic background, it is safe to assume, under certain circumstances, that the observations are independent.

Let \(X_i\) be random variables, and \(x_i\) be their realizations (actual observed values). Let’s assume that the \(X_i\) are distributed according to a certain probability distribution \(D\) with density \(f(\theta)\) where \(\theta\) is a parameter of said distribution. Because our sample is composed of i.i.d. random variables, the probability that it was generated by our distribution \(D(\theta)\) is:

\[\prod_{i=1}^N Pr(X_i = x_i)\]

It is customary to take the log of this expression:

\[\log(\prod_{i=1}^N Pr(X_i = x_i)) = \sum_{i=1}^N \log(Pr(X_i = x_i))\]

The expression above is called the *log-likelihood*, \(logL(\theta; x_1, ..., x_N)\). Maximizing this
function yields \(\theta^*\), the value of the parameter that makes the sample the most probable.
In the case of linear regression, the density function to use is the one from the Normal distribution.

Hadley Wickham’s seminal paper, The Split-Apply-Combine Strategy for Data Analysis
presents the *split-apply-combine* strategy, which should remind the reader of the map-reduce
framework from Google. The idea is to recognize that in some cases big problems are simply an
aggregation of smaller problems. This is the case for Maximum Likelihood Estimation of the linear
model as well.
The picture below illustrates how Maximum Likelihood works, in the standard case:

Let’s use R to do exactly this. Let’s first start by simulating some data:

```
library("tidyverse")
size <- 500000
x1 <- rnorm(size)
x2 <- rnorm(size)
x3 <- rnorm(size)
dep_y <- 1.5 + 2*x1 + 3*x2 + 4*x3 + rnorm(size)
x_data <- cbind(dep_y, 1, x1, x2, x3)
x_df <- as.data.frame(x_data) %>%
rename(iota = V2)
head(x_df)
```

```
## dep_y iota x1 x2 x3
## 1 1.637044 1 0.2287198 0.91609653 -0.4006215
## 2 -1.684578 1 1.2780291 -0.02468559 -1.4020914
## 3 1.289595 1 1.0524842 0.30206515 -0.3553641
## 4 -3.769575 1 -2.5763576 0.13864796 -0.3181661
## 5 13.110239 1 -0.9376462 0.77965301 3.0351646
## 6 5.059152 1 0.7488792 -0.10049061 0.1307225
```

Now that this is done, let’s write a function to perform Maximum Likelihood Estimation:

```
loglik_linmod <- function(parameters, x_data){
sum_log_likelihood <- x_data %>%
mutate(log_likelihood =
dnorm(dep_y,
mean = iota*parameters[1] + x1*parameters[2] + x2*parameters[3] + x3*parameters[4],
sd = parameters[5],
log = TRUE)) %>%
summarise(sum(log_likelihood))
-1 * sum_log_likelihood
}
```

The function returns minus the log likelihood, because `optim()`

which I will be using to optimize
the log-likelihood function minimizes functions by default (minimizing the opposite of a function is the
same as maximizing a function). Let’s optimize the function and see if we’re able to find the
parameters of the data generating process, `1.5, 2, 3, 4`

and `1`

(the standard deviation of the
error term):

`optim(c(1,1,1,1,1), loglik_linmod, x_data = x_df)`

We successfully find the parameters of our data generating process!

Now, what if I’d like to distribute the computation of the contribution to the likelihood of each observations across my 12 cores? The goal is not necessarily to speed up the computations but to be able to handle larger than RAM data. If I have data that is too large to fit in memory, I could split it into chunks, compute the contributions to the likelihood of each chunk, sum everything again, and voila! This is illustrated below:

To do this, I use the `{disk.frame}`

package, and only need to change my `loglik_linmod()`

function
slightly:

```
library("disk.frame")
x_diskframe <- as.disk.frame(x_df) #Convert the data frame to a disk.frame
loglik_linmod_df <- function(parameters, x_data){
sum_log_likelihood <- x_data %>%
mutate(log_likelihood =
dnorm(dep_y,
mean = iota*parameters[1] + x1*parameters[2] + x2*parameters[3] + x3*parameters[4],
sd = parameters[5],
log = TRUE)) %>%
chunk_summarise(sum(log_likelihood))
out <- sum_log_likelihood %>%
collect() %>%
pull() %>%
sum()
-out
}
```

The function is applied to each chunk, and `chunk_summarise()`

computes the sum of the contributions
inside each chunk. Thus, I first need to use `collect()`

to transfer the chunk-wise sums in memory
and then use `pull()`

to convert it to an atomic vector, and finally sum them all again.

Let’s now optimize this function:

`optim(rep(1, 5), loglik_linmod_df, x_data = x_diskframe)`

```
## $par
## [1] 1.5351722 1.9566144 3.0067978 4.0202956 0.9889412
##
## $value
## [1] 709977.2
##
## $counts
## function gradient
## 502 NA
##
## $convergence
## [1] 1
##
## $message
## NULL
```

This is how you can use the split-apply-combine approach for maximum likelihood estimation of a
linear model! This approach is quite powerful, and the familiar `map()`

and `reduce()`

functions
included in `{purrr}`

can also help with this task. However, this only works if you can split your
problem into chunks, which is sometimes quite hard to achieve.

However, as usual, there is rarely a need to write your own functions, as `{disk.frame}`

includes
the `dfglm()`

function which can be used to estimate any generalized linear model using `disk.frame`

objects!

Hope you enjoyed! If you found this blog post useful, you might want to follow me on twitter for blog post updates and buy me an espresso or paypal.me, or buy my ebook on Leanpub