Optimization & MLE
EMSE 6035: Marketing Analytics for Design Decisions
John Paul Helveston
October 21, 2026

Quiz 3

10:00
Write your name on the quiz!
Rules:
  • Work alone; no outside help of any kind is allowed.
  • No calculators, no notes, no books, no computers, no phones.

Week 9: Optimization & MLE

1. Maximum likelihood estimation

2. Optimization (in general)

3. Joins

4. Pilot data cleaning

Week 9: Optimization & MLE

1. Maximum likelihood estimation

2. Optimization (in general)

3. Joins

4. Pilot data cleaning

Computing the likelihood

x: an observation

f(x): probability of observing x

Computing the likelihood

x: an observation

f(x): probability of observing x

\mathcal{L}(\theta | x): probability that \theta are the true parameters, given that observed x

We want to estimate \theta

We actually compute the log-likelihood
(converts multiplication to addition)

Practice Question 1

Observations - Height of students (inches):

#>  [1] 65 69 66 67 68 72 68 69 63 70
  1. Let’s say we know that the height of students, \tilde{x}, in a classroom follows a normal distribution. A professor obtains the above height measurements students in her classroom. What is the log-likelihood that \tilde{x} \sim \mathcal{N} (68, 4)? In other words, compute \ln \mathcal{L} (\mu = 68, \sigma = 4).
  1. Compute the log-likelihood function using the same standard deviation (\sigma = 4) but with the following different values for the mean, \mu: 66, 67, 68, 69, 70. How do the results compare? Which value for \mu produces the highest log-likelihood?

Week 9: Optimization & MLE

1. Maximum likelihood estimation

2. Optimization (in general)

3. Joins

4. Pilot data cleaning

f(x)

Practice Question 2

Consider the following function:

f(x) = x^2 - 6x

The gradient is:

\nabla f(x) = 2x - 6

Using the starting point x = 1 and the step size \gamma = 0.3, apply the gradient descent method to compute the next three points in the search algorithm.

Practice Question 3

Consider the following function:

f(\underline{x}) = x_1^2 + 4x_2^2

The gradient is:

\nabla f(\underline{x}) = \begin{bmatrix} 2x_1 \\ 8x_2 \end{bmatrix}

Using the starting point \underline{x}_0 = [1, 1] and the step size \gamma = 0.15, apply the gradient descent method to compute the next three points in the search algorithm.

Download the logitr-cars repo from GitHub

Estimating utility models


1. Open the logitr-cars folder in Positron
2. Open code/3.1-model-mnl.R

Maximum likelihood estimation

Break

05:00

Week 9: Optimization & MLE

1. Maximum likelihood estimation

2. Optimization (in general)

3. Joins

4. Pilot data cleaning

What’s wrong with this map?

Likely culprit: Merging two columns

head(names)
#>              state_name
#> 1               Alabama
#> 2                Alaska
#> 3               Arizona
#> 4              Arkansas
#> 5   Armed Forces Africa
#> 6 Armed Forces Americas
head(abbs)
#>   state_abb
#> 1        AA
#> 2        AE
#> 3        AE
#> 4        AE
#> 5        AE
#> 6        AK
result <- cbind(names, abbs)
head(result)
#>              state_name state_abb
#> 1               Alabama        AA
#> 2                Alaska        AE
#> 3               Arizona        AE
#> 4              Arkansas        AE
#> 5   Armed Forces Africa        AE
#> 6 Armed Forces Americas        AK

Joins

  1. inner_join()
  2. left_join() / right_join()
  3. full_join()

Example: band_members & band_instruments

band_members
#> # A tibble: 3 × 2
#>   name  band   
#>   <chr> <chr>  
#> 1 Mick  Stones 
#> 2 John  Beatles
#> 3 Paul  Beatles
band_instruments
#> # A tibble: 3 × 2
#>   name  plays 
#>   <chr> <chr> 
#> 1 John  guitar
#> 2 Paul  bass  
#> 3 Keith guitar
inner_join() 
band_members %>%
    inner_join(band_instruments)
#> # A tibble: 2 × 3
#>   name  band    plays 
#>   <chr> <chr>   <chr> 
#> 1 John  Beatles guitar
#> 2 Paul  Beatles bass

full_join() 
band_members %>%
    full_join(band_instruments)
#> # A tibble: 4 × 3
#>   name  band    plays 
#>   <chr> <chr>   <chr> 
#> 1 Mick  Stones  <NA>  
#> 2 John  Beatles guitar
#> 3 Paul  Beatles bass  
#> 4 Keith <NA>    guitar

left_join() 
band_members %>%
    left_join(band_instruments)
#> # A tibble: 3 × 3
#>   name  band    plays 
#>   <chr> <chr>   <chr> 
#> 1 Mick  Stones  <NA>  
#> 2 John  Beatles guitar
#> 3 Paul  Beatles bass

right_join() 
band_members %>%
    right_join(band_instruments)
#> # A tibble: 3 × 3
#>   name  band    plays 
#>   <chr> <chr>   <chr> 
#> 1 John  Beatles guitar
#> 2 Paul  Beatles bass  
#> 3 Keith <NA>    guitar

Specify the joining variable name

band_members %>%
    left_join(band_instruments)
#> Joining with `by = join_by(name)`
#> # A tibble: 3 × 3
#>   name  band    plays 
#>   <chr> <chr>   <chr> 
#> 1 Mick  Stones  <NA>  
#> 2 John  Beatles guitar
#> 3 Paul  Beatles bass
band_members %>%
    left_join(
        band_instruments,
        by = 'name'
    )
#> # A tibble: 3 × 3
#>   name  band    plays 
#>   <chr> <chr>   <chr> 
#> 1 Mick  Stones  <NA>  
#> 2 John  Beatles guitar
#> 3 Paul  Beatles bass

Specify the joining variable name

If the names differ, use by = c("left_name" = "joining_name")

band_members
#> # A tibble: 3 × 2
#>   name  band   
#>   <chr> <chr>  
#> 1 Mick  Stones 
#> 2 John  Beatles
#> 3 Paul  Beatles
band_instruments2
#> # A tibble: 3 × 2
#>   artist plays 
#>   <chr>  <chr> 
#> 1 John   guitar
#> 2 Paul   bass  
#> 3 Keith  guitar
band_members %>%
    left_join(
        band_instruments2,
        by = c("name" = "artist")
    )
#> # A tibble: 3 × 3
#>   name  band    plays 
#>   <chr> <chr>   <chr> 
#> 1 Mick  Stones  <NA>  
#> 2 John  Beatles guitar
#> 3 Paul  Beatles bass

Specify the joining variable name

Or just rename the joining variable in a pipe

band_members
#> # A tibble: 3 × 2
#>   name  band   
#>   <chr> <chr>  
#> 1 Mick  Stones 
#> 2 John  Beatles
#> 3 Paul  Beatles
band_instruments2
#> # A tibble: 3 × 2
#>   artist plays 
#>   <chr>  <chr> 
#> 1 John   guitar
#> 2 Paul   bass  
#> 3 Keith  guitar
band_members %>%
    rename(artist = name) %>%
    left_join(
        band_instruments2,
        by = "artist"
    )
#> # A tibble: 3 × 3
#>   artist band    plays 
#>   <chr>  <chr>   <chr> 
#> 1 Mick   Stones  <NA>  
#> 2 John   Beatles guitar
#> 3 Paul   Beatles bass

15:00

Your turn

  1. Create a new data frame called state_data by joining the state_abbs and state_regions data frames. The result should be a data frame with variables state_name, state_abb, and state_region. It should look like this:
head(state_data)
#> # A tibble: 6 × 3
#>   state_name state_abb state_region
#>   <chr>      <chr>     <chr>       
#> 1 Alabama    AL        Southeast   
#> 2 Alaska     AK        Pacific     
#> 3 Arizona    AZ        Mountain    
#> 4 Arkansas   AR        Delta States
#> 5 California CA        Pacific     
#> 6 Colorado   CO        Mountain
  1. Join the state_data data frame to the wildlife_impacts data frame, adding the variables state_region and state_name.
glimpse(wildlife_impacts)
#> Rows: 56,978
#> Columns: 23
#> $ state_abb             <chr> "FL", "IN", NA, NA, NA, "FL", "FL", NA, NA, "FL", NA, "TX", NA, NA, "NY", NA, NA, "MD", "CA", "AZ", "NC", "TX", NA, NA, "CA", NA, NA, "NM", NA, NA, NA, NA, "CA", "NC", "FL", "FL", "CA", NA, "TX", "CA", "PA", NA, "TX", …
#> $ state_name            <chr> "Florida", "Indiana", NA, NA, NA, "Florida", "Florida", NA, NA, "Florida", NA, "Texas", NA, NA, "New York", NA, NA, "Maryland", "California", "Arizona", "North Carolina", "Texas", NA, NA, "California", NA, NA, "New Mex…
#> $ state_region          <chr> "Southeast", "Corn Belt", NA, NA, NA, "Southeast", "Southeast", NA, NA, "Southeast", NA, "Southern Plains", NA, NA, "Northeast", NA, NA, "Northeast", "Pacific", "Mountain", "Appalachian", "Southern Plains", NA, NA, "Pa…
#> $ incident_date         <dttm> 2018-12-31, 2018-12-29, 2018-12-29, 2018-12-27, 2018-12-27, 2018-12-27, 2018-12-27, 2018-12-26, 2018-12-23, 2018-12-23, 2018-12-23, 2018-12-22, 2018-12-22, 2018-12-22, 2018-12-22, 2018-12-22, 2018-12-21, 2018-12-21, 2…
#> $ airport_id            <chr> "KMIA", "KIND", "ZZZZ", "ZZZZ", "ZZZZ", "KMIA", "KMCO", "ZZZZ", "ZZZZ", "KFLL", "ZZZZ", "KGRK", "ZZZZ", "ZZZZ", "KJFK", "MDPP", "MNMG", "KBWI", "KSMF", "KPHX", "KCLT", "KDFW", "ZZZZ", "ZZZZ", "KSNA", "ZZZZ", "ZZZZ", "K…
#> $ airport               <chr> "MIAMI INTL", "INDIANAPOLIS INTL ARPT", "UNKNOWN", "UNKNOWN", "UNKNOWN", "MIAMI INTL", "ORLANDO INTL", "UNKNOWN", "UNKNOWN", "FORT LAUDERDALE/HOLLYWOOD INTL", "UNKNOWN", "KILLEEN/FT HOOD REGIONAL", "UNKNOWN", "UNKNOWN"…
#> $ operator              <chr> "AMERICAN AIRLINES", "AMERICAN AIRLINES", "AMERICAN AIRLINES", "AMERICAN AIRLINES", "AMERICAN AIRLINES", "AMERICAN AIRLINES", "AMERICAN AIRLINES", "AMERICAN AIRLINES", "AMERICAN AIRLINES", "AMERICAN AIRLINES", "AMERICA…
#> $ atype                 <chr> "B-737-800", "B-737-800", "UNKNOWN", "B-737-900", "B-737-800", "A-319", "A-321", "B-737-800", "A-321", "B-737-800", "B-737-800", "EMB-145", "A-319", "A-319", "B-737-800", "B-737-800", "B-737-800", "A-319", "A-319", "B-…
#> $ type_eng              <chr> "D", "D", NA, "D", "D", "D", "D", "D", "D", "D", "D", "D", "D", "D", "D", "D", "D", "D", "D", "D", "D", "D", "D", "D", "D", "D", "D", "D", NA, "D", "D", "D", "D", "D", "D", "D", "D", "D", "D", "D", "D", "D", "D", "D", …
#> $ species_id            <chr> "UNKBL", "R", "R2004", "N5205", "J2139", "UNKB", "UNKBS", "ZT001", "ZT101", "I1301", "UNKB", "O22", "ZX010", "ZX303", "K5114", "UNKBS", "UNKBS", "UNKB", "J2141", "UNKBS", "ZT101", "UNKB", "UNKB", "ZS009", "UNKB", "UNKB…
#> $ species               <chr> "Unknown bird - large", "Owls", "Short-eared owl", "Southern lapwing", "Lesser scaup", "Unknown bird", "Unknown bird - small", "Eastern meadowlark", "Red-winged blackbird", "Cattle egret", "Unknown bird", "Doves", "Pin…
#> $ damage                <chr> "M?", "N", NA, "M?", "M?", "N", "N", "N", "N", "N", "N", "N", "N", "N", NA, "N", "N", "N", "N", "N", "N", "N", "N", "N", "N", "N", NA, "N", NA, "N", "N", "N", "N", "N", "N", "N", "N", "N", "N", NA, "N", "N", "N", NA, "…
#> $ num_engs              <dbl> 2, 2, NA, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, NA, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, NA, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,…
#> $ incident_month        <dbl> 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 11, 11, 11, 11, 11, 11, 11, 11…
#> $ incident_year         <dbl> 2018, 2018, 2018, 2018, 2018, 2018, 2018, 2018, 2018, 2018, 2018, 2018, 2018, 2018, 2018, 2018, 2018, 2018, 2018, 2018, 2018, 2018, 2018, 2018, 2018, 2018, 2018, 2018, 2018, 2018, 2018, 2018, 2018, 2018, 2018, 2018, 20…
#> $ time_of_day           <chr> "Day", "Night", NA, NA, NA, "Day", "Night", NA, NA, "Day", "Night", "Day", NA, NA, "Day", "Day", "Day", "Day", "Day", "Night", "Day", "Dawn", NA, NA, "Day", NA, NA, "Day", NA, NA, NA, NA, "Day", "Day", "Day", "Day", "D…
#> $ time                  <dbl> 1207, 2355, NA, NA, NA, 955, 948, NA, NA, 1321, 15, 1612, NA, NA, 905, 1457, 1418, 1628, 627, 2130, 719, 747, NA, NA, 1348, NA, NA, 1305, NA, NA, NA, NA, 1345, 944, 1400, 1415, 1150, 800, 1400, 1505, 1731, NA, NA, 1733…
#> $ height                <dbl> 700, 0, NA, NA, NA, NA, 600, NA, NA, 0, NA, 0, NA, NA, 0, 500, 100, 0, 1000, 4500, 300, NA, NA, NA, NA, NA, NA, 0, NA, NA, NA, NA, 0, NA, 0, 100, 0, 3027, 0, NA, 0, NA, 200, NA, 0, 300, 8, NA, 500, NA, NA, NA, NA, NA, …
#> $ speed                 <dbl> 200, NA, NA, NA, NA, NA, 145, NA, NA, 130, NA, NA, NA, NA, NA, 160, 150, NA, NA, 250, NA, NA, NA, NA, NA, NA, NA, 160, NA, NA, NA, NA, 100, NA, NA, 100, 150, 130, 130, NA, 150, NA, 150, NA, NA, 140, 144, NA, 145, NA, N…
#> $ phase_of_flt          <chr> "Climb", "Landing Roll", NA, NA, NA, "Approach", "Approach", NA, NA, "Take-off run", NA, "Landing Roll", NA, NA, "Landing Roll", "Approach", "Approach", "Take-off run", "Climb", "Climb", "Approach", "Approach", NA, NA,…
#> $ sky                   <chr> "Some Cloud", NA, NA, NA, NA, NA, "Some Cloud", NA, NA, "No Cloud", NA, "Some Cloud", NA, NA, "Some Cloud", "No Cloud", "No Cloud", NA, "No Cloud", "No Cloud", "No Cloud", "Some Cloud", NA, NA, NA, NA, NA, "Some Cloud"…
#> $ precip                <chr> "None", NA, NA, NA, NA, NA, "None", NA, NA, "None", NA, "None", NA, NA, "None", "None", "None", NA, "None", "None", "None", "None", NA, NA, NA, NA, NA, "None", NA, NA, NA, NA, "None", "None", NA, "None", "None", "None"…
#> $ cost_repairs_infl_adj <dbl> NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA…

Week 9: Optimization & MLE

1. Maximum likelihood estimation

2. Optimization (in general)

3. Joins

4. Pilot data cleaning

Download the conjoint_buttons template repo

Cleaning surveydown survey data


For the rest of class, work with your team to pull in your pilot survey responses and clean your pilot data.


1. Open the survey folder in Positron
2. Open code/2_get_raw_data.R
3. Open code/3_data_cleaning.R