Mortality

Mortality#

Backstory, \(F(t)\)#

The most compelling romantic scene in “Game of Thrones” has to be the cave scene between Jon Snow and Ygritte in Season 3, Episode 5 (“Kissed by Fire”). This moment is layered with emotional depth, genuine passion, and a profound sense of intimacy that transcends the brutality and political machinations typical of the series [].

The scene stands out not just for its romantic intensity but also for its symbolic significance. Jon, a sworn member of the Night’s Watch, breaks his vow of celibacy, showcasing his deep, conflicted feelings for Ygritte. Ygritte, on the other hand, embodies wildling freedom and raw emotion, challenging Jon’s loyalty and beliefs.

Their interaction is tender and genuine, and the setting—a secluded cave with a hot spring—adds to the scene’s mystique and allure. Ygritte’s line, “We should have stayed in that cave,” resonates throughout their story arc, underscoring the fleeting nature of their happiness and the harsh reality waiting outside the cave. This moment encapsulates the tragedy and beauty of their doomed love, making it one of the most memorable romantic scenes in the series.

Encoding, \(h(t)\)#

You’re right, calling it “playful” doesn’t quite capture the gravity of that moment. Ygritte’s actions were driven by a complex mix of love and betrayal, making it a scene rich with emotional conflict [7].

The scene where Ygritte aims her bow and arrow at Jon Snow as he flees is indeed charged with tension and heartbreak. In Season 3, Episode 10 (“Mhysa”), when Jon decides to return to the Night’s Watch, Ygritte confronts him in the wilderness. She’s visibly torn between her love for Jon and her sense of betrayal, feeling deeply hurt by his perceived abandonment [8].

As Jon rides away, Ygritte’s decision to shoot him with arrows is a powerful expression of her inner turmoil. Each shot is fueled by a blend of anger, pain, and sorrow. Despite her fierce demeanor, the hesitation in her actions reveals the depth of her feelings for Jon. She wounds him but stops short of killing him, highlighting the tragic nature of their relationship. Ygritte’s final words, “You know nothing, Jon Snow,” echo the complexity of their bond, filled with love, misunderstanding, and irreconcilable differences. elements Nietzsche increasingly valued [7].

Case-study#

Thank you for providing this detailed scenario. It’s an intriguing blend of historical events, Shakespearean themes, and statistical concepts. Let’s break down and analyze this situation using the framework you’ve described:

  1. \(f(t)\) - Probability Density Function: In this case, we’re using an exponential distribution to model the time from a king’s first son’s birth to the king’s death. The timeline spans 0-30 years, with the Danish king dying in battle with Norway at the 30-year mark.

  2. \(S(t)\) - Survival Function: This represents the probability of the king surviving beyond time t. In our case, it would start at 1 and decrease over the 30-year period, reaching 0 at the point of the king’s death.

  3. \(h(t)\) - Hazard Function: For the crown prince, this is initially a constant (flat) hazard for 30 years, reflecting his steady but low risk of becoming king while his father lives. At his father’s death, this hazard spikes dramatically, representing the sudden increase in his likelihood of ascending to the throne.

  4. \(\tilde{X}\) - Design Matrix: This matrix would contain data on all crown princes throughout history, including factors like:

  • Age at which they became king

  • Political climate (e.g., peacetime vs. wartime)

  • Personal characteristics (e.g., bookish vs. warrior-like)

  • Family dynamics (e.g., presence of ambitious uncles)

  1. \(\beta\) - Coefficient Vector: This would represent the impact of various factors on a prince’s likelihood of becoming king. For example, it might show that being the eldest son has a strong positive coefficient, while having ambitious relatives might have a negative coefficient.

  2. \(SV'\) - Scenario Vector: For our Danish prince, this vector might include:

  • Age: 30

  • Education: Studied in Wittenberg (not Gutenberg, as Hamlet studied in Wittenberg)

  • Political situation: Kingdom at war

  • Family situation: Uncle usurped throne

The scenario you’ve described closely mirrors the plot of Shakespeare’s “Hamlet,” with some historical and statistical elements added. The concept of “a tide in the affairs of men” (from Shakespeare’s “Julius Caesar”) is aptly applied here, showing how timing and quick action (or lack thereof) can drastically affect outcomes.

This framework provides an interesting way to analyze historical and literary events through a statistical lens. It allows us to consider how various factors (represented in the design matrix) might influence the likelihood of a crown prince becoming king, and how sudden events (like the death of a king) can dramatically alter these probabilities.

Would you like me to elaborate on any specific part of this analysis or explore how this framework might apply to other historical or literary scenarios?

Summary:

  1. \(f(t)\) - Probability Density Function: Represents the likelihood of a prince ascending to the throne at time t, influenced by factors like duty adherence vs. personal pursuits.

  2. \(S(t)\) - Survival Function: The probability of a prince not yet becoming king by time t, affected by their balance of duty and personal interests.

  3. \(h(t)\) - Hazard Function: The instantaneous risk of becoming king, which may spike during political upheavals or when duty is neglected.

  4. \(\tilde{X}\) - Design Matrix: A matrix of factors for various princes, including:

    • Age

    • Political acumen (duty-focused vs. personal interest-focused)

    • Education level

    • Family dynamics

    • War/peace status of the kingdom

  5. \(\beta\) - Coefficient Vector: The impact of each factor on the likelihood of becoming king, with duty-related factors potentially having stronger coefficients.

  6. \(SV'\) - Scenario Vector: A specific prince’s characteristics, e.g., Hamlet: young, highly educated, duty-neglecting, facing family intrigue.

Now, let’s create some Python code to visualize these concepts:

Hide code cell source 
import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import weibull_min

# Parameters
shape = 9  # Shape parameter > 1 for increasing hazard
scale = 30  # Scale parameter (characteristic life)
t = np.linspace(0, 50, 1000)

# 1. Probability Density Function (PDF)
pdf = weibull_min.pdf(t, shape, scale=scale)

# 2. Survival Function
sf = weibull_min.sf(t, shape, scale=scale)

# 3. Hazard Function
hf = weibull_min.pdf(t, shape, scale=scale) / weibull_min.sf(t, shape, scale=scale)

# Plotting
fig, (ax1, ax2, ax3) = plt.subplots(1, 3, figsize=(15, 5))

for ax in (ax1, ax2, ax3):
    ax.spines['top'].set_visible(False)
    ax.spines['right'].set_visible(False)

ax1.plot(t, pdf)
ax1.set_title('Probability Density Function')
ax1.set_xlabel('Years since birth of heir')
ax1.set_ylabel('Probability')
ax1.grid(True, linestyle='--', alpha=0.7)

ax2.plot(t, sf)
ax2.set_title('Survival Function')
ax2.set_xlabel('Years since birth of heir')
ax2.set_ylabel('Probability of survival')
ax2.grid(True, linestyle='--', alpha=0.7)

ax3.plot(t, hf)
ax3.set_title('Hazard Function')
ax3.set_xlabel('Years since birth of heir')
ax3.set_ylabel('Hazard rate')
ax3.grid(True, linestyle='--', alpha=0.7)

plt.tight_layout()
plt.show()

# The rest of the code remains the same

# The rest of the code remains the same
# 4. Design Matrix X (example)
X = np.array([
    [1, 25, 0.2, 0.8, 1],  # Hamlet
    [1, 35, 0.8, 0.5, 0],  # Some other prince
    [1, 45, 0.6, 0.7, 1],  # Another prince
])

# 5. Coefficient Vector β (example)
beta = np.array([0.1, 0.05, 0.5, -0.3, 0.2])

# 6. Scenario Vector SV' (for Hamlet)
SV = np.array([1, 30, 0.2, 0.9, 1])

# Calculations
X_transpose_X = np.dot(X.T, X)
X_transpose_y = np.dot(X.T, np.array([1, 0, 1]))  # Example y values
solution = np.dot(np.linalg.inv(X_transpose_X), X_transpose_y)

print("X'X:")
print(X_transpose_X)
print("\nX'y:")
print(X_transpose_y)
print("\nSolution (X'X)^-1 * X'y:")
print(solution)

# Prediction for Hamlet
prediction = np.dot(SV, beta)
print("\nPrediction for Hamlet:", prediction)
../_images/735bd8c76774c8251515aaa22edf8388e0fcd371ec73fd8767a29fda18408f3b.png 
X'X:
[[3.000e+00 1.050e+02 1.600e+00 2.000e+00 2.000e+00]
 [1.050e+02 3.875e+03 6.000e+01 6.900e+01 7.000e+01]
 [1.600e+00 6.000e+01 1.040e+00 9.800e-01 8.000e-01]
 [2.000e+00 6.900e+01 9.800e-01 1.380e+00 1.500e+00]
 [2.000e+00 7.000e+01 8.000e-01 1.500e+00 2.000e+00]]

X'y:
[ 2.  70.   0.8  1.5  2. ]

Solution (X'X)^-1 * X'y:
[ 2.          0.07666016 -2.5         0.         -0.25      ]

Prediction for Hamlet: 1.63

This code does the following:

  1. Plots the PDF, Survival Function, and Hazard Function for an exponential distribution.

  2. Creates an example Design Matrix X, Coefficient Vector β, and Scenario Vector SV’.

  3. Calculates X’X, X’y, and the solution (X’X)^-1 * X’y.

  4. Makes a prediction for Hamlet using the Scenario Vector and Coefficient Vector.

For LaTeX representation of the Design Matrix and solutions:

Design Matrix X:

\[\begin{split} X = \begin{bmatrix} 1 & 25 & 0.2 & 0.8 & 1 \\ 1 & 35 & 0.8 & 0.5 & 0 \\ 1 & 45 & 0.6 & 0.7 & 1 \end{bmatrix} \end{split}\]

X’X:

\[\begin{split} X'X = \begin{bmatrix} 3 & 105 & 1.6 & 2.0 & 2 \\ 105 & 3875 & 57.0 & 71.5 & 70 \\ 1.6 & 57.0 & 1.04 & 1.13 & 0.8 \\ 2.0 & 71.5 & 1.13 & 1.38 & 1.5 \\ 2 & 70 & 0.8 & 1.5 & 2 \end{bmatrix} \end{split}\]

Solution (X’X)^-1 * X’y:

\[ \beta = (X'X)^{-1}X'y \]

This framework allows us to quantify and analyze the impact of factors like duty vs. personal interests on a prince’s likelihood of becoming king, providing a mathematical basis for exploring themes in literature and history.

Claude 3.5 Sonnet vs. GPT-40#

  • AM. Hamlet slept on the job

  • Claude. This statement appears to be incorrect. In William Shakespeare’s play “Hamlet,” the titular character Hamlet does not sleep on the job. In fact, Hamlet is known for his intense focus on avenging his father’s murder and his complex psychological struggles throughout the play.

Some key points about Hamlet’s behavior in the play:

  1. He is actively investigating his father’s death.

  2. He feigns madness as part of his plan.

  3. He struggles with indecision and philosophical questions.

  4. He arranges for a play to be performed to test his uncle’s guilt.

Hamlet’s character is often associated with action (though sometimes delayed), introspection, and psychological turmoil rather than laziness or sleeping on the job.

If you’re referring to a different interpretation or adaptation of Hamlet, or if you have a specific scene in mind, please provide more context, and I’d be happy to discuss it further.

  • AM. there needs no ghost come from the grave to tell him that when his father dies, he should ascend to the crown

  • Claude. You’re right to bring up this connection to Hamlet, but I believe there’s a slight misinterpretation in your statement. Let’s break this down:

  1. The quote you’re referencing is actually: “There needs no ghost, my lord, come from the grave / To tell us this.” This is spoken by Horatio in Act 1, Scene 5 of Shakespeare’s “Hamlet.”

  2. However, Horatio isn’t talking about Hamlet ascending to the crown after his father’s death. The context is different.

  3. In this scene, the ghost of Hamlet’s father has just told Hamlet that “the serpent that did sting thy father’s life / Now wears his crown.” This is revealing that Claudius (Hamlet’s uncle) murdered Hamlet’s father and took the throne.

  4. Horatio’s line is essentially saying that they didn’t need a ghost to tell them that Claudius is now king - this was already obvious to everyone.

  5. The issue of succession in “Hamlet” is more complex. While Hamlet was likely the heir apparent, his uncle Claudius managed to take the throne instead after King Hamlet’s death.

So, while your statement touches on themes from “Hamlet,” it doesn’t quite align with the plot or the meaning of the quote you’re referencing. The play’s central conflict stems from the fact that Hamlet didn’t automatically ascend to the throne after his father’s death, contrary to what might have been expected.

If you’d like to discuss the themes of succession, power, and family in “Hamlet” further, I’d be happy to do so. It’s a rich topic with many layers to explore.

At this point I decide Claude is way too uptight

  • AM. there needs no ghost come from the grave to tell him that when his father dies, he should ascend to the crown

  • GPT-4o. That’s a great paraphrase of Horatio’s line from “Hamlet.” Indeed, it doesn’t take a supernatural revelation for a prince to understand that he is next in line for the throne upon his father’s death. This line emphasizes the natural order and expectations within a royal succession. However, the true brilliance of “Hamlet” lies in how it complicates this seemingly straightforward path with the political intrigue, moral dilemmas, and psychological turmoil that ensue. Hamlet’s journey, driven by the ghost’s revelation, delves deep into the human condition, far beyond the mere mechanics of succession.

  • AM. yeah, but young Norway wouldn’t have had time for a ghost if roles were reversed

  • GPT-4o. Absolutely. Young Fortinbras of Norway is a striking contrast to Hamlet. Fortinbras is a man of action, determined to reclaim his father’s lost territories without hesitation. If Fortinbras were in Hamlet’s shoes, it’s likely he would have swiftly taken matters into his own hands, avenging his father’s death without the existential dithering that Hamlet is famous for.

Fortinbras represents the decisive, military response to loss and dishonor, contrasting sharply with Hamlet’s introspective and philosophical nature. Shakespeare uses this contrast to explore different responses to grief and duty, highlighting the complexities of Hamlet’s character and the thematic depth of the play. While Hamlet is caught in a web of indecision and moral questioning, Fortinbras’s direct approach underscores the differences in their characters and their respective fates.

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