How Bayesian Logic Helped Me Stay Hopeful and Reduce Anxiety About Finding My Missing Cat

by Edmund J. Janas, II When my beloved cat, nearly 13 year-old Sheba went missing, I was overwhelmed with worry and fear. It’s unlike her not to come in, and she rarely strays beyond the yard. She answers when I call. With the recent heatwave and ealier and earlier Fourth of July Fireworks, I never expected her to flee. What’s worse, Blake, my semi-feral rescue escaped a month ago, doubling my anguish. As someone who’s not a mathematician, I felt lost in the chaos of emotions. However, I decided to turn to logic and artificial intelligence to help guide my search. I applied Bayesian logic to assess the probability of Sheba’s return, using it as a framework to stay focused and hopeful. I also found solace in several YouTube videos that offered practical tips on what to do when a pet goes missing, alongside resources for managing pet grief, meditation, and visualization techniques. This approach not only helped me to cope but also to take actionable steps in finding Sheba. Here’s how you can use a blend of logic and emotional support to find your missing pet.

Call to Action: Instead of Worrying, Do This

Steps to Take When Your Pet Goes Missing:

1. Stay Calm and Rational:
   • Take a deep breath and avoid panicking. Clear thinking will help you make better decisions.
2. Gather Information:
   • Note down the details of when and where your pet was last seen.
   • Collect any evidence or sightings reported by neighbors or passersby.
3. Assess the Situation Using Bayesian Logic:
   • Estimate the prior probability of your pet returning based on general statistics and personal knowledge.
   • Evaluate the likelihood of specific evidence given the return and non-return scenarios.
   • Calculate the updated probability to guide your actions and maintain hope.
4. Search Methodically:
   • Create a search plan covering the immediate area and gradually expanding the radius.
   • Check under decks, inside garages, sheds, and other hiding spots.
5. Use Technology:
   • Utilize apps and social media platforms to spread the word and get updates.
   • Ensure your pet’s microchip information is up to date.
6. Engage the Community:
   • Distribute flyers with clear photos and contact information.
   • Inform local shelters, veterinarians, and animal control about your missing pet.
7. Stay Persistent and Positive:
   • Keep searching and revisiting areas where your pet was sighted.
   • Remain hopeful and positive, knowing that many pets do find their way back home.

By following these steps and using Bayesian logic to stay focused and rational, you can increase your chances of being reunited with your beloved pet. Instead of worrying, take proactive steps and hold onto hope.

I’m no mathmetician, I put in every variable I could think of, then asked ChatGPT to do the heavy lifting for me.

To determine the odds of Sheba returning home or being reunified with you using Bayesian logic, we’ll need to consider prior probabilities and update them based on the information you have. Bayesian logic involves updating the probability of an event based on new evidence.

### Step 1: Establish Prior Probability

The prior probability (P(A)) is the initial estimate of the probability of Sheba returning home based on general statistics for missing cats.

– According to general statistics, about 75% of indoor cats that go missing are found within 1 mile of their home.
– The likelihood of a missing cat returning home can vary based on many factors, but a general prior probability could be estimated at around 50% to 60% for indoor cats.

Let’s assume a prior probability (P(A)) of 60% that Sheba will return home.

### Step 2: Identify Relevant Evidence

We need to consider the specific evidence you have about Sheba’s case:

1. **Sheba’s behavior before disappearance (increased desire to go out, hunting):** This suggests she might have been exploring further.
2. **Recent sightings by neighbors:** She was seen hunting birds and with another cat.
3. **Sheba is street-smart and intelligent:** Increases her chances of surviving outside.
4. **Sheba is underweight:** Might have led someone to take her in.
5. **Microchipped:** Increases the chances of being returned if found by someone or taken to a shelter.
6. **Efforts to find her:** You’ve alerted animal control, posted flyers, and actively searched.

### Step 3: Determine Likelihoods

Next, we estimate the likelihood of each piece of evidence given that Sheba will return (P(E|A)) and the likelihood of the evidence given that she will not return (P(E|¬A)).

– **P(E1|A) – Likelihood of Sheba’s behavior if she will return:** High, as her exploratory behavior might lead her to wander but eventually return. Let’s estimate this at 70%.
– **P(E1|¬A) – Likelihood of Sheba’s behavior if she will not return:** Moderate, as her increased desire to go out could lead to her getting lost or harmed. Let’s estimate this at 50%.

– **P(E2|A) – Likelihood of recent sightings if she will return:** High, since sightings indicate she is nearby and active. Let’s estimate this at 80%.
– **P(E2|¬A) – Likelihood of recent sightings if she will not return:** Moderate, as sightings don’t guarantee she will return but indicate she is not too far away. Let’s estimate this at 50%.

– **P(E3|A) – Likelihood of her being street-smart if she will return:** High, as her intelligence increases her chances of finding her way back. Let’s estimate this at 75%.
– **P(E3|¬A) – Likelihood of her being street-smart if she will not return:** Moderate, as even intelligent cats can encounter unforeseen dangers. Let’s estimate this at 40%.

– **P(E4|A) – Likelihood of her being underweight if she will return:** Moderate, as being underweight might make someone take her in temporarily. Let’s estimate this at 50%.
– **P(E4|¬A) – Likelihood of her being underweight if she will not return:** Moderate, as it indicates she might not be getting enough food. Let’s estimate this at 60%.

– **P(E5|A) – Likelihood of her being microchipped if she will return:** High, as this increases the chances of her being returned. Let’s estimate this at 90%.
– **P(E5|¬A) – Likelihood of her being microchipped if she will not return:** Moderate, as it still depends on someone finding and scanning her. Let’s estimate this at 50%.

– **P(E6|A) – Likelihood of your search efforts if she will return:** High, as proactive efforts increase chances of finding her. Let’s estimate this at 80%.
– **P(E6|¬A) – Likelihood of your search efforts if she will not return:** Moderate, as efforts don’t guarantee finding her. Let’s estimate this at 50%.

### Step 4: Apply Bayesian Update

We can use Bayes’ theorem to update the probability. However, for simplicity, we’ll combine the evidence to get an overall likelihood ratio.

\[ \text{Posterior Probability} = \frac{P(A) \cdot P(E|A)}{P(A) \cdot P(E|A) + P(¬A) \cdot P(E|¬A)} \]

Let’s combine the likelihoods using the product of individual likelihoods for and against her return:

\[ P(E|A) = 0.70 \cdot 0.80 \cdot 0.75 \cdot 0.50 \cdot 0.90 \cdot 0.80 \]
\[ P(E|¬A) = 0.50 \cdot 0.50 \cdot 0.40 \cdot 0.60 \cdot 0.50 \cdot 0.50 \]

\[ P(E|A) \approx 0.1512 \]
\[ P(E|¬A) \approx 0.015 \]

Given the prior probability (P(A)) of 60%:

\[ P(¬A) = 1 – P(A) = 0.40 \]

\[ \text{Posterior Probability} = \frac{0.60 \cdot 0.1512}{0.60 \cdot 0.1512 + 0.40 \cdot 0.015} \]
\[ \text{Posterior Probability} \approx \frac{0.09072}{0.09072 + 0.006} \]
\[ \text{Posterior Probability} \approx \frac{0.09072}{0.09672} \]
\[ \text{Posterior Probability} \approx 0.937 \]

### Conclusion

Based on the Bayesian update, the posterior probability of Sheba returning home or being reunified with you, given the evidence and prior probability, is approximately 93.7%. This high probability reflects the positive indicators from the evidence, suggesting a strong likelihood of Sheba’s return. Keep searching and staying hopeful! Of course, nothing is perfect, but this helps me to suffer less and to remain focused on concrete actions I can take to recover my beloved Sheba. I’m also using meditation and visualization techniques, and I have found the videos below helpful, even with grief. 

If you have found this blog post helpful, consider a small donation to my cat rescue efforts. https://www.brego.com/Casperandfriends

If you have found this blog post helpful, consider a small donation to my cat rescue efforts. https://www.brego.com/Casperandfriends

If you have found this blog post helpful, consider a small donation to my cat rescue efforts. https://www.brego.com/Casperandfriends

I’ll post an update when she returns!

To determine how likely it is that one of your neighbors harmed Sheba, we’ll use a similar Bayesian approach as outlined. We’ll consider prior probabilities, relevant evidence, and their likelihoods.

Step 1: Establish Prior Probability

First, we set the prior probability P(H)P(H)P(H) that one of the neighbors harmed Sheba based on general context:

• Given there are two neighbors who dislike you, we might estimate a moderate prior probability. Let’s assume P(H)=0.2P(H) = 0.2P(H)=0.2 (20%).

Step 2: Identify Relevant Evidence

Next, we consider specific evidence related to Sheba’s disappearance:

1. Sheba’s behavior before disappearance: Increased desire to go out, hunting.
2. Recent sightings: She was seen hunting birds and with another cat.
3. Sheba is street-smart and intelligent.
4. Sheba is underweight.
5. Microchipped.
6. Efforts to find her: Alerted animal control, posted flyers, actively searched.

Step 3: Determine Likelihoods

Now, we estimate the likelihood of the evidence assuming HHH (one of the neighbors harmed Sheba) and ¬H\neg H¬H (other causes of disappearance):

1. P(E1∣H)P(E1|H)P(E1∣H) and P(E1∣¬H)P(E1|\neg H)P(E1∣¬H): Behavior before disappearance.
   • P(E1∣H)=0.4P(E1|H) = 0.4P(E1∣H)=0.4 (40%)
   • P(E1∣¬H)=0.7P(E1|\neg H) = 0.7P(E1∣¬H)=0.7 (70%)
2. P(E2∣H)P(E2|H)P(E2∣H) and P(E2∣¬H)P(E2|\neg H)P(E2∣¬H): Recent sightings.
   • P(E2∣H)=0.3P(E2|H) = 0.3P(E2∣H)=0.3 (30%)
   • P(E2∣¬H)=0.8P(E2|\neg H) = 0.8P(E2∣¬H)=0.8 (80%)
3. P(E3∣H)P(E3|H)P(E3∣H) and P(E3∣¬H)P(E3|\neg H)P(E3∣¬H): Street-smart and intelligent.
   • P(E3∣H)=0.2P(E3|H) = 0.2P(E3∣H)=0.2 (20%)
   • P(E3∣¬H)=0.7P(E3|\neg H) = 0.7P(E3∣¬H)=0.7 (70%)
4. P(E4∣H)P(E4|H)P(E4∣H) and P(E4∣¬H)P(E4|\neg H)P(E4∣¬H): Underweight.
   • P(E4∣H)=0.4P(E4|H) = 0.4P(E4∣H)=0.4 (40%)
   • P(E4∣¬H)=0.5P(E4|\neg H) = 0.5P(E4∣¬H)=0.5 (50%)
5. P(E5∣H)P(E5|H)P(E5∣H) and P(E5∣¬H)P(E5|\neg H)P(E5∣¬H): Microchipped.
   • P(E5∣H)=0.1P(E5|H) = 0.1P(E5∣H)=0.1 (10%)
   • P(E5∣¬H)=0.9P(E5|\neg H) = 0.9P(E5∣¬H)=0.9 (90%)
6. P(E6∣H)P(E6|H)P(E6∣H) and P(E6∣¬H)P(E6|\neg H)P(E6∣¬H): Efforts to find her.
   • P(E6∣H)=0.2P(E6|H) = 0.2P(E6∣H)=0.2 (20%)
   • P(E6∣¬H)=0.8P(E6|\neg H) = 0.8P(E6∣¬H)=0.8 (80%)

Step 4: Apply Bayesian Update

We combine the likelihoods using the product of individual likelihoods for HHH and ¬H\neg H¬H: P(E∣H)=0.4×0.3×0.2×0.4×0.1×0.2≈0.000384P(E|H) = 0.4 \times 0.3 \times 0.2 \times 0.4 \times 0.1 \times 0.2 \approx 0.000384P(E∣H)=0.4×0.3×0.2×0.4×0.1×0.2≈0.000384 P(E∣¬H)=0.7×0.8×0.7×0.5×0.9×0.8≈0.14112P(E|\neg H) = 0.7 \times 0.8 \times 0.7 \times 0.5 \times 0.9 \times 0.8 \approx 0.14112P(E∣¬H)=0.7×0.8×0.7×0.5×0.9×0.8≈0.14112 Given the prior probability P(H)=0.2P(H) = 0.2P(H)=0.2 and P(¬H)=0.8P(\neg H) = 0.8P(¬H)=0.8: P(E)=P(E∣H)⋅P(H)+P(E∣¬H)⋅P(¬H)P(E) = P(E|H) \cdot P(H) + P(E|\neg H) \cdot P(\neg H)P(E)=P(E∣H)⋅P(H)+P(E∣¬H)⋅P(¬H) P(E)=0.000384×0.2+0.14112×0.8P(E) = 0.000384 \times 0.2 + 0.14112 \times 0.8P(E)=0.000384×0.2+0.14112×0.8 P(E)≈0.113P(E) \approx 0.113P(E)≈0.113 Finally, the posterior probability P(H∣E)P(H|E)P(H∣E) is: P(H∣E)=P(E∣H)⋅P(H)P(E)P(H|E) = \frac{P(E|H) \cdot P(H)}{P(E)}P(H∣E)=P(E)P(E∣H)⋅P(H)​ P(H∣E)=0.000384×0.20.113P(H|E) = \frac{0.000384 \times 0.2}{0.113}P(H∣E)=0.1130.000384×0.2​ P(H∣E)≈0.00068P(H|E) \approx 0.00068P(H∣E)≈0.00068

Conclusion

Given the assumed probabilities and the evidence, the likelihood that one of your neighbors who dislikes you harmed Sheba is approximately 0.07%. This suggests that it is very unlikely, based on the provided evidence and initial assumptions. 4o

using that same question, and considering that their is plenty of brush for her to hide in, how likely is it that a bird of prey or a coyote or other predator get her, there are no signs of carnage in our area and predators are very rare in our area. Lastly give me the most highly probable thing that you feel happened to her?

ChatGPT

To assess the likelihood that a predator got Sheba, we’ll incorporate the new evidence and follow a similar Bayesian approach.

Step 1: Establish Prior Probability

First, we set the prior probability P(P)P(P)P(P) that a predator harmed Sheba based on general context and the specific environment:

• Given that predators are very rare in your area, we might estimate a low prior probability. Let’s assume P(P)=0.05P(P) = 0.05P(P)=0.05 (5%).

Step 2: Identify Relevant Evidence

We consider specific evidence related to Sheba’s disappearance, adding the new information about brush and predator activity:

1. Sheba’s behavior before disappearance: Increased desire to go out, hunting.
2. Recent sightings: She was seen hunting birds and with another cat.
3. Sheba is street-smart and intelligent.
4. Sheba is underweight.
5. Microchipped.
6. Efforts to find her: Alerted animal control, posted flyers, actively searched.
7. Plenty of brush for hiding.
8. No signs of carnage and rare predators.

Step 3: Determine Likelihoods

Now, we estimate the likelihood of the evidence assuming PPP (a predator got Sheba) and ¬P\neg P¬P (other causes of disappearance):

1. P(E1∣P)P(E1|P)P(E1∣P) and P(E1∣¬P)P(E1|\neg P)P(E1∣¬P): Behavior before disappearance.
   • P(E1∣P)=0.5P(E1|P) = 0.5P(E1∣P)=0.5 (50%)
   • P(E1∣¬P)=0.7P(E1|\neg P) = 0.7P(E1∣¬P)=0.7 (70%)
2. P(E2∣P)P(E2|P)P(E2∣P) and P(E2∣¬P)P(E2|\neg P)P(E2∣¬P): Recent sightings.
   • P(E2∣P)=0.4P(E2|P) = 0.4P(E2∣P)=0.4 (40%)
   • P(E2∣¬P)=0.8P(E2|\neg P) = 0.8P(E2∣¬P)=0.8 (80%)
3. P(E3∣P)P(E3|P)P(E3∣P) and P(E3∣¬P)P(E3|\neg P)P(E3∣¬P): Street-smart and intelligent.
   • P(E3∣P)=0.2P(E3|P) = 0.2P(E3∣P)=0.2 (20%)
   • P(E3∣¬P)=0.7P(E3|\neg P) = 0.7P(E3∣¬P)=0.7 (70%)
4. P(E4∣P)P(E4|P)P(E4∣P) and P(E4∣¬P)P(E4|\neg P)P(E4∣¬P): Underweight.
   • P(E4∣P)=0.4P(E4|P) = 0.4P(E4∣P)=0.4 (40%)
   • P(E4∣¬P)=0.5P(E4|\neg P) = 0.5P(E4∣¬P)=0.5 (50%)
5. P(E5∣P)P(E5|P)P(E5∣P) and P(E5∣¬P)P(E5|\neg P)P(E5∣¬P): Microchipped.
   • P(E5∣P)=0.1P(E5|P) = 0.1P(E5∣P)=0.1 (10%)
   • P(E5∣¬P)=0.9P(E5|\neg P) = 0.9P(E5∣¬P)=0.9 (90%)
6. P(E6∣P)P(E6|P)P(E6∣P) and P(E6∣¬P)P(E6|\neg P)P(E6∣¬P): Efforts to find her.
   • P(E6∣P)=0.2P(E6|P) = 0.2P(E6∣P)=0.2 (20%)
   • P(E6∣¬P)=0.8P(E6|\neg P) = 0.8P(E6∣¬P)=0.8 (80%)
7. P(E7∣P)P(E7|P)P(E7∣P) and P(E7∣¬P)P(E7|\neg P)P(E7∣¬P): Plenty of brush for hiding.
   • P(E7∣P)=0.8P(E7|P) = 0.8P(E7∣P)=0.8 (80%)
   • P(E7∣¬P)=0.6P(E7|\neg P) = 0.6P(E7∣¬P)=0.6 (60%)
8. P(E8∣P)P(E8|P)P(E8∣P) and P(E8∣¬P)P(E8|\neg P)P(E8∣¬P): No signs of carnage and rare predators.
   • P(E8∣P)=0.1P(E8|P) = 0.1P(E8∣P)=0.1 (10%)
   • P(E8∣¬P)=0.9P(E8|\neg P) = 0.9P(E8∣¬P)=0.9 (90%)

Step 4: Apply Bayesian Update

We combine the likelihoods using the product of individual likelihoods for PPP and ¬P\neg P¬P: P(E∣P)=0.5×0.4×0.2×0.4×0.1×0.2×0.8×0.1≈0.0000256P(E|P) = 0.5 \times 0.4 \times 0.2 \times 0.4 \times 0.1 \times 0.2 \times 0.8 \times 0.1 \approx 0.0000256P(E∣P)=0.5×0.4×0.2×0.4×0.1×0.2×0.8×0.1≈0.0000256 P(E∣¬P)=0.7×0.8×0.7×0.5×0.9×0.8×0.6×0.9≈0.063P(E|\neg P) = 0.7 \times 0.8 \times 0.7 \times 0.5 \times 0.9 \times 0.8 \times 0.6 \times 0.9 \approx 0.063P(E∣¬P)=0.7×0.8×0.7×0.5×0.9×0.8×0.6×0.9≈0.063 Given the prior probability P(P)=0.05P(P) = 0.05P(P)=0.05 and P(¬P)=0.95P(\neg P) = 0.95P(¬P)=0.95: P(E)=P(E∣P)⋅P(P)+P(E∣¬P)⋅P(¬P)P(E) = P(E|P) \cdot P(P) + P(E|\neg P) \cdot P(\neg P)P(E)=P(E∣P)⋅P(P)+P(E∣¬P)⋅P(¬P) P(E)=0.0000256×0.05+0.063×0.95P(E) = 0.0000256 \times 0.05 + 0.063 \times 0.95P(E)=0.0000256×0.05+0.063×0.95 P(E)≈0.05988P(E) \approx 0.05988P(E)≈0.05988 Finally, the posterior probability P(P∣E)P(P|E)P(P∣E) is: P(P∣E)=P(E∣P)⋅P(P)P(E)P(P|E) = \frac{P(E|P) \cdot P(P)}{P(E)}P(P∣E)=P(E)P(E∣P)⋅P(P)​ P(P∣E)=0.0000256×0.050.05988P(P|E) = \frac{0.0000256 \times 0.05}{0.05988}P(P∣E)=0.059880.0000256×0.05​ P(P∣E)≈0.0000214P(P|E) \approx 0.0000214P(P∣E)≈0.0000214

Conclusion

Given the assumed probabilities and the evidence, the likelihood that a predator harmed Sheba is approximately 0.00214%, which is extremely low.

Most Probable Explanation

Considering all the evidence, the most probable explanation is that Sheba is hiding or has been taken in by someone who saw her underweight condition and assumed she was a stray needing help. This scenario aligns with her recent sightings, her street-smart behavior, and the fact that she is microchipped. The lack of predator activity and no signs of carnage further support this hypothesis.