基于历史事件的未来事件概率计算及贝叶斯随机模型构建咨询

Hey there, let's dive into these two questions—they're great examples of moving from theoretical probability to practical Bayesian modeling.

First, there's no one-size-fits-all answer, but here's a structured approach to tackle this:

• Define your event type: Are you counting how many times an event happens (e.g., number of machine failures per year) or just whether it happens at all (e.g., did a failure occur this year)? This dictates the probability distribution you'll use.
• Check for dependencies: Is the event independent across time (e.g., rare asteroid sightings) or does past occurrence affect future chances (e.g., wildfires following a drought)? If there's time dependence, you'll need models like Markov chains instead of simple static distributions.
• Pick a probability distribution:
  • For count data: Start with Poisson (if mean ≈ variance) or Negative Binomial (if variance > mean, to account for overdispersion).
  • For binary "occur/not occur" data: Use Bernoulli or Binomial distributions.
• Estimate parameters:
  • Frequentist approach: Use historical data to calculate point estimates (e.g., Poisson's λ = average number of events per period).
  • Bayesian approach: Combine historical data with a prior belief about the parameter to get a posterior distribution, which quantifies uncertainty around the true parameter.
• Validate your model: Test if your chosen distribution fits the historical data (e.g., check if observed counts match the distribution's expected frequencies).

Your dataset is [2,0,0,1,0,3,1,0,1,0]—10 years of event counts, with a total of 8 events and an average of 0.8 events per year. Let's build a Bayesian model step by step:

Step 1: Choose the Likelihood Function #

Since we're dealing with count data, the Poisson distribution is a natural starting point. For each year (i), the count (y_i) follows:
[ y_i \sim \text{Poisson}(\lambda) ]
where (\lambda) is the average number of events per year. Our data's variance (~0.995) is close to its mean (~0.8), so Poisson is a reasonable fit (if variance were much higher, we'd switch to Negative Binomial to handle overdispersion).

Step 2: Select a Prior Distribution #

For Poisson's parameter (\lambda), the Gamma distribution is a conjugate prior—this means the posterior distribution will also be Gamma, making calculations straightforward. We'll use a weak-information prior to let the data speak for itself:
[ \lambda \sim \text{Gamma}(\alpha=0.001, \beta=0.001) ]
This prior is intentionally non-informative, so it won't skew our results away from what the historical data tells us.

Step 3: Compute the Posterior Distribution #

Using Bayes' Theorem, the posterior distribution of (\lambda) combines the prior and the likelihood:
[ P(\lambda | y) \propto P(y | \lambda) \times P(\lambda) ]
With our data, the posterior parameters become:

• (\alpha_{\text{post}} = \alpha_{\text{prior}} + \sum y_i = 0.001 + 8 = 8.001)
• (\beta_{\text{post}} = \beta_{\text{prior}} + n = 0.001 + 10 = 10.001)
  So (\lambda \sim \text{Gamma}(8.001, 10.001)) after updating with our historical data.

Step 4: Predict This Year's Event Probability #

To predict the count this year ((y_{\text{new}})), we use the posterior distribution of (\lambda) to generate a predictive distribution. Since (y_{\text{new}} \sim \text{Poisson}(\lambda)) and (\lambda) is Gamma-distributed, the predictive distribution simplifies to a Negative Binomial distribution.

Here are key probabilities you might care about:

• Probability of 0 events this year: (\left(\frac{\beta_{\text{post}}}{\beta_{\text{post}} + 1}\right)^{\alpha_{\text{post}}} \approx 0.4305)
• Probability of at least 1 event this year: (1 - 0.4305 = 0.5695) (≈57%)
• Probability of exactly 1 event: ~0.30, exactly 2 events: ~0.13 (you can calculate these via the Negative Binomial PMF or by sampling from the posterior)

Step 5: Quick Code Implementation #

If you want to run this yourself, here's a Python snippet using scipy:

import numpy as np
from scipy.stats import gamma, poisson

# Historical data
y = np.array([2, 0, 0, 1, 0, 3, 1, 0, 1, 0])
sum_y = y.sum()
n_years = len(y)

# Prior parameters (weak information)
alpha_prior = 0.001
beta_prior = 0.001

# Posterior parameters
alpha_post = alpha_prior + sum_y
beta_post = beta_prior + n_years

# Calculate key probabilities
p_zero = (beta_post / (beta_post + 1)) ** alpha_post
p_at_least_one = 1 - p_zero

print(f"Probability of 0 events this year: {p_zero:.4f}")
print(f"Probability of at least 1 event this year: {p_at_least_one:.4f}")

# Generate predictive samples for full distribution
n_samples = 10000
lambda_samples = gamma.rvs(alpha_post, scale=1/beta_post, size=n_samples)
y_new_samples = poisson.rvs(lambda_samples)

# Print probabilities for counts 0-3
for k in range(4):
    prob = np.mean(y_new_samples == k)
    print(f"Probability of {k} events: {prob:.4f}")

内容的提问来源于stack exchange，提问作者ARK