Bayesian Estimation

Bayesian estimation is a powerful approach to parameter estimation that leverages the principles of Bayesian inference. Unlike traditional methods such as Maximum Likelihood Estimation (MLE), Bayesian estimation incorporates prior knowledge and updates beliefs based on observed data. This article explores estimation techniques within the Bayesian framework, including point estimates and posterior predictive distributions, and compares Bayesian estimation with MLE through detailed examples.

Understanding Bayesian Estimation

What is Bayesian Estimation?

Bayesian estimation involves estimating the parameters of a statistical model by using Bayes’ Theorem to update prior beliefs about the parameters based on observed data. The result is a posterior distribution that reflects the updated beliefs about the parameters after considering the evidence provided by the data.

Mathematically, given a parameter $\theta$, prior distribution $P(\theta)$, likelihood $P(X|\theta)$, and observed data $X$, the posterior distribution is given by:

$$P(\theta|X) = \frac{P(X|\theta) \cdot P(\theta)}{P(X)}$$

Where:

• $P(\theta|X)$ is the posterior distribution: the probability distribution of the parameter $\theta$ after observing the data $X$.
• $P(X|\theta)$ is the likelihood: the probability of observing the data $X$ given the parameter $\theta$.
• $P(\theta)$ is the prior distribution: the initial belief about the parameter before observing the data.
• $P(X)$ is the marginal likelihood or evidence: the total probability of the data under all possible parameter values.

Key Concepts in Bayesian Estimation

1. Prior Distribution ($P(\theta)$):

• Reflects prior knowledge or beliefs about the parameter before observing any data. Priors can be informative (based on expert knowledge) or non-informative (reflecting complete uncertainty).

2. Posterior Distribution ($P(\theta|X)$):

• Combines the prior distribution and the likelihood to produce an updated belief about the parameter after observing the data.

3. Bayesian Point Estimates:

• Maximum A Posteriori (MAP) Estimate: The mode of the posterior distribution, representing the most probable value of the parameter given the data.
• Posterior Mean: The expected value of the posterior distribution, representing the average of the parameter values weighted by their posterior probabilities.
• Posterior Median: The value that divides the posterior distribution into two equal halves, representing a central estimate of the parameter.

4. Posterior Predictive Distribution:

• The distribution of a new data point predicted using the posterior distribution of the parameters. It accounts for both the uncertainty in the parameter estimates and the variability in the data.

Bayesian Estimation Techniques

Point Estimates: MAP and Posterior Mean

Example 1: Estimating the Mean of a Normal Distribution

Problem Setup

Suppose you have a set of data points that you believe are drawn from a normal distribution with an unknown mean $\mu$ and known variance $\sigma^2 = 4$. Assume that the data points $x_1, x_2, \dots, x_n$ are independent and identically distributed (i.i.d.). You have a prior belief that $\mu$ is normally distributed with mean $\mu_0 = 0$ and variance $\tau^2 = 1$. You observe a sample of data $X = \{x_1, x_2, \dots, x_n\}$, where $n = 10$, and you want to estimate $\mu$ using Bayesian estimation.

Step 1: Specify the Prior and Likelihood

• Prior Distribution: $P(\mu) \sim \mathcal{N}(\mu_0, \tau^2) = \mathcal{N}(0, 1)$
• Likelihood: $P(X|\mu) = \prod_{i=1}^n \frac{1}{\sqrt{2\pi\sigma^2}} \exp\left(-\frac{(x_i - \mu)^2}{2\sigma^2}\right)$

Step 2: Derive the Posterior Distribution

Using Bayes' Theorem, the posterior distribution of $\mu$ is:

$$P(\mu|X) \propto P(X|\mu) \cdot P(\mu)$$

Since both the prior and the likelihood are normal, the posterior distribution is also normal. The posterior mean $\mu_{\text{post}}$ and posterior variance $\sigma_{\text{post}}^2$ are given by:

$$\mu_{\text{post}} = \frac{\sigma^2 \mu_0 + n \tau^2 \bar{x}}{\sigma^2 + n \tau^2}$$ $$\sigma_{\text{post}}^2 = \frac{\sigma^2 \tau^2}{\sigma^2 + n \tau^2}$$

Where $\bar{x}$ is the sample mean. Thus,

$$P(\mu|X) \sim \mathcal{N}\left(\frac{\sigma^2 \mu_0 + n\tau^2 \bar{x}}{\sigma^2 + n\tau^2}, \frac{\sigma^2 \tau^2}{\sigma^2 + n\tau^2}\right)$$

Step 3: Calculate the MAP and Posterior Mean

• MAP Estimate: Since the posterior distribution is normal, the MAP estimate is the same as the posterior mean:

  $$\mu_{\text{MAP}} = \frac{\sigma^2 \mu_0 + n\tau^2 \bar{x}}{\sigma^2 + n\tau^2}$$

• Posterior Mean: In this case, the posterior mean and the MAP estimate coincide, so:

  $$\hat{\mu} = \mu_{\text{MAP}} = \frac{\sigma^2 \mu_0 + n\tau^2 \bar{x}}{\sigma^2 + n\tau^2}$$

Note on MAP and Posterior Mean: In this example, because the posterior distribution is symmetric (normal), the MAP estimate coincides with the posterior mean. However, in cases where the posterior distribution is skewed or multimodal, the MAP and posterior mean can differ.

Step 4: Interpretation

The MAP and posterior mean provide a point estimate of the mean $\mu$ that balances the prior belief with the evidence provided by the data. As the sample size $n$ increases, the influence of the prior diminishes, and the estimate converges to the sample mean $\bar{x}$.

Posterior Predictive Distribution

Example 2: Predicting a New Observation in a Normal Model

Problem Setup

Continuing from the previous example, suppose you want to predict a new observation $x_{n+1}$ from the same normal distribution after observing the data $X$.

Step 1: Derive the Posterior Predictive Distribution

The posterior predictive distribution is the distribution of $x_{n+1}$ given the observed data $X$:

$$P(x_{n+1}|X) = \int P(x_{n+1}|\mu) P(\mu|X) \, d\mu$$

Since $P(x_{n+1}|\mu) \sim \mathcal{N}(\mu, \sigma^2)$ and $P(\mu|X) \sim \mathcal{N}(\mu_{\text{post}}, \sigma_{\text{post}}^2)$, the posterior predictive distribution is also normal:

$$P(x_{n+1}|X) \sim \mathcal{N}\left(\mu_{\text{post}}, \sigma^2 + \sigma_{\text{post}}^2\right)$$

Where $\mu_{\text{post}}$ is the posterior mean and $\sigma_{\text{post}}^2$ is the variance of the posterior distribution.

Step 2: Calculate the Predictive Mean and Variance

The mean and variance of the posterior predictive distribution are given by:

• Predictive Mean:

  $$\mathbb{E}[x_{n+1}|X] = \mu_{\text{post}}$$

• Predictive Variance:

  $$\text{Var}(x_{n+1}|X) = \sigma^2 + \sigma_{\text{post}}^2$$

Step 3: Interpretation

The posterior predictive distribution accounts for both the uncertainty in the parameter $\mu$ and the variability in the data. It provides a more complete picture of the uncertainty associated with predicting new observations compared to point estimates alone.

Practical Implications: The posterior predictive distribution is crucial for tasks such as forecasting and decision-making, where understanding the range of possible future outcomes is essential. It allows for the incorporation of parameter uncertainty into predictions, leading to more robust and reliable inferences.

Comparison of Bayesian Estimation and MLE

Bayesian estimation and Maximum Likelihood Estimation (MLE) are both powerful methods for estimating parameters, but they differ fundamentally in how they incorporate uncertainty and prior information.

Key Differences

1. Incorporation of Prior Information:

   • Bayesian Estimation: Incorporates prior knowledge or beliefs about the parameters through the prior distribution. The posterior distribution updates these beliefs based on the observed data.
   • MLE: Does not incorporate prior information. It relies solely on the observed data to estimate parameters.

2. Output:

   • Bayesian Estimation: Produces a full posterior distribution, which can be summarized using point estimates (e.g., MAP, posterior mean) and used to generate predictive distributions.
   • MLE: Produces a single point estimate that maximizes the likelihood function.

3. Interpretation:

   • Bayesian Estimation: Provides a probabilistic interpretation of the parameter estimates, reflecting the uncertainty in the estimates.
   • MLE: Provides a deterministic estimate without explicitly accounting for uncertainty in the parameters.

4. Handling of Small Data Sets:

   • Bayesian Estimation: Can be more robust with small data sets due to the influence of the prior, which can guide the estimation process.
   • MLE: May perform poorly with small data sets, as it relies entirely on the observed data, which may be sparse or unrepresentative.

Example 3: Comparing Bayesian Estimation and MLE for a Binomial Model

Problem Setup

Suppose you are flipping a biased coin and want to estimate the probability $p$ of getting heads. You conduct $n = 10$ flips and observe $k = 7$ heads.

• Prior: You have a prior belief that $p$ is likely close to 0.5, so you use a Beta distribution with parameters $\alpha = 2$ and $\beta = 2$.
• Likelihood: The likelihood of observing $k$ heads in $n$ flips follows a Binomial distribution.

Step 1: Bayesian Estimation

• Posterior Distribution:

  The posterior distribution of $p$ is:

  $$P(p|X) \sim \text{Beta}(\alpha + k, \beta + n - k) = \text{Beta}(2 + 7, 2 + 3) = \text{Beta}(9, 5)$$

• MAP Estimate:

  The MAP estimate is the mode of the Beta distribution:

  $$p_{\text{MAP}} = \frac{\alpha + k - 1}{\alpha + \beta + n - 2} = \frac{9 - 1}{14 - 2} = \frac{8}{12} = \frac{2}{3} \approx 0.67$$

• Posterior Mean:

  The posterior mean is:

  $$\mathbb{E}[p|X] = \frac{\alpha + k}{\alpha + \beta + n} = \frac{9}{14} \approx 0.64$$

Step 2: MLE

• MLE Estimate:

  The MLE estimate is simply the observed proportion of heads:

  $$\hat{p}_{\text{MLE}} = \frac{k}{n} = \frac{7}{10} = 0.7$$

Step 3: Comparison

• Interpretation:

  The MLE estimate of $0.7$ reflects the observed data but does not incorporate any prior knowledge. In contrast, the Bayesian estimates (MAP = $0.67$, posterior mean = $0.64$) are slightly lower, reflecting the prior belief that $p$ is likely close to $0.5$.

  This example highlights how Bayesian estimation provides a more nuanced estimate that balances prior knowledge with observed data, while MLE provides a straightforward, data-driven estimate.

Conclusion

Bayesian estimation offers a flexible and powerful approach to parameter estimation that integrates prior knowledge with observed data to produce a posterior distribution. This posterior distribution provides a comprehensive summary of our beliefs about the parameters, allowing for point estimates, uncertainty quantification, and predictive modeling.

Through detailed examples, we explored how to apply Bayesian estimation techniques, including the derivation of MAP estimates, posterior means, and posterior predictive distributions. We also compared Bayesian estimation with Maximum Likelihood Estimation, highlighting their differences in handling prior information, uncertainty, and small data sets.

By mastering Bayesian estimation, data scientists can enhance their ability to model complex problems, make more informed decisions, and provide richer insights based on probabilistic reasoning.