L16.5 Example: The LMS Estimate

TL;DR

Bayesian estimation involves finding the posterior distribution of a random variable given a conditional distribution, and in this example, the posterior distribution of the unknown variable Theta is a uniform distribution.

Transcript

Let us now go through an example. Suppose that we have an unknown random variable Theta that has a uniform distribution between 4 and 10. We observe some other random variable X that's related to Theta according to the following model. This is the conditional distribution of X given Theta. For any given value of theta, X is going to take values bet... Read More

Key Insights

• 💁 Bayesian estimation involves finding the posterior distribution based on the conditional distribution and prior information.
• ✖️ The joint distribution is obtained by multiplying the constant prior and conditional distributions.
• 🧡 The posterior distribution, in this example, is a uniform distribution on the range corresponding to the observed value of X.
• 🥋 The conditional expectation of Theta is the midpoint of the uniform distribution given the observed value of X.
• ❓ The estimator function in Bayesian estimation is determined by the conditional expectation of Theta for different values of X.
• 💁 The resulting estimator function forms a curve that represents the conditional expectation of Theta based on different values of X.
• 💁 The information obtained from the observed data is used to estimate the unknown variable Theta.

Questions & Answers

Q: What is the first step in Bayesian estimation?

The first step in Bayesian estimation is finding the posterior distribution of the random variable given the conditional distribution.

Q: What constraints are there on the possible values of X and Theta in the example?

In the example, Theta must be less than or equal to X+1 and larger than or equal to X-1. Additionally, Theta is between 4 and 10.

Q: What is the shape of the joint PDF in the given example?

The joint PDF is constant over the set of possible X and Theta values, forming a shape between two lines representing Theta = X+1 and Theta = X-1.

Q: What is the posterior distribution of Theta in the example?

The posterior distribution of Theta, given a specific value of X, is a uniform distribution on the range corresponding to the observed X value.

Summary & Key Takeaways

• The example involves an unknown random variable Theta with a uniform distribution between 4 and 10, and a related variable X with a conditional distribution that is uniform between Theta-1 and Theta+1.

• The joint distribution of X and Theta is constant over a specific range, resulting in a uniform joint PDF.

• The posterior distribution of Theta, given a specific value of X, is also a uniform distribution on the range corresponding to the observed X value.