Understanding Bayesian Inference & Gaussian Distribution

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In summary, the conversation discusses Bayesian Inference and its application in making predictions based on past data. The participants also inquire about the Bayesian formalism, the mean of a gaussian distribution, and recommended books on the topic. The main concept of Bayesian Inference is the weighting of prior beliefs and actual observations in order to calculate a posterior mean.
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I am reading a topic on Bayesian Inference.I read books from different authors but they are all the same. I cannot see how the terms are derived.

Could anyone briefly explain what is going on and what is it that we are trying to find using this Bayesian. Bayesian is a combination of belief from past data. So I am thinking that we are making the prediction. But I am not sure of what sort of prediction. Is ther an example to this.

I need help in understanding the 1) Bayesian formalism and
2) the mean of a gaussian distribution - how are the parameters; mu and the variance are derived and how is the posterior distri bution derived as well.

By the way could anyone suggest any recommended books for this topic. Thx
 
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Kapur and Kesevan.
 
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In practice Bayesian inference comes down to a weighting of prior beliefs vs. actual observations. I.e. the posterior mean is a weighted average of the prior beliefs (e.g. [itex]\mu_0[/itex]) and the data average.
 

1. What is Bayesian Inference?

Bayesian Inference is a statistical method used to update our beliefs (or prior knowledge) about a particular event or hypothesis in light of new evidence. It allows us to quantify uncertainty and make predictions based on probabilities.

2. How does Bayesian Inference differ from traditional statistical methods?

Traditional statistical methods, such as frequentist inference, rely on fixed parameters and do not take into account prior knowledge. Bayesian Inference, on the other hand, incorporates prior beliefs and updates them based on new evidence, resulting in a more flexible and personalized approach to statistical analysis.

3. What is the role of Gaussian Distribution in Bayesian Inference?

Gaussian Distribution, also known as normal distribution, is a probability distribution that is often used to model real-world data. It is a key component of Bayesian Inference as it allows us to make predictions about the likelihood of an event or hypothesis based on the mean and variance of the data.

4. How is Bayesian Inference useful in scientific research?

Bayesian Inference allows scientists to incorporate prior knowledge and uncertainty into their research, making their predictions and conclusions more accurate and robust. It also allows for the incorporation of new data as it becomes available, resulting in a continuous improvement of our understanding of the world.

5. Are there any limitations to Bayesian Inference?

One limitation of Bayesian Inference is the subjective nature of setting prior beliefs. Different individuals may have different prior beliefs, leading to potentially different conclusions. Additionally, the computation can be complex and time-consuming, especially for large datasets.

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