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I'm relatively new to bayesian inference, and was trying to apply a bayesian model in a real-world scenario.

Let me describe the model in brief:

We have $N$ i.i.d. random variables $D =(X_1, X_2, ... X_N)$ ~ $N(\mu, \sigma^2)$.

In a bayesian framework, we want to find the posterior distribution of $\mu | D$.

$$ P(\mu|D) \propto P(D|\mu)*P(\mu) \\ P(\mu|D) \propto \Pi_{i=1}^{N} P(X_i|\mu)*P(\mu) \\ $$

We put a Gaussian prior distribution on $\mu$ : $P(\mu)$ ~ $N(\mu_o, \sigma_o^2)$

In the above setting, $\sigma, \mu_o, \sigma_o$ are known quantities.

My notions of statistical variability are still weak, and I wanted to know how are the values of $\sigma$ and $\sigma_o$ chosen in real world scenarios?

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In the above setting, $\sigma, \mu_0, \sigma_0$ are known quantities.

Though the language "assumed known" appears in the video you reference, it's a bit misleading to lump all three of these together.

To answer your question, let's first distinguish between the parameters of your prior distribution, $\mu_0$ and $\sigma_0$. These are not really "assumed": They're chosen to reflect the state of knowledge about the parameter $\mu$. This describes the process nicely: "The prior distribution can be viewed as representing the current state of knowledge, or current state of uncertainty, about the model parameters prior to data being observed."

How one represents the "current state of knowledge" is rather domain- and problem-specific. For instance, it depends on factors like parameter interpretation, what is known and published about them, and what you can convince your skeptical audience are reasonable priors. For example, consider this from Bayesian Data Analysis (3rd ed., p. 478):

Solving the problem of estimating metabolism from indirect data is facilitated by using [a model] in which the individual and population parameters have direct physical interpretations (for example, blood flow through the fatty tissue, or tissue/blood partition coefficients). These models permit the identification of many of their parameter values through prior (for example, published) physiological data.

On the opposite end of that spectrum, if little or nothing is known about your parameters, then you may use a weakly informative or non-informative prior.

As for $\sigma$, this you may consider "assumed known," but I struggle to give an illustrative, real-world example, in part because it's such a large assumption. I've only ever seen it in examples to demonstrate conjugacy, usually as a stepping stone to models that do not assume known variance. (E.g., here.) I've read examples describing an instrument with known measurement error, but I don't know that this is a realistic scenario.

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