Cross Validated Asked by Dom Jo on December 7, 2020

The question is as follows:

Consider an SDOF mass-spring system. The value of the mass is known and is equal to 1 kg.

The value of the spring stiffness is unknow and based on the experience and judgement the following is assumed. Value of stiffness is in the following range [0.5, 1.5] N/m.

To have a more accurate estimate of the value of the stiffness an experiment is performed where in the natural frequency of the system is observed. The following observation are made:

```
Observation 1 Freq = 1.021 rad/sec
Observation 2 Freq = 1.015 rad/sec
Observation 3 Freq = 0.994 rad/sec
Observation 4 Freq = 1.005 rad/sec
Observation 5 Freq = 0.989 rad/sec
```

- Based on the information provided write the functional form of prior PDF.
- Plot the likelihood function with different number of observations.
- Based on the information provided write the functional form of the posterior PDF.
- Plot the posterior distribution.

**My work so far:**

spring constant $$k = sqrt{{w}/{m}}$$

m = 1kg, so $$w = k^{2}$$.

$$k sim Uniform(0.5, 1.5)$$,

so pdf of w = $$ f(w) = 2w$$

where $$w epsilon [sqrt{0.5},sqrt{1.5}] $$

So prior distribution is linear in the range root(0.5), root(1.5).

$$Likelihood = L = 2^{5}(1.021*1.015..*0.989) approx 2.04772 $$

This is what I have done so far. I am new to Bayesian inference and I am not sure how to proceed after this or if what I have done so far is correct. Pleas advice on how to find the posterior function.

I gave up my reputation for a bounty so unable to comment.

The posterior is the prior multiplied by the likelihood. If you use a conjugate prior, these types of problems will work out nicely.

What is the sampling distribution in this case? Normal?

Not sure how to handle the bounds but you could use a normal prior for $k$ w/ infinite variance to resemble a uniform distribution or you could just do a normal distribution centered at 1 w/ some large variance out to 0.5 and 1.5.

You say you are not interested in $k$ though? Can you work back to it?

Answered by Quantoisseur on December 7, 2020

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