- #1

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**Summary::**Hello there, I'm a mechanical engineer pursuing my graduate degree and I'm taking a class on machine learning. Coding is a skill of mine, but statistics is not... anyway, I have a homework problem on Bernoulli and Bayesian probabilities. I believe I've done the first few parts correctly, but the final question asks me to explain why one is more accurate than another, and the inverse as well. I am not sure, so I figured I'd reach out here and ask. The work and appropriate equations are below:

**1. (10 pts) Consider 20 values randomly sampled from the Bernoulli Distribution with parameter :**

Matlab:

```
x = [1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1];
N = length(x);
```

**(a) Estimate the parameter using the maximum likelihood approach and the 20 data values.**

Matlab:

```
u = sum(x==1)/N; % u = 0.75
bern = (u.^x).*(1-u).^(1-x)
p = 0;
for n = 1:N
pTemp = x(n)*log(u) + (1-x(n))*log(1-u);
p = p+pTemp;
end
%ln(a) = b <--> a = e^b
p = exp(p); % p = 1.3050e-05
```

**(b) Estimate the parameter using the Bayesian approach. Use the beta distribution Beta(a=8, b=4).**

Matlab:

```
% a + sum(xn),b + N - sum(xn)
% (8 + 15 - 1) / (12 + 20 - 2) = 22/30
u = 22/30; % u = 0.7333
```

**(c) Estimate the parameter using the Bayesian approach. Use the beta distribution Beta(a=4, b=8).**

Matlab:

```
% (4 + 15 - 1) / (12 + 20 - 2) = 18/30
u = 18/30; % u = 0.6
```

**(d) Discuss why the estimation from (b) is more accurate than that from (a) and why the estimation from (c) is worse than that from (a).**

Matlab:

```
uA = 0.75;
uB = 0.7333;
uC = 0.6;
```

Thanks in advance for any help!!