Chapter 5: Summary added

05_prob_prog_intro/05_prob_prog_intro.jl

@@ -1,5 +1,5 @@
 ### A Pluto.jl notebook ###
-# v0.12.18
+# v0.12.21
 
 using Markdown
 using InteractiveUtils
@@ -37,6 +37,14 @@ begin
 	title!("Prior distribution for p")
 end
 
+# ╔═╡ 3a8615d2-8be3-11eb-3913-b97d04bb053a
+md"### To do list
+
+We are currently working on:
+
+";
+
+
 # ╔═╡ 9608da48-1acd-11eb-102a-27fbac1ec88c
 md" # Probabilistic Programming"
 
@@ -152,7 +160,7 @@ We can see in the plot below, showing the posterior or updated distribution of *
 # ╔═╡ 0c570210-1af6-11eb-1d5d-5f78f2a000fd
 begin
 	p_summary = chain[:p]
-	plot(p_summary, seriestype = :histogram, normed=true, legend=false, size=(400, 250), color="purple", alpha=0.7)
+	plot(p_summary, seriestype = :histogram, normed=true, legend=false, size=(500, 250), color="purple", alpha=0.7)
 	title!("Posterior distribution of p after getting tails")
 	ylabel!("Probability")
 	xlabel!("p")
@@ -235,7 +243,18 @@ plot(plots[3], plots_[3], title = ["Posterior for Uniform prior and 4 outcomes"
 md"So in this case, incorporating our beliefs in the prior distribution we saw the model reached faster the more plausible values for *p*, needing less outcomes to reach a very similar posterior distribution. When we used an uniform prior, we were conservative, meaning that we said we didn't know anything about *p* so we assign equal probability for all values. Sometimes these kind of distribution (uniform distributions), called a non-informative prior, can be maybe too conservative, being in some cases not helpful at all. They even can slow the convergence of the model to the more plausible values for our posterior, as shown."
 
 # ╔═╡ 92a7cfaa-1a2e-11eb-06f2-f50e91cfbba0
-md" ### Bayesian Bandits "
+md" ### Summary 
+
+In this chapter, we gave an introduction to probabilistic programming languages and explored the classic coin flipping example in a Bayesian way.
+
+First, we saw that in this kind of Bernoulli trial scenario, where the experiment has two possible outcomes 0 or 1, it is a good idea to set our likelihood to a binomial distribution.
+We also learned what sampling is and saw why we use it to update our beliefs.
+Then we used the Julia library Turing.jl to create a probabilistic model, setting our prior probability to a uniform distribution and the likelihood to a binomial one. 
+We sampled our model with the Markov chain Monte Carlo algorithm and saw how the posterior probability was updated every time we input a new coin flip result.
+
+Finally, we created a new model with the prior probability set to a normal distribution centered on *p* = 0.5 which gave us more accurate results.
+
+"
 
 # ╔═╡ 95efe302-35a4-11eb-17ac-3f0ad66fb164
 md"
@@ -244,7 +263,26 @@ md"
 * [Not a monad tutorial article about Soss.jl](https://notamonadtutorial.com/soss-probabilistic-programming-with-julia-6acc5add5549)
 "
 
+# ╔═╡ a77d01ca-8b78-11eb-34dc-b18bf984bf22
+md" ### Give us feedback
+
+
+This book is currently in a beta version. We are looking forward to getting feedback and criticism:
+  * Submit a GitHub issue **[here](https://github.com/unbalancedparentheses/data_science_in_julia_for_hackers/issues)**.
+  * Mail us to **[email protected]**
+
+Thank you!
+"
+
+
+# ╔═╡ a8fda810-8b78-11eb-21af-d18ab85709b7
+md"
+[Next chapter](https://datasciencejuliahackers.com/06_gravity.jl.html)
+"
+
+
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