This is the (updated) code I’m using, which gives results like I’d expect:
$args = array(
'post_type' => 'entry',
'orderby' => 'bayesian_average',
'order' => 'DESC',
'post_status' => 'publish'
);
$loop = new WP_Query($args);
$number_of_entrants = $loop->post_count;
$total_ratings = $total_num_votes = 0;
foreach ($loop->posts as $query_post) {
$count = $query_post->ratings_count;
$average = $query_post->ratings_average;
$total_num_votes += $count;
$total_ratings += $average;
}
$average_rating = $total_ratings / $number_of_entrants;
$avg_num_votes = $total_num_votes / $number_of_entrants;
if ($loop>have_posts()):
?>
<table class="wp-list-table widefat fixed striped pages">
<thead>
<tr>
<th scope="col" id="entry"><?php _e('Entry', 'textdomain'); ?></th>
<th scope="col" id="rating-average"><?php _e('Rating average', 'textdomain'); ?></th>
<th scope="col" id="rating-count"><?php _e('Rating count', 'textdomain'); ?></th>
<th scope="col" id="beyesian-rating"><?php _e('Bayesian Rating', 'textdomain'); ?></th>
</tr>
</thead>
<tbody><?php
global $post;
while ($loop->have_posts()) :
$loop->the_post();
$title = get_the_title();
$this_num_votes = $post->ratings_count;
$this_avg_rating = $post->ratings_average;
$bayesian_average = (($avg_num_votes * $average_rating) + ($this_num_votes * $this_avg_rating)) / ($avg_num_votes + $this_num_votes);
update_post_meta(get_the_ID(), 'bayesian_average', $bayesian_average); ?>
<tr>
<td><a href="https://wordpress.stackexchange.com/questions/259601/<?php echo get_permalink(); ?>" target="_blank"
title="<?php echo $title; ?>"><?php echo $title; ?>
</a></td>
<td>
<?php echo $post->ratings_average; ?>
</td>
<td>
<?php echo $post->ratings_count; ?>
</td>
<td>
<?php echo round($bayesian_average, 3); ?>
</td>
</tr><?php
endwhile; ?>
</tbody>
</table><?php
else:
_e('No Entries', 'textdomain');
endif;